%!PS-Adobe-3.0 %%Title: (Microsoft Word - smr8) %%Creator: (Microsoft Word: LaserWriter 8 8.3.4) %%CreationDate: (3:28 PM Tuesday, October 21, 1997) %%For: (peter) %%Pages: 34 %%DocumentFonts: Times-Roman Symbol Times-Bold Times-BoldItalic Times-Italic %%DocumentNeededFonts: Times-Roman Symbol Times-Bold Times-BoldItalic Times-Italic %%DocumentSuppliedFonts: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Portrait %%DocumentMedia: Default 612 792 0 () () %ADO_ImageableArea: 31 31 583 761 %%EndComments userdict begin/dscInfo 5 dict dup begin /Title(Microsoft Word - smr8)def /Creator(Microsoft Word: LaserWriter 8 8.3.4)def /CreationDate(3:28 PM Tuesday, October 21, 1997)def /For(peter)def /Pages 34 def end def end save /version23-manualfeedpatch where { pop false } { true }ifelse % we don't do an explicit 'get' since product and version MAY % be in systemdict or statusdict - this technique gets the lookup % without failure statusdict begin product (LaserWriter) eq % true if LaserWriter version cvr 23.0 eq % true if version 23 end and % only install this patch if both are true and % true only if patch is not installed and is for this printer % save object and boolean on stack dup { exch restore }if % either true OR saveobject false dup { /version23-manualfeedpatch true def /oldversion23-showpage /showpage load def /showpage % this showpage will wait extra time if manualfeed is true {% statusdict /manualfeed known {% manualfeed known in statusdict statusdict /manualfeed get {% if true then we loop for 5 seconds usertime 5000 add % target usertime { % loop dup usertime sub 0 lt { exit }if }loop pop % pop the usertime off the stac }if }if oldversion23-showpage }bind def }if not{ restore }if /md 223 dict def md begin/currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if %%BeginFile: adobe_psp_basic %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /bd{bind def}bind def /xdf{exch def}bd /xs{exch store}bd /ld{load def}bd /Z{0 def}bd /T/true /F/false /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 14{ld}repeat /$m matrix def /av 83 def /por true def /normland false def /psb-nosave{}bd /pse-nosave{}bd /us Z /psb{/us save store}bd /pse{us restore}bd /level2 /languagelevel where { pop languagelevel 2 ge }{ false }ifelse def /featurecleanup { stopped cleartomark countdictstack exch sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /setcopies { userdict/#copies 3 -1 roll put }bd level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /setcopies { 1 dict begin/NumCopies xdf currentdict end setpagedevice }bd level2 not endnoload /pm Z /mT Z /sD Z /realshowpage Z /initializepage { /pm save store mT concat }bd /endp { pm restore showpage }def /$c/DeviceRGB def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile %%BeginFile: adobe_psp_colorspace_level1 %%Copyright: Copyright 1991-1993 Adobe Systems Incorporated. All Rights Reserved. /G/setgray ld /:F1/setgray ld /:F/setrgbcolor ld /:F4/setcmykcolor where { pop /setcmykcolor ld }{ { 3 { dup 3 -1 roll add dup 1 gt{pop 1}if 1 exch sub 4 1 roll }repeat pop setrgbcolor }bd }ifelse /:Fx { counttomark {0{G}0{:F}{:F4}} exch get exec pop }bd /:rg{/DeviceRGB :ss}bd /:sc{$cs :ss}bd /:dc{/$cs xdf}bd /:sgl{}def /:dr{}bd /:fCRD{pop}bd /:ckcs{}bd /:ss{/$c xdf}bd /$cs Z %%EndFile %%BeginFile: adobe_psp_uniform_graphics %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /@a { np :M 0 rl :L 0 exch rl 0 rl :L fill }bd /@b { np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill }bd /arct where { pop }{ /arct { arcto pop pop pop pop }bd }ifelse /x1 Z /x2 Z /y1 Z /y2 Z /rad Z /@q { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct fill }bd /@s { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct :K stroke }bd /@i { np 0 360 arc fill }bd /@j { gS np :T scale 0 0 .5 0 360 arc fill gR }bd /@e { np 0 360 arc :K stroke }bd /@f { np $m currentmatrix pop :T scale 0 0 .5 0 360 arc :K $m setmatrix stroke }bd /@k { gS np :T 0 0 :M 0 0 5 2 roll arc fill gR }bd /@l { gS np :T 0 0 :M scale 0 0 .5 5 -2 roll arc fill gR }bd /@m { np arc stroke }bd /@n { np $m currentmatrix pop :T scale 0 0 .5 5 -2 roll arc $m setmatrix stroke }bd %%EndFile %%BeginFile: adobe_psp_customps %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /$t Z /$p Z /$s Z /$o 1. def /2state? false def /ps Z level2 startnoload /pushcolor/currentrgbcolor ld /popcolor/setrgbcolor ld /setcmykcolor where { pop/currentcmykcolor where { pop/pushcolor/currentcmykcolor ld /popcolor/setcmykcolor ld }if }if level2 endnoload level2 not startnoload /pushcolor { currentcolorspace $c eq { currentcolor currentcolorspace true }{ currentcmykcolor false }ifelse }bd /popcolor { { setcolorspace setcolor }{ setcmykcolor }ifelse }bd level2 not endnoload /pushstatic { ps 2state? $o $t $p $s $cs }bd /popstatic { /$cs xs /$s xs /$p xs /$t xs /$o xs /2state? xs /ps xs }bd /pushgstate { save errordict/nocurrentpoint{pop 0 0}put currentpoint 3 -1 roll restore pushcolor currentlinewidth currentlinecap currentlinejoin currentdash exch aload length np clippath pathbbox $m currentmatrix aload pop }bd /popgstate { $m astore setmatrix 2 index sub exch 3 index sub exch rC array astore exch setdash setlinejoin setlinecap lw popcolor np :M }bd /bu { pushgstate gR pushgstate 2state? { gR pushgstate }if pushstatic pm restore mT concat }bd /bn { /pm save store popstatic popgstate gS popgstate 2state? { gS popgstate }if }bd /cpat{pop 64 div setgray 8{pop}repeat}bd %%EndFile %%BeginFile: adobe_psp_basic_text %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /S/show ld /A{ 0.0 exch ashow }bd /R{ 0.0 exch 32 exch widthshow }bd /W{ 0.0 3 1 roll widthshow }bd /J{ 0.0 32 4 2 roll 0.0 exch awidthshow }bd /V{ 0.0 4 1 roll 0.0 exch awidthshow }bd /fcflg true def /fc{ fcflg{ vmstatus exch sub 50000 lt{ (%%[ Warning: Running out of memory ]%%\r)print flush/fcflg false store }if pop }if }bd /$f[1 0 0 -1 0 0]def /:ff{$f :mf}bd /MacEncoding StandardEncoding 256 array copy def MacEncoding 39/quotesingle put MacEncoding 96/grave put /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis /dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash /infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation /product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash /questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft /guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl /daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave /Oacute/Ocircumflex/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde /macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron MacEncoding 128 128 getinterval astore pop level2 startnoload /copyfontdict { findfont dup length dict begin { 1 index/FID ne{def}{pop pop}ifelse }forall }bd level2 endnoload level2 not startnoload /copyfontdict { findfont dup length dict copy begin }bd level2 not endnoload md/fontname known not{ /fontname/customfont def }if /Encoding Z /:mre { copyfontdict /Encoding MacEncoding def fontname currentdict end definefont :ff def }bd /:bsr { copyfontdict /Encoding Encoding 256 array copy def Encoding dup }bd /pd{put dup}bd /:esr { pop pop fontname currentdict end definefont :ff def }bd /scf { scalefont def }bd /scf-non { $m scale :mf setfont }bd /ps Z /fz{/ps xs}bd /sf/setfont ld /cF/currentfont ld /mbf { /makeblendedfont where { pop makeblendedfont /ABlend exch definefont }{ pop }ifelse def }def %%EndFile %%BeginFile: adobe_psp_derived_styles %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /wi version(23.0)eq { { gS 0 0 0 0 rC stringwidth gR }bind }{ /stringwidth load }ifelse def /$o 1. def /gl{$o G}bd /ms{:M S}bd /condensedmtx[.82 0 0 1 0 0]def /:mc { condensedmtx :mf def }bd /extendedmtx[1.18 0 0 1 0 0]def /:me { extendedmtx :mf def }bd /basefont Z /basefonto Z /dxa Z /dxb Z /dxc Z /dxd Z /dsdx2 Z /bfproc Z /:fbase { dup/FontType get 0 eq{ dup length dict begin dup{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall /FDepVector exch/FDepVector get[exch/:fbase load forall]def }/bfproc load ifelse /customfont currentdict end definefont }bd /:mo { /bfproc{ dup dup length 2 add dict begin { 1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse }forall /PaintType 2 def /StrokeWidth .012 0 FontMatrix idtransform pop def /customfont currentdict end definefont 8 dict begin /basefonto xdf /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin basefont setfont ( )dup 0 4 -1 roll put dup wi setcharwidth 0 0 :M gS gl dup show gR basefonto setfont show end }def }store :fbase }bd /:mso { /bfproc{ 7 dict begin /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxa 1 ps div def basefont setfont ( )dup 0 4 -1 roll put dup wi 1 index 0 ne { exch dxa add exch }if setcharwidth dup 0 0 ms dup dxa 0 ms dup dxa dxa ms dup 0 dxa ms gl dxa 2. div dup ms end end }def }store :fbase }bd /:ms { /bfproc{ dup dup length 2 add dict begin { 1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse }forall /PaintType 2 def /StrokeWidth .012 0 FontMatrix idtransform pop def /customfont currentdict end definefont 8 dict begin /basefonto xdf /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxb .05 def basefont setfont ( )dup 0 4 -1 roll put dup wi exch dup 0 ne { dxb add }if exch setcharwidth dup dxb .01 add 0 ms 0 dxb :T gS gl dup 0 0 ms gR basefonto setfont 0 0 ms end end }def }store :fbase }bd /:mss { /bfproc{ 7 dict begin /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxc 1 ps div def /dsdx2 .05 dxc 2 div add def basefont setfont ( )dup 0 4 -1 roll put dup wi exch dup 0 ne { dsdx2 add }if exch setcharwidth dup dsdx2 .01 add 0 ms 0 .05 dxc 2 div sub :T dup 0 0 ms dup dxc 0 ms dup dxc dxc ms dup 0 dxc ms gl dxc 2 div dup ms end end }def }store :fbase }bd /:msb { /bfproc{ 7 dict begin /basefont xdf /FontType 3 def /FontMatrix[1 0 0 1 0 0]def /FontBBox[0 0 1 1]def /Encoding StandardEncoding def /BuildChar { exch begin sD begin /dxd .03 def basefont setfont ( )dup 0 4 -1 roll put dup wi 1 index 0 ne { exch dxd add exch }if setcharwidth dup 0 0 ms dup dxd 0 ms dup dxd dxd ms 0 dxd ms end end }def }store :fbase }bd /italicmtx[1 0 -.212557 1 0 0]def /:mi { italicmtx :mf def }bd /:v { [exch dup/FontMatrix get exch dup/FontInfo known { /FontInfo get dup/UnderlinePosition known { dup/UnderlinePosition get 2 index 0 3 1 roll transform exch pop }{ .1 }ifelse 3 1 roll dup/UnderlineThickness known { /UnderlineThickness get exch 0 3 1 roll transform exch pop abs }{ pop pop .067 }ifelse }{ pop pop .1 .067 }ifelse ] }bd /$t Z /$p Z /$s Z /:p { aload pop 2 index mul/$t xs 1 index mul/$p xs .012 mul/$s xs }bd /:m {gS 0 $p rm $t lw 0 rl stroke gR }bd /:n { gS 0 $p rm $t lw 0 rl gS gl stroke gR strokepath $s lw /setstrokeadjust where{pop currentstrokeadjust true setstrokeadjust stroke setstrokeadjust }{ stroke }ifelse gR }bd /:o {gS 0 $p rm $t 2 div dup rm $t lw dup 0 rl stroke gR :n }bd %%EndFile /currentpacking where {pop sc_oldpacking setpacking}if end %%EndProlog %%BeginSetup md begin countdictstack[{ %%BeginFeature: *ManualFeed False statusdict /manualfeed false put %%EndFeature }featurecleanup countdictstack[{ %%BeginFeature: *InputSlot Cassette %%EndFeature }featurecleanup countdictstack[{ %%BeginFeature: *PageRegion LetterSmall lettersmall %%EndFeature }featurecleanup (peter)setjob /mT[1 0 0 -1 31 761]def /sD 16 dict def 300 level2{1 dict dup/WaitTimeout 4 -1 roll put setuserparams}{statusdict/waittimeout 3 -1 roll put}ifelse %%IncludeFont: Times-Roman %%IncludeFont: Symbol %%IncludeFont: Times-Bold %%IncludeFont: Times-BoldItalic %%IncludeFont: Times-Italic /f0_1/Times-Roman :mre /f0_12 f0_1 12 scf /f0_10 f0_1 10 scf /f0_9 f0_1 9 scf /f0_8 f0_1 8 scf /f0_7 f0_1 7 scf /f0_6 f0_1 6 scf /f0_5 f0_1 5 scf /f1_1/Symbol :bsr 240/apple pd :esr /f1_12 f1_1 12 scf /f1_10 f1_1 10 scf /f1_7 f1_1 7 scf /f2_1/Times-Bold :mre /f2_16 f2_1 16 scf /f2_14 f2_1 14 scf /f2_12 f2_1 12 scf /f2_10 f2_1 10 scf /f2_7 f2_1 7 scf /f3_1 f1_1 def /f3_12 f3_1 12 scf /f4_1/Times-BoldItalic :mre /f4_12 f4_1 12 scf /f5_1/Times-Italic :mre /f5_12 f5_1 12 scf /f5_10 f5_1 10 scf /f6_1 f1_1 :mi /f6_10 f6_1 10 scf /Courier findfont[10 0 0 -10 0 0]:mf setfont %%EndSetup %%Page: 1 1 %%BeginPageSetup initializepage (peter; page: 1 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (1)S gR gS 0 0 552 730 rC 90 62 :M f2_16 sf -.066(Using Path Diagrams as a Structural Equation Modelling)A 260 80 :M -.372(Tool)A 85 104 :M f2_12 sf 2.902 .29(by Peter Spirtes, Thomas Richardson, Chris Meek, Richard Scheines,)J 220 122 :M 1.946 .195(and Clark Glymour)J 59 152 :M f2_14 sf (1)S 67 152 :M (.)S 72 152 :M 19.5 1.95( )J 95 152 :M .617(Introduction)A 77 179 :M f0_12 sf -.123(Linear structural equation models \(SEMs\) )A 278 179 :M -.215(are )A 296 179 :M -.11(widely )A 332 179 :M (used )S 358 179 :M -.167(in )A 371 179 :M .17 .017(sociology, )J 425 179 :M -.081(econometrics,)A 59 197 :M -.057(biology, and other sciences. A SEM )A 234 197 :M -.082(\(without )A 278 197 :M -.16(free )A 300 197 :M -.147(parameters\) )A 359 197 :M (has )S 379 197 :M (two )S 401 197 :M -.109(parts: )A 431 197 :M -.326(a )A 440 197 :M -.166(probability)A 59 215 :M -.111(distribution )A 118 215 :M -.109(\(in )A 136 215 :M -.22(the )A 155 215 :M -.108(Normal )A 196 215 :M -.161(case )A 221 215 :M -.145(specified )A 268 215 :M (by )S 285 215 :M -.326(a )A 295 215 :M -.109(set )A 314 215 :M (of )S 330 215 :M -.219(linear )A 362 215 :M -.131(structural )A 412 215 :M -.11(equations )A 463 215 :M -.109(and )A 486 215 :M (a)S 59 233 :M -.196(covariance )A 115 233 :M -.22(matrix )A 151 233 :M -.132(among )A 189 233 :M -.22(the )A 209 233 :M -.137(\322error\323 )A 248 233 :M (or )S 264 233 :M -.151(\322disturbance\323 )A 334 233 :M .207 .021(terms\), )J 374 233 :M -.109(and )A 398 233 :M -.163(an )A 416 233 :M -.131(associated )A 471 233 :M -.22(path)A 59 251 :M -.113(diagram corresponding to the functional composition of variables specified by the )A 447 251 :M -.145(structural)A 59 269 :M -.11(equations )A 109 269 :M -.109(and )A 131 269 :M -.22(the )A 150 269 :M -.136(correlations )A 210 269 :M -.132(among )A 247 269 :M -.22(the )A 266 269 :M -.061(error )A 294 269 :M .216 .022(terms. )J 329 269 :M -.164(It )A 342 269 :M (is )S 356 269 :M -.131(often )A 386 269 :M -.095(thought )A 428 269 :M -.249(that )A 451 269 :M -.22(the )A 471 269 :M -.22(path)A 59 287 :M -.188(diagram )A 102 287 :M (is )S 115 287 :M -.095(nothing )A 156 287 :M -.163(more )A 185 287 :M -.165(than )A 210 287 :M -.326(a )A 220 287 :M -.146(heuristic )A 265 287 :M -.219(device )A 300 287 :M (for )S 320 287 :M -.166(illustrating )A 376 287 :M -.22(the )A 396 287 :M -.03(assumptions )A 461 287 :M (of )S 477 287 :M -.33(the)A 59 305 :M (model. )S 96 305 :M .571 .057(However, )J 148 305 :M -.167(in )A 161 305 :M -.084(this )A 182 305 :M .222 .022(paper, )J 216 305 :M (we )S 234 305 :M -.166(will )A 256 305 :M .479 .048(show )J 286 305 :M .259 .026(how )J 311 305 :M -.165(path )A 335 305 :M -.123(diagrams )A 382 305 :M -.217(can )A 402 305 :M -.163(be )A 417 305 :M (used )S 443 305 :M -.167(to )A 456 305 :M -.065(solve )A 486 305 :M (a)S 59 323 :M -.108(number of important problems in structural equation modelling.)A 77 347 :M -.195(There )A 108 347 :M -.215(are )A 126 347 :M -.326(a )A 135 347 :M -.109(number )A 175 347 :M (of )S 189 347 :M -.082(problems )A 237 347 :M -.131(associated )A 290 347 :M -.083(with )A 316 347 :M -.131(structural )A 365 347 :M -.165(equation )A 410 347 :M -.036(modeling. )A 463 347 :M -.162(These)A 59 365 :M -.144(problems include:)A 77 389 :M f1_12 sf S 83 389 :M ( )S 83 389 :M f0_12 sf ( )S 89 389 :M .518 .052(How )J 119 389 :M -.165(much )A 151 389 :M (do )S 169 389 :M -.165(sample )A 208 389 :M -.247(data )A 233 389 :M -.164(underdetermine )A 312 389 :M -.22(the )A 333 389 :M -.186(correct )A 372 389 :M -.199(model )A 408 389 :M -.188(specification? )A 479 389 :M -.656(Of)A 59 407 :M .405 .04(course, )J 98 407 :M -.109(one )A 119 407 :M -.084(must )A 146 407 :M -.219(decide )A 181 407 :M .259 .026(how )J 207 407 :M -.165(much )A 238 407 :M -.203(credence )A 284 407 :M -.167(to )A 298 407 :M -.165(give )A 323 407 :M -.239(alternative )A 376 407 :M -.137(explanations )A 440 407 :M -.249(that )A 462 407 :M -.061(afford)A 59 425 :M -.144(different )A 103 425 :M -.082(fits )A 122 425 :M -.167(to )A 135 425 :M -.109(any )A 156 425 :M -.197(particular )A 204 425 :M -.247(data )A 227 425 :M .481 .048(set. )J 248 425 :M -.195(There )A 279 425 :M -.215(are )A 297 425 :M -.326(a )A 306 425 :M -.188(variety )A 342 425 :M (of )S 356 425 :M -.131(techniques )A 410 425 :M (for )S 428 425 :M -.249(that )A 449 425 :M .145(purpose,)A 59 443 :M -.148(including )A 108 443 :M -.123(Bayesian )A 156 443 :M (updating, )S 206 443 :M -.109(and )A 228 443 :M -.326(a )A 238 443 :M -.188(variety )A 275 443 :M (of )S 290 443 :M -.22(fit )A 306 443 :M -.08(measures )A 356 443 :M -.083(with )A 383 443 :M -.164(well )A 409 443 :M -.032(understood )A 468 443 :M -.245(large)A 59 461 :M -.056(sample properties . But )A 173 461 :M -.081(what )A 200 461 :M -.132(about )A 230 461 :M ( )S 234 461 :M (two )S 256 461 :M (or )S 270 461 :M -.163(more )A 298 461 :M -.239(alternative )A 350 461 :M -.11(models )A 388 461 :M -.249(that )A 409 461 :M -.22(fit )A 423 461 :M -.326(a )A 432 461 :M -.163(specific )A 472 461 :M -.329(data)A 59 479 :M -.044(set equally well, or, subject )A 193 479 :M ( )S 197 479 :M -.167(to )A 210 479 :M -.234(certain )A 245 479 :M -.024(restrictions, )A 305 479 :M -.22(fit )A 319 479 :M -.109(any )A 340 479 :M -.247(data )A 363 479 :M -.109(set )A 380 479 :M -.236(meeting )A 421 479 :M -.22(the )A 439 479 :M -.119(restrictions)A 59 497 :M -.174(equally well? The )A 146 497 :M -.109(number )A 186 497 :M (of )S 200 497 :M (such )S 226 497 :M -.15(equivalents )A 283 497 :M (for )S 301 497 :M -.326(a )A 310 497 :M -.132(given )A 340 497 :M -.219(linear )A 370 497 :M -.131(structural )A 418 497 :M -.165(equation )A 462 497 :M -.249(model)A 59 515 :M -.082(may be very large. Even if there are sources of knowledge about )A 367 515 :M -.108(structure )A 412 515 :M -.08(from )A 439 515 :M -.094(outside )A 477 515 :M -.33(the)A 59 533 :M -.106(data set, the number of equivalent )A 222 533 :M -.11(models )A 260 533 :M -.331(all )A 275 533 :M -.236(meeting )A 316 533 :M -.065(those )A 345 533 :M -.072(knowledge )A 401 533 :M -.089(constraints )A 456 533 :M -.22(may )A 480 533 :M -.326(be)A 59 551 :M -.071(considerable, and the structures )A 212 551 :M -.165(they )A 236 551 :M -.147(postulate )A 282 551 :M -.22(may )A 306 551 :M -.163(have )A 332 551 :M -.181(importantly )A 390 551 :M -.144(different )A 434 551 :M -.211(implications)A 59 569 :M (for )S 77 569 :M (policy. )S 114 569 :M .214 .021(Unless )J 151 569 :M (we )S 169 569 :M -.245(characterize )A 228 569 :M (such )S 254 569 :M -.093(equivalencies, )A 325 569 :M -.183(selection )A 370 569 :M (of )S 384 569 :M -.326(a )A 393 569 :M -.197(particular )A 441 569 :M -.199(model )A 475 569 :M -.326(can)A 59 587 :M -.122(only involve an element of arbitrary choice.)A 77 611 :M f1_12 sf S 83 611 :M ( )S 83 611 :M f0_12 sf -.119( Given that there are equivalent models, is it possible to extract the features common to)A 59 629 :M -.1(those models? Under some circumstances, every member of a set of equivalent models may)A 59 647 :M -.116(share some of the same linear coefficients or )A 272 647 :M -.196(correlated )A 322 647 :M .604 .06(errors. )J 358 647 :M (If )S 370 647 :M -.249(that )A 391 647 :M (is )S 403 647 :M -.22(the )A 421 647 :M .236 .024(case, )J 449 647 :M -.165(then )A 473 647 :M -.334(it )A 483 647 :M (is)S 59 665 :M -.041(possible )A 102 665 :M -.249(that )A 123 665 :M -.163(even )A 149 665 :M -.056(though )A 186 665 :M -.22(the )A 204 665 :M -.247(data )A 227 665 :M -.22(may )A 251 665 :M -.111(not )A 270 665 :M -.165(help )A 294 665 :M .277 .028(us )J 309 665 :M -.053(choose )A 346 665 :M -.139(between )A 389 665 :M -.22(the )A 408 665 :M -.144(different )A 453 665 :M .057(models,)A 59 683 :M -.114(the data may provide evidence for features common to all of the best models.)A endp %%Page: 2 2 %%BeginPageSetup initializepage (peter; page: 2 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (2)S gR gS 0 0 552 730 rC 77 56 :M f1_12 sf S 83 56 :M ( )S 83 56 :M f0_12 sf ( )S 88 56 :M -.163(When )A 121 56 :M -.326(a )A 131 56 :M -.188(modeler )A 174 56 :M .234 .023(draws )J 208 56 :M -.059(conclusions )A 269 56 :M -.132(about )A 300 56 :M -.163(coefficients )A 359 56 :M -.167(in )A 373 56 :M -.163(an )A 389 56 :M .198 .02(unknown )J 440 56 :M -.11(underlying)A 59 74 :M -.131(structural )A 107 74 :M -.165(equation )A 151 74 :M -.199(model )A 184 74 :M -.08(from )A 211 74 :M -.326(a )A 220 74 :M -.248(multivariate )A 280 74 :M .33 .033(regression, )J 338 74 :M -.145(precisely )A 385 74 :M -.081(what )A 413 74 :M -.03(assumptions )A 477 74 :M -.323(are)A 59 92 :M -.112(being made about the structural equation model? For example, when does a non-zero partial)A 59 110 :M -.12(regression coefficient correspond to a non-zero coefficient in a structural equation?)A 77 134 :M -.064(These questions have been addressed many times, though usually only for )A 432 134 :M -.11(models )A 470 134 :M -.11(with)A 59 152 :M -.188(special )A 95 152 :M .171 .017(structures, )J 149 152 :M -.109(and )A 170 152 :M -.094(usually )A 208 152 :M -.141(relying )A 245 152 :M (on )S 261 152 :M -.219(linear )A 291 152 :M -.038(algebra, )A 333 152 :M -.22(the )A 351 152 :M -.24(mathematics )A 413 152 :M -.249(that )A 434 152 :M -.064(seems )A 468 152 :M -.112(most)A 59 170 :M -.188(natural )A 96 170 :M (for )S 115 170 :M -.326(a )A 125 170 :M (study )S 156 170 :M (of )S 171 170 :M -.219(linear )A 202 170 :M .199 .02(models. )J 245 170 :M -.219(The )A 268 170 :M -.331(aim )A 290 170 :M (of )S 305 170 :M -.084(this )A 327 170 :M -.129(paper )A 358 170 :M (is )S 371 170 :M -.167(to )A 385 170 :M -.189(explain )A 424 170 :M .259 .026(how )J 451 170 :M -.22(the )A 471 170 :M -.22(path)A 59 188 :M -.109(diagram provides much more than heuristics )A 272 188 :M (for )S 290 188 :M -.188(special )A 326 188 :M -.108(cases; )A 358 188 :M -.22(the )A 376 188 :M -.109(theory )A 410 188 :M (of )S 424 188 :M -.165(path )A 448 188 :M -.14(diagrams)A 59 206 :M -.065(helps )A 90 206 :M -.167(to )A 105 206 :M -.187(clarify )A 141 206 :M -.139(several )A 180 206 :M (of )S 196 206 :M -.22(the )A 216 206 :M .212 .021(issues )J 251 206 :M -.084(just )A 274 206 :M .212 .021(noted, )J 310 206 :M .212 .021(issues )J 345 206 :M -.249(that )A 369 206 :M -.163(have )A 398 206 :M -.163(been )A 427 206 :M -.22(the )A 448 206 :M (focus )S 481 206 :M (of)S 59 224 :M -.1(intelligent--if, in our )A 159 224 :M -.036(judgment, )A 211 224 :M -.266(ultimately )A 261 224 :M -.111(too )A 280 224 :M -.03(sweeping-- )A 337 224 :M -.22(criticism )A 381 224 :M (of )S 395 224 :M -.22(the )A 413 224 :M (use )S 433 224 :M (of )S 447 224 :M -.145(structural)A 59 242 :M -.165(equation )A 103 242 :M .199 .02(models. )J 145 242 :M -.247(What )A 174 242 :M (follows )S 214 242 :M (is )S 226 242 :M -.326(a )A 235 242 :M -.108(report )A 267 242 :M -.249(that )A 288 242 :M -.071(describes )A 336 242 :M -.082(some )A 366 242 :M (of )S 381 242 :M -.081(what )A 409 242 :M (has )S 430 242 :M -.163(been )A 457 242 :M -.218(learned)A 59 260 :M -.132(about )A 91 260 :M -.131(these )A 121 260 :M .212 .021(issues )J 156 260 :M (by )S 174 260 :M -.073(following )A 226 260 :M -.326(a )A 237 260 :M -.144(different )A 283 260 :M -.109(set )A 302 260 :M (of )S 318 260 :M -.303(mathematical )A 385 260 :M -.131(ideas )A 415 260 :M -.249(that )A 438 260 :M -.19(exploit )A 477 260 :M -.33(the)A 59 278 :M -.13(graphical structure implicit in structural equation models.)A 77 302 :M (In )S 94 302 :M -.088(particular, )A 149 302 :M (we )S 170 302 :M -.166(will )A 195 302 :M -.092(present )A 236 302 :M .21 .021(answers )J 282 302 :M -.167(to )A 298 302 :M -.131(these )A 329 302 :M -.036(questions )A 381 302 :M -.249(that )A 406 302 :M -.109(depend )A 448 302 :M (upon )S 480 302 :M -.326(an)A 59 320 :M -.075(understanding )A 130 320 :M (of )S 144 320 :M -.22(the )A 162 320 :M -.137(relationship )A 221 320 :M -.139(between )A 264 320 :M -.22(the )A 282 320 :M -.165(path )A 306 320 :M -.188(diagram )A 348 320 :M (used )S 375 320 :M -.167(to )A 389 320 :M -.107(represent )A 437 320 :M -.326(a )A 447 320 :M -.145(structural)A 59 338 :M -.165(equation )A 103 338 :M (model, )S 140 338 :M -.109(and )A 161 338 :M -.22(the )A 179 338 :M -.161(zero )A 203 338 :M -.235(partial )A 236 338 :M -.136(correlations )A 295 338 :M -.248(entailed )A 335 338 :M (by )S 351 338 :M -.249(that )A 372 338 :M -.165(path )A 396 338 :M -.188(diagram )A 438 338 :M -.219(\(entailed )A 482 338 :M -.334(in)A 59 356 :M -.121(the sense that every structural equation model that shares the path diagram has a )A 438 356 :M -.161(zero )A 462 356 :M -.275(partial)A 59 374 :M -.074(correlation\). )A 122 374 :M -.326(We )A 143 374 :M -.166(will )A 166 374 :M -.122(describe )A 210 374 :M -.326(a )A 220 374 :M -.182(graphical )A 268 374 :M -.072(relation, )A 312 374 :M -.22(the )A 332 374 :M -.162(Pearl-Geiger-Verma )A 433 374 :M -.118(d-separation)A 59 392 :M -.099(criterion, among a pair )A 170 392 :M (of )S 184 392 :M -.145(variables )A 230 392 :M .306 .031(X )J 243 392 :M -.109(and )A 264 392 :M .281 .028(Y, )J 280 392 :M -.109(and )A 301 392 :M -.326(a )A 310 392 :M -.109(set )A 327 392 :M (of )S 341 392 :M -.145(variables )A 387 392 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 403 392 :M -.249(that )A 424 392 :M (is )S 436 392 :M -.326(a )A 445 392 :M -.079(necessary)A 59 410 :M -.109(and )A 80 410 :M -.131(sufficient )A 128 410 :M -.148(condition )A 176 410 :M (for )S 194 410 :M -.326(a )A 203 410 :M -.131(structural )A 251 410 :M -.165(equation )A 295 410 :M -.199(model )A 328 410 :M -.167(to )A 341 410 :M -.276(entail )A 370 410 :M -.326(a )A 379 410 :M -.161(zero )A 403 410 :M -.235(partial )A 437 410 :M -.088(correlation.)A 59 428 :M (Such )S 88 428 :M -.07(necessary )A 139 428 :M -.109(and )A 161 428 :M -.131(sufficient )A 210 428 :M -.1(conditions )A 265 428 :M -.163(have )A 293 428 :M -.163(been )A 321 428 :M .224 .022(known )J 360 428 :M (for )S 380 428 :M -.165(path )A 406 428 :M -.123(diagrams )A 455 428 :M -.111(without)A 59 446 :M -.085(correlated errors, but we will extend the conditions to path diagrams with correlated errors.)A 77 470 :M (In )S 92 470 :M -.141(section )A 130 470 :M (2 )S 141 470 :M (we )S 160 470 :M -.166(will )A 183 470 :M -.248(motivate )A 228 470 :M -.164(interest )A 267 470 :M -.167(in )A 281 470 :M -.22(the )A 300 470 :M -.108(d-separation )A 363 470 :M -.206(relation )A 404 470 :M (by )S 422 470 :M -.098(describing )A 477 470 :M -.33(the)A 59 488 :M -.102(problems that it helps to solve in more detail. Then in section 3 we )A 376 488 :M -.166(will )A 398 488 :M .479 .048(show )J 428 488 :M .259 .026(how )J 453 488 :M -.22(the )A 471 488 :M -.215(zero)A 59 506 :M -.235(partial )A 92 506 :M -.136(correlations )A 151 506 :M -.248(entailed )A 191 506 :M (by )S 208 506 :M -.326(a )A 218 506 :M -.131(structural )A 267 506 :M -.165(equation )A 312 506 :M -.199(model )A 346 506 :M -.217(can )A 367 506 :M -.163(be )A 383 506 :M -.161(read )A 408 506 :M (off )S 427 506 :M -.08(from )A 455 506 :M -.112(its )A 471 506 :M -.22(path)A 59 524 :M -.039(diagram, )A 106 524 :M -.109(and )A 128 524 :M -.167(in )A 142 524 :M -.141(section )A 180 524 :M (4 )S 192 524 :M (use )S 214 524 :M -.22(the )A 234 524 :M -.182(machinery )A 289 524 :M -.146(developed )A 343 524 :M -.167(in )A 358 524 :M -.141(section )A 397 524 :M (3 )S 409 524 :M -.167(to )A 424 524 :M -.093(provide )A 466 524 :M -.109(some)A 59 542 :M -.037(solutions )A 107 542 :M -.167(to )A 121 542 :M -.082(problems )A 170 542 :M -.108(described )A 220 542 :M -.167(in )A 234 542 :M -.141(section )A 272 542 :M .833 .083(2. )J 287 542 :M (In )S 302 542 :M -.141(section )A 340 542 :M (6 )S 351 542 :M (we )S 370 542 :M -.064(prove )A 402 542 :M -.22(the )A 421 542 :M -.249(main )A 449 542 :M -.045(theorem,)A 59 560 :M -.072(hitherto unpublished, which justifies the )A 253 560 :M (use )S 273 560 :M (of )S 287 560 :M -.108(d-separation )A 349 560 :M -.167(in )A 362 560 :M -.165(path )A 386 560 :M -.123(diagrams )A 433 560 :M -.118(representing)A 59 578 :M -.196(correlated )A 110 578 :M (errors )S 143 578 :M -.107(\(represented )A 206 578 :M (by )S 224 578 :M -.064(edges )A 257 578 :M (of )S 273 578 :M -.22(the )A 293 578 :M -.08(form )A 322 578 :M f1_12 sf 1.011A f0_12 sf .441 .044(, )J 345 578 :M -.065(which )A 380 578 :M (we )S 400 578 :M -.33(call )A 422 578 :M -.136(double-headed)A 59 596 :M .195(arrows\).)A 59 626 :M f2_14 sf (2)S 67 626 :M (.)S 72 626 :M 19.5 1.95( )J 95 626 :M 3.201 .32(Problems in SEM Modeling)J 77 653 :M f0_12 sf -.099(In order to describe the problems listed in section 1 )A 322 653 :M -.167(in )A 335 653 :M -.163(more )A 363 653 :M -.093(detail, )A 396 653 :M (we )S 414 653 :M -.166(will )A 436 653 :M -.064(first )A 459 653 :M -.128(review)A 59 671 :M .259 .026(how )J 88 671 :M -.165(path )A 116 671 :M -.123(diagrams )A 167 671 :M -.215(are )A 189 671 :M (used )S 219 671 :M -.167(to )A 236 671 :M -.107(represent )A 287 671 :M -.131(structural )A 339 671 :M -.165(equation )A 387 671 :M -.11(models )A 429 671 :M -.095(without )A 473 671 :M -.213(free)A endp %%Page: 3 3 %%BeginPageSetup initializepage (peter; page: 3 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (3)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.119(parameters. The path diagram contains a directed edge from B to )A 367 56 :M -.663(A )A 379 56 :M -.164(if )A 390 56 :M -.109(and )A 411 56 :M -.083(only )A 436 56 :M -.164(if )A 447 56 :M -.196(there )A 474 56 :M (is )S 486 56 :M (a)S 59 74 :M -.08(non-zero )A 106 74 :M -.208(coefficient )A 160 74 :M (for )S 179 74 :M (B )S 193 74 :M -.167(in )A 208 74 :M -.22(the )A 228 74 :M -.165(equation )A 274 74 :M (for )S 294 74 :M -.499(A; )A 311 74 :M -.109(and )A 334 74 :M -.196(there )A 363 74 :M (is )S 377 74 :M -.326(a )A 388 74 :M -.125(double-headed )A 463 74 :M (arrow)S 59 92 :M -.151(between A and )A 133 92 :M (B )S 145 92 :M -.164(if )A 156 92 :M -.109(and )A 177 92 :M -.083(only )A 202 92 :M -.164(if )A 213 92 :M -.22(the )A 231 92 :M -.061(error )A 258 92 :M -.247(term )A 283 92 :M (for )S 301 92 :M -.663(A )A 313 92 :M -.109(and )A 334 92 :M -.22(the )A 352 92 :M -.061(error )A 379 92 :M -.247(term )A 404 92 :M (for )S 422 92 :M (B )S 434 92 :M -.163(have )A 460 92 :M -.326(a )A 469 92 :M (non-)S 59 110 :M -.161(zero )A 84 110 :M -.088(correlation.)A 138 107 :M f0_8 sf (1)S 142 110 :M f0_12 sf ( )S 147 110 :M -.219(The )A 170 110 :M -.165(path )A 195 110 :M -.188(diagram )A 238 110 :M -.131(associated )A 291 110 :M -.083(with )A 317 110 :M -.326(a )A 327 110 :M -.223(SEM )A 356 110 :M -.22(may )A 381 110 :M -.189(contain )A 420 110 :M -.205(directed )A 462 110 :M -.196(cycles)A 59 128 :M -.099(\(representing )A 126 128 :M -.029(feedback\), )A 181 128 :M -.109(and )A 203 128 :M -.125(double-headed )A 277 128 :M .223 .022(arrows )J 315 128 :M -.099(\(representing )A 382 128 :M -.196(correlated )A 433 128 :M .574 .057(errors.\) )J 475 128 :M -.652(We)A 59 146 :M -.143(will call a path diagram which contains )A 246 146 :M (no )S 262 146 :M -.125(double-headed )A 335 146 :M .223 .022(arrows )J 372 146 :M -.326(a )A 381 146 :M f2_12 sf 2.237 .224(directed )J 431 146 :M .39(graph)A f0_12 sf .347 .035(. )J 471 146 :M -.323(\(We)A 59 164 :M -.262(place )A 87 164 :M (sets )S 109 164 :M (of )S 123 164 :M -.145(variables )A 169 164 :M -.109(and )A 190 164 :M -.14(defined )A 229 164 :M -.131(terms )A 259 164 :M -.167(in )A 273 164 :M -.03(boldface.\) )A 326 164 :M (In )S 341 164 :M -.326(a )A 351 164 :M -.223(SEM )A 380 164 :M .277 .028(M, )J 399 164 :M (we )S 418 164 :M -.166(will )A 441 164 :M -.164(denote )A 477 164 :M -.33(the)A 59 182 :M -.179(correlation )A 114 182 :M -.22(matrix )A 149 182 :M -.132(among )A 186 182 :M -.22(the )A 205 182 :M -.033(non-error )A 255 182 :M -.145(variables )A 302 182 :M (by )S 319 182 :M f3_12 sf .786(S)A f2_12 sf .712(\(M\))A f0_12 sf .603 .06(, )J 357 182 :M -.109(and )A 379 182 :M -.22(the )A 398 182 :M -.049(corresponding )A 471 182 :M -.22(path)A 59 200 :M -.188(diagram )A 102 200 :M .4(by)A f2_12 sf .2 .02( )J 120 200 :M .693(G\(M\))A f0_12 sf .528 .053(. )J 160 200 :M -.326(We )A 181 200 :M -.166(will )A 204 200 :M .259 .026(now )J 230 200 :M -.107(review )A 267 200 :M -.22(the )A 286 200 :M -.082(problems )A 335 200 :M -.184(mentioned )A 389 200 :M -.167(in )A 403 200 :M -.141(section )A 441 200 :M (1 )S 452 200 :M -.167(in )A 467 200 :M -.218(more)A 59 218 :M -.109(detail.)A 95 248 :M f4_12 sf (2)S 102 248 :M (.)S 107 248 :M (1)S 114 248 :M (.)S 119 248 :M 9 .9( )J 131 248 :M 4.525 .452(Covariance Equivalence)J 77 269 :M f0_12 sf -.094(Consider the following example. The graph in Figure 1\(a\) is the path diagram of a SEM)A 59 287 :M -.667(M )A 73 287 :M (proposed )S 121 287 :M (by )S 137 287 :M -.274(Aberle )A 172 287 :M -.035(\(Blalock, )A 220 287 :M (1961\) )S 253 287 :M (as )S 268 287 :M -.326(a )A 278 287 :M -.199(model )A 312 287 :M (for )S 331 287 :M -.137(evolutionary )A 395 287 :M -.188(culture )A 432 287 :M -.167(in )A 446 287 :M -.329(American)A 59 305 :M -.043(Indian tribes, where W )A 172 305 :M (is )S 184 305 :M -.204(matridominant )A 256 305 :M -.084(division )A 298 305 :M (of )S 312 305 :M .217 .022(labor, )J 344 305 :M .306 .031(X )J 357 305 :M (is )S 369 305 :M -.264(matrilocal )A 419 305 :M -.03(residence, )A 471 305 :M -.663(Y )A 483 305 :M (is)S 59 323 :M -.111(matricentered land tenure, and Z is matrilinear system of descent.)A 77 341 :M -.074(Suppose for the moment )A 197 341 :M -.249(that )A 218 341 :M -.196(there )A 245 341 :M (is )S 257 341 :M -.326(a )A 266 341 :M -.223(SEM )A 294 341 :M -.083(with )A 319 341 :M -.22(the )A 337 341 :M -.165(path )A 361 341 :M -.188(diagram )A 403 341 :M -.167(in )A 416 341 :M -.054(Figure )A 451 341 :M -.078(1\(a\) )A 474 341 :M -.163(and)A 59 359 :M -.22(the )A 79 359 :M .32(p\()A f1_12 sf .422(c)A f0_7 sf 0 -5 rm .224(2)A 0 5 rm f0_12 sf .534 .053(\), )J 114 359 :M -.22(the )A 134 359 :M -.22(AIC )A 160 359 :M -.235(\(Aikake )A 203 359 :M -.119(Information )A 266 359 :M -.028(Criterion\), )A 323 359 :M -.109(and )A 347 359 :M -.22(the )A 368 359 :M (BIC )S 395 359 :M -.053(\(Bayes )A 435 359 :M -.131(Information)A 59 377 :M -.102(Criterion\) score )A 137 377 :M (for )S 155 377 :M -.084(this )A 176 377 :M -.223(SEM )A 204 377 :M -.215(are )A 222 377 :M -.331(all )A 237 377 :M -.111(high)A 258 374 :M f0_8 sf (2)S 262 377 :M f0_12 sf ( )S 266 377 :M -.079(\(See )A 291 377 :M -.139(Raftery )A 330 377 :M (1995 )S 358 377 :M (for )S 376 377 :M -.326(a )A 385 377 :M (discussion )S 439 377 :M (of )S 453 377 :M -.22(the )A 471 377 :M (BIC)S 59 395 :M .408 .041(score.\) )J 96 395 :M (In )S 110 395 :M -.062(order )A 139 395 :M -.167(to )A 152 395 :M -.247(evaluate )A 194 395 :M .259 .026(how )J 219 395 :M -.164(well )A 243 395 :M -.22(the )A 261 395 :M -.247(data )A 284 395 :M .188 .019(supports )J 329 395 :M -.084(this )A 350 395 :M (model, )S 387 395 :M -.334(it )A 397 395 :M (is )S 409 395 :M -.201(still )A 431 395 :M -.07(necessary )A 482 395 :M -.334(to)A 59 413 :M -.101(know whether or not there are other models compatible with background knowledge that )A 481 413 :M -.331(fit)A 59 431 :M -.185(the data equally )A 136 431 :M .229 .023(well. )J 164 431 :M -.244(\(Lee )A 189 431 :M -.109(and )A 210 431 :M -.026(Hershberger )A 273 431 :M .667 .067(1990, )J 305 431 :M -.046(Stelzl, )A 339 431 :M .629 .063(1986\). )J 375 431 :M (In )S 389 431 :M -.084(this )A 410 431 :M .236 .024(case, )J 438 431 :M (for )S 456 431 :M -.245(each )A 481 431 :M (of)S 59 449 :M -.095(the path diagrams in Figure 1, and for )A f5_12 sf -.116(any)A f0_12 sf -.098( data set D, there is a SEM with that path diagram)A 59 467 :M -.116(that fits D as well as M does \(in the )A 229 467 :M (sense )S 259 467 :M -.249(that )A 280 467 :M -.245(each )A 305 467 :M -.223(SEM )A 333 467 :M (has )S 353 467 :M -.22(the )A 371 467 :M -.163(same )A 399 467 :M .171(p\()A f1_12 sf .226(c)A f0_7 sf 0 -5 rm .12(2)A 0 5 rm f0_12 sf .218 .022(\) )J 428 467 :M -.109(and )A 449 467 :M -.22(the )A 467 467 :M -.218(same)A 59 485 :M -.076(BIC and AIC scores.\) If )A f2_12 sf -.144(O)A f0_12 sf -.074( represent the set of measured variables in path )A 410 485 :M -.123(diagrams )A 457 485 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 474 485 :M -.163(and)A 59 503 :M .542(G)A f0_7 sf 0 3 rm .219(2)A 0 -3 rm f0_12 sf .857 .086(, then G)J f0_7 sf 0 3 rm .219(1)A 0 -3 rm f0_12 sf .77 .077( and G)J f0_7 sf 0 3 rm .219(2)A 0 -3 rm f0_12 sf .517 .052( are )J f2_12 sf 1.643 .164(covariance equivalent over O)J 335 500 :M f0_8 sf (3)S 339 503 :M f0_12 sf -.078( if and only if for every )A 453 503 :M -.223(SEM )A 481 503 :M (M)S 59 530 :M ( )S 59 527.48 -.48 .48 203.48 527 .48 59 527 @a 59 540 :M f0_8 sf (1)S 63 543 :M f0_10 sf .5 .05( )J 67 543 :M .517 .052(This )J 89 543 :M .694 .069(is )J 100 543 :M .832 .083(slightly )J 135 543 :M -.213(different )A 171 543 :M .202 .02(than )J 192 543 :M .218 .022(the )J 208 543 :M .263 .026(usual )J 233 543 :M .033 .003(convention )J 281 543 :M .601 .06(in )J 293 543 :M .043 .004(which )J 321 543 :M .328 .033(if )J 331 543 :M f1_10 sf .305(e)A f0_6 sf 0 2 rm .301(A)A 0 -2 rm f0_10 sf .173 .017( )J 344 543 :M -.313(and )A 361 543 :M f1_10 sf .447(e)A f0_6 sf 0 2 rm .407(B)A 0 -2 rm f0_10 sf .254 .025( )J 375 543 :M -.235(are )A 391 543 :M -.179(correlated, )A 436 543 :M .202 .02(then )J 458 543 :M .202 .02(they )J 480 543 :M -.602(are)A 59 555 :M .366 .037(explicitly )J 101 555 :M -.242(included )A 137 555 :M .601 .06(in )J 149 555 :M .218 .022(the )J 165 555 :M .146 .015(graph, )J 194 555 :M -.097(there )A 217 555 :M .694 .069(is )J 228 555 :M .056 .006(a )J 236 555 :M -.337(directed )A 269 555 :M -.344(edge )A 290 555 :M .047 .005(from )J 313 555 :M f1_10 sf .305(e)A f0_6 sf 0 2 rm .301(A)A 0 -2 rm f0_10 sf .173 .017( )J 327 555 :M .601 .06(to )J 340 555 :M .65 .065(A, )J 355 555 :M .056 .006(a )J 364 555 :M -.337(directed )A 398 555 :M -.344(edge )A 420 555 :M .047 .005(from )J 444 555 :M f1_10 sf .527(e)A f0_6 sf 0 2 rm .481(B)A 0 -2 rm f0_10 sf .546 .055(, )J 461 555 :M -.313(and )A 479 555 :M -.108(the)A 59 567 :M -.112(double-headed arrow is placed between)A f6_10 sf -.065( )A f1_10 sf -.115(e)A f0_6 sf 0 2 rm -.113(A)A 0 -2 rm f0_10 sf -.102( and )A f1_10 sf -.115(e)A f0_6 sf 0 2 rm -.105(B)A 0 -2 rm f0_10 sf -.131(. )A 257 567 :M -.08(However, )A 299 567 :M .218 .022(the )J 315 567 :M .033 .003(convention )J 363 567 :M -.308(adopted )A 396 567 :M -.176(here )A 416 567 :M .676 .068(will )J 436 567 :M .65 .065(simplify )J 474 567 :M -.19(later)A 59 579 :M -.06(theorems and proofs.)A 59 588 :M f0_8 sf (2)S 63 591 :M f0_10 sf .5 .05( )J 68 591 :M .144 .014(In )J 81 591 :M .28 .028(counting )J 121 591 :M -.29(degrees )A 154 591 :M .144 .014(of )J 167 591 :M -.164(freedom, )A 206 591 :M -.079(we )A 222 591 :M .676 .068(will )J 243 591 :M .041 .004(assume )J 277 591 :M .361 .036(that )J 297 591 :M .056 .006(a )J 306 591 :M .726 .073(SEM )J 332 591 :M .517 .052(with )J 356 591 :M -.258(free )A 376 591 :M -.136(parameters )A 424 591 :M -.316(\(and )A 446 591 :M .417 .042(no )J 462 591 :M -.061(latents\))A 59 603 :M -.035(associates a )A 109 603 :M -.044(linear )A 135 603 :M -.113(coefficient )A 180 603 :M -.163(parameter )A 222 603 :M .517 .052(with )J 244 603 :M -.204(each )A 265 603 :M -.337(directed )A 298 603 :M -.344(edge )A 319 603 :M .637 .064(\(i.e. )J 339 603 :M f1_10 sf .191A f0_10 sf .103 .01(\) )J 356 603 :M .601 .06(in )J 368 603 :M .811 .081(its )J 382 603 :M .202 .02(path )J 403 603 :M -.095(diagram, )A 441 603 :M .056 .006(a )J 449 603 :M -.131(correlation)A 59 615 :M -.053(parameter with )A 122 615 :M -.204(each )A 143 615 :M -.335(double-headed )A 201 615 :M -.163(arrow )A 227 615 :M .637 .064(\(i.e. )J 247 615 :M f1_10 sf -.159A f0_10 sf -.089(\) )A 264 615 :M .601 .06(in )J 276 615 :M .811 .081(its )J 290 615 :M .202 .02(path )J 311 615 :M -.095(diagram, )A 349 615 :M -.313(and )A 366 615 :M .056 .006(a )J 374 615 :M -.17(variance )A 410 615 :M -.163(parameter )A 452 615 :M .517 .052(with )J 474 615 :M -.438(each)A 59 627 :M (vertex. )S 91 627 :M -.188(We )A 109 627 :M .281 .028(also )J 130 627 :M .041 .004(assume )J 164 627 :M .361 .036(that )J 184 627 :M .417 .042(no )J 199 627 :M -.097(extra )A 223 627 :M .087 .009(constraints )J 271 627 :M -.031(\(such )A 297 627 :M .144 .014(as )J 310 627 :M -.089(equality )A 347 627 :M .087 .009(constraints )J 396 627 :M .189 .019(among )J 429 627 :M -.153(parameters\) )A 480 627 :M -.602(are)A 59 639 :M -.055(imposed.)A 59 648 :M f0_8 sf (3)S 63 651 :M f0_10 sf .331 .033( For technical reasons, a more formal definition requires a slight complication. G is a)J f2_10 sf .468 .047( sub-path)J f0_10 sf .05 .005( )J f2_10 sf .133(diagram)A 59 663 :M f0_10 sf .144 .014(of )J 71 663 :M <47D520>S 85 663 :M -.039(when )A 110 663 :M .255 .026(G )J 122 663 :M -.313(and )A 140 663 :M <47D520>S 155 663 :M -.094(have )A 178 663 :M .218 .022(the )J 195 663 :M (same )S 220 663 :M -.01(vertices, )A 258 663 :M -.313(and )A 276 663 :M .255 .026(G )J 288 663 :M .133 .013(has )J 306 663 :M .056 .006(a )J 315 663 :M .315 .032(subset )J 345 663 :M .144 .014(of )J 358 663 :M .218 .022(the )J 375 663 :M -.253(edges )A 401 663 :M .601 .06(in )J 414 663 :M .349 .035(G\325. )J 432 663 :M .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .055 .006( )J 447 663 :M -.313(and )A 465 663 :M .159(G)A f0_6 sf 0 2 rm .066(2)A 0 -2 rm f0_10 sf .055 .006( )J 480 663 :M -.602(are)A 59 675 :M f2_10 sf 1.159 .116(covariance equivalent over O)J f0_10 sf .638 .064( if for every SEM M such that G\(M\) = G)J f0_6 sf 0 2 rm .162(1)A 0 -2 rm f0_10 sf .654 .065(, there is a SEM M\325 with path)J 59 687 :M -.02(diagram G\(M\325\) that is a sub-path diagram of G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf -.019(, and the marginal over )A f2_10 sf (O)S f0_10 sf -.016( of )A f1_10 sf (S)S f0_10 sf -.021(\(M\325\) equals the )A 435 687 :M (marginal )S 474 687 :M -.255(over)A endp %%Page: 4 4 %%BeginPageSetup initializepage (peter; page: 4 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (4)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.053(such that G\(M\) = G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.051(, there is a SEM M\325 with )A 278 56 :M -.165(path )A 302 56 :M -.188(diagram )A 344 56 :M -.062(G\(M\325\) )A 379 56 :M .211 .021(= )J 390 56 :M .876(G)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 411 56 :M -.109(and )A 432 56 :M -.22(the )A 450 56 :M -.235(marginal)A 59 74 :M -.053(of )A f1_12 sf -.087(S)A f0_12 sf -.061(\(M\325\) over )A f2_12 sf -.115(O)A f0_12 sf -.057( equals the the marginal of )A f1_12 sf -.087(S)A f0_12 sf -.063(\(M\) over )A f2_12 sf -.115(O)A f0_12 sf -.074(, )A 335 74 :M -.109(and )A 356 74 :M -.027(vice-versa. )A 412 74 :M -.025(\(Informally, )A 474 74 :M -.163(any)A 59 92 :M -.196(covariance )A 114 92 :M -.22(matrix )A 149 92 :M -.08(over )A 175 92 :M f2_12 sf -.253(O)A f0_12 sf ( )S 190 92 :M -.181(generated )A 241 92 :M (by )S 259 92 :M -.326(a )A 270 92 :M -.226(parameterization )A 353 92 :M (of )S 369 92 :M -.165(path )A 395 92 :M -.188(diagram )A 439 92 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 458 92 :M -.217(can )A 480 92 :M -.326(be)A 59 110 :M -.091(generated by a parameterization of path diagram G)A f0_7 sf 0 3 rm -.064(2)A 0 -3 rm f0_12 sf -.084(, and vice-versa.\) If )A 398 110 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 415 110 :M -.109(and )A 436 110 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 453 110 :M -.163(have )A 479 110 :M (no)S 59 128 :M -.098(latent variables, \(i.e all of the variables in their path diagrams are in )A f2_12 sf -.208(O)A f0_12 sf -.111(\), then we will simply)A 59 146 :M -.013(say that G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.014( and G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.014( are )A 168 146 :M f2_12 sf 2.845 .285(covariance )J 233 146 :M .51(equivalent)A f0_12 sf .522 .052(. )J 299 146 :M (If )S 311 146 :M (two )S 333 146 :M -.196(covariance )A 387 146 :M -.198(equivalent )A 439 146 :M -.11(models )A 477 146 :M -.323(are)A 59 164 :M -.189(equally )A 97 164 :M -.231(compatible )A 152 164 :M -.083(with )A 177 164 :M -.065(background )A 237 164 :M .175 .018(knowledge, )J 297 164 :M -.109(and )A 318 164 :M -.163(have )A 344 164 :M -.22(the )A 363 164 :M -.163(same )A 392 164 :M -.091(degrees )A 433 164 :M (of )S 448 164 :M (freedom,)S 59 182 :M -.097(the data does not help distinguish them, so it is important to be able to find )A 414 182 :M -.22(the )A 432 182 :M -.248(complete )A 478 182 :M -.164(set)A 59 200 :M -.093(of path )A 96 200 :M -.123(diagrams )A 143 200 :M -.249(that )A 164 200 :M -.215(are )A 182 200 :M -.196(covariance )A 236 200 :M -.198(equivalent )A 288 200 :M -.167(to )A 301 200 :M -.326(a )A 310 200 :M -.132(given )A 340 200 :M -.165(path )A 364 200 :M -.039(diagram. )A 410 200 :M -.107(\(Every )A 446 200 :M -.223(SEM )A 474 200 :M -.331(that)A 59 218 :M -.083(contains a path diagram in Figure 1 has the same number of degrees of freedom.\))A 251 404 :M f2_12 sf 2.949 .295(Figure 1)J 77 428 :M f0_12 sf -.093(It is often far from obvious what constitutes a complete set of )A 370 428 :M -.165(path )A 394 428 :M -.123(diagrams )A 441 428 :M -.218(covariance)A 59 446 :M -.198(equivalent )A 113 446 :M -.167(to )A 128 446 :M -.326(a )A 139 446 :M -.132(given )A 171 446 :M -.165(path )A 197 446 :M -.039(diagram. )A 245 446 :M -.326(We )A 267 446 :M -.166(will )A 291 446 :M -.33(call )A 313 446 :M (such )S 341 446 :M -.326(a )A 352 446 :M -.248(complete )A 400 446 :M -.109(set )A 419 446 :M -.326(a )A 431 446 :M f2_12 sf .521(covariance)A 59 464 :M 1.972 .197(equivalence class over O)J f0_12 sf 1.129 .113(. \(Again, if )J 256 464 :M (we )S 274 464 :M -.081(consider )A 318 464 :M -.083(only )A 343 464 :M -.084(SEMs )A 376 464 :M -.095(without )A 416 464 :M -.276(latent )A 445 464 :M -.034(variables,)A 59 482 :M .493 .049(we will call such a complete set a )J f2_12 sf 1.047 .105(covariance equivalence class)J f0_12 sf .282 .028(.\) If it is )J 423 482 :M -.326(a )A 432 482 :M -.248(complete )A 478 482 :M -.164(set)A 59 500 :M (of )S 73 500 :M -.165(path )A 97 500 :M -.123(diagrams )A 144 500 :M -.095(without )A 184 500 :M -.196(correlated )A 234 500 :M (errors )S 266 500 :M (or )S 280 500 :M -.205(directed )A 321 500 :M (cycles, )S 358 500 :M .957 .096(i.e. )J 378 500 :M -.205(directed )A 419 500 :M -.282(acyclic )A 455 500 :M .169(graphs,)A 59 518 :M -.249(that )A 80 518 :M -.215(are )A 98 518 :M -.196(covariance )A 152 518 :M -.198(equivalent )A 204 518 :M (we )S 222 518 :M -.166(will )A 244 518 :M -.33(call )A 264 518 :M -.334(it )A 274 518 :M -.326(a )A 283 518 :M f2_12 sf 2.922 .292(simple )J 325 518 :M 2.845 .285(covariance )J 391 518 :M 3.34 .334(equivalence )J 463 518 :M 1.001(class)A 59 536 :M 2.391 .239(over )J 89 536 :M .325(O)A f0_12 sf .29 .029(.\) )J 110 536 :M -.165(As )A 127 536 :M .447 .045(shown )J 163 536 :M -.167(in )A 176 536 :M -.141(section )A 213 536 :M .833 .083(4, )J 227 536 :M -.22(the )A 245 536 :M -.165(path )A 269 536 :M -.123(diagrams )A 316 536 :M -.167(in )A 329 536 :M -.054(Figure )A 365 536 :M (1 )S 376 536 :M -.215(are )A 395 536 :M -.326(a )A 405 536 :M -.166(simple )A 441 536 :M -.218(covariance)A 59 554 :M -.095(equivalence class.)A 77 572 :M -.1(Another example of a case where it is not obvious whether or not two path diagrams are)A 59 590 :M -.196(covariance )A 115 590 :M -.198(equivalent )A 169 590 :M -.08(over )A 196 590 :M f2_12 sf -.253(O)A f0_12 sf ( )S 211 590 :M (is )S 225 590 :M .447 .045(shown )J 263 590 :M .423 .042(below. )J 302 590 :M -.164(It )A 315 590 :M (is )S 329 590 :M -.131(often )A 359 590 :M -.095(thought )A 401 590 :M -.249(that )A 425 590 :M -.22(the )A 446 590 :M (two )S 471 590 :M -.22(path)A 59 608 :M -.123(diagrams )A 107 608 :M -.167(in )A 121 608 :M -.054(Figure )A 157 608 :M (2 )S 169 608 :M -.194(\(each )A 200 608 :M (of )S 216 608 :M -.065(which )A 251 608 :M (is )S 265 608 :M -.163(part )A 289 608 :M (of )S 305 608 :M -.326(a )A 316 608 :M -.154(just-identified )A 387 608 :M -.165(SEM\) )A 421 608 :M -.215(are )A 441 608 :M -.218(covariance)A 59 626 :M (equivalent over )S f2_12 sf (O)S f0_12 sf .007 .001( = {X,Y,Z}. However, as shown in Spirtes et al. \(1996\), there is a SEM)J 59 644 :M -.102(with path diagram in Figure 2\(b\) with the covariance matrix )A f1_12 sf -.212(S )A 355 644 :M f0_12 sf -.08(over )A 380 644 :M 1.114 .111(X, )J 397 644 :M .281 .028(Y, )J 413 644 :M -.109(and )A 434 644 :M .558 .056(Z, )J 449 644 :M -.111(but )A 468 644 :M -.245(there)A 59 662 :M ( )S 59 659.48 -.48 .48 491.48 659 .48 59 659 @a 59 675 :M f2_10 sf (O)S f0_10 sf .024 .002( of )J f1_10 sf (S)S f0_10 sf .059 .006(\(M\), and for every SEM M\325 such that G\(M\325\) = G)J f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf .044 .004(, there )J 315 675 :M .694 .069(is )J 326 675 :M .056 .006(a )J 334 675 :M .726 .073(SEM )J 359 675 :M .555 .056(M )J 372 675 :M .517 .052(with )J 394 675 :M .202 .02(path )J 415 675 :M -.18(diagram )A 450 675 :M -.066(G\(M\) )A 476 675 :M (that)S 59 687 :M (is a sub-path diagram of G)S f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (, and the marginal over )S f2_10 sf (O)S f0_10 sf ( of )S f1_10 sf (S)S f0_10 sf (\(M\) equals the marginal over )S f2_10 sf (O)S f0_10 sf ( of )S f1_10 sf (S)S f0_10 sf <284DD5292E>S 145 227 18 19 rC 145 242 :M f0_12 sf (X)S gR gS 96 228 18 19 rC 96 243 :M f0_12 sf (W)S gR gS 102 266 18 19 rC 102 281 :M f0_12 sf (Y)S gR gS 145 267 18 19 rC 145 282 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 113 240.75 -.75 .75 136.75 240 .75 113 240 @a np 134 236 :M 134 244 :L 142 240 :L 134 236 :L .75 lw eofill -.75 -.75 134.75 244.75 .75 .75 134 236 @b -.75 -.75 134.75 244.75 .75 .75 142 240 @b 134 236.75 -.75 .75 142.75 240 .75 134 236 @a 122 276.75 -.75 .75 140.75 276 .75 122 276 @a np 124 280 :M 124 272 :L 116 276 :L 124 280 :L eofill -.75 -.75 124.75 280.75 .75 .75 124 272 @b -.75 -.75 116.75 276.75 .75 .75 124 272 @b 116 276.75 -.75 .75 124.75 280 .75 116 276 @a -.75 -.75 148.75 260.75 .75 .75 148 240 @b np 152 258 :M 144 258 :L 148 266 :L 152 258 :L eofill 144 258.75 -.75 .75 152.75 258 .75 144 258 @a 144 258.75 -.75 .75 148.75 266 .75 144 258 @a -.75 -.75 148.75 266.75 .75 .75 152 258 @b 111 243.75 -.75 .75 135.75 262 .75 111 243 @a np 136 258 :M 132 264 :L 140 266 :L 136 258 :L eofill -.75 -.75 132.75 264.75 .75 .75 136 258 @b 132 264.75 -.75 .75 140.75 266 .75 132 264 @a 136 258.75 -.75 .75 140.75 266 .75 136 258 @a 236 230 18 19 rC 236 245 :M f0_12 sf (X)S gR gS 192 229 18 19 rC 192 244 :M f0_12 sf (W)S gR gS 193 269 18 19 rC 193 284 :M f0_12 sf (Y)S gR gS 236 270 18 19 rC 236 285 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 204 243.75 -.75 .75 226.75 243 .75 204 243 @a np 224 239 :M 224 247 :L 232 243 :L 224 239 :L .75 lw eofill -.75 -.75 224.75 247.75 .75 .75 224 239 @b -.75 -.75 224.75 247.75 .75 .75 232 243 @b 224 239.75 -.75 .75 232.75 243 .75 224 239 @a 213 279.75 -.75 .75 231.75 279 .75 213 279 @a np 215 283 :M 215 275 :L 207 279 :L 215 283 :L eofill -.75 -.75 215.75 283.75 .75 .75 215 275 @b -.75 -.75 207.75 279.75 .75 .75 215 275 @b 207 279.75 -.75 .75 215.75 283 .75 207 279 @a -.75 -.75 239.75 269.75 .75 .75 239 252 @b np 235 254 :M 243 254 :L 239 246 :L 235 254 :L eofill 235 254.75 -.75 .75 243.75 254 .75 235 254 @a 239 246.75 -.75 .75 243.75 254 .75 239 246 @a -.75 -.75 235.75 254.75 .75 .75 239 246 @b 207 250.75 -.75 .75 231.75 269 .75 207 250 @a np 206 254 :M 211 248 :L 202 246 :L 206 254 :L eofill -.75 -.75 206.75 254.75 .75 .75 211 248 @b 202 246.75 -.75 .75 211.75 248 .75 202 246 @a 202 246.75 -.75 .75 206.75 254 .75 202 246 @a 324 229 18 19 rC 324 244 :M f0_12 sf (X)S gR gS 280 228 18 19 rC 280 243 :M f0_12 sf (W)S gR gS 281 268 18 19 rC 281 283 :M f0_12 sf (Y)S gR gS 324 269 18 19 rC 324 284 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 298 242.75 -.75 .75 320.75 242 .75 298 242 @a np 300 246 :M 300 238 :L 292 242 :L 300 246 :L .75 lw eofill -.75 -.75 300.75 246.75 .75 .75 300 238 @b -.75 -.75 292.75 242.75 .75 .75 300 238 @b 292 242.75 -.75 .75 300.75 246 .75 292 242 @a 301 278.75 -.75 .75 319.75 278 .75 301 278 @a np 303 282 :M 303 274 :L 295 278 :L 303 282 :L eofill -.75 -.75 303.75 282.75 .75 .75 303 274 @b -.75 -.75 295.75 278.75 .75 .75 303 274 @b 295 278.75 -.75 .75 303.75 282 .75 295 278 @a -.75 -.75 327.75 262.75 .75 .75 327 245 @b np 331 260 :M 323 260 :L 327 268 :L 331 260 :L eofill 323 260.75 -.75 .75 331.75 260 .75 323 260 @a 323 260.75 -.75 .75 327.75 268 .75 323 260 @a -.75 -.75 327.75 268.75 .75 .75 331 260 @b 295 249.75 -.75 .75 319.75 268 .75 295 249 @a np 294 253 :M 299 247 :L 290 245 :L 294 253 :L eofill -.75 -.75 294.75 253.75 .75 .75 299 247 @b 290 245.75 -.75 .75 299.75 247 .75 290 245 @a 290 245.75 -.75 .75 294.75 253 .75 290 245 @a 410 229 18 19 rC 410 244 :M f0_12 sf (X)S gR gS 366 228 18 19 rC 366 243 :M f0_12 sf (W)S gR gS 367 268 18 19 rC 367 283 :M f0_12 sf (Y)S gR gS 410 269 18 19 rC 410 284 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 378 242.75 -.75 .75 400.75 242 .75 378 242 @a np 398 238 :M 398 246 :L 406 242 :L 398 238 :L .75 lw eofill -.75 -.75 398.75 246.75 .75 .75 398 238 @b -.75 -.75 398.75 246.75 .75 .75 406 242 @b 398 238.75 -.75 .75 406.75 242 .75 398 238 @a 387 278.75 -.75 .75 405.75 278 .75 387 278 @a np 389 282 :M 389 274 :L 381 278 :L 389 282 :L eofill -.75 -.75 389.75 282.75 .75 .75 389 274 @b -.75 -.75 381.75 278.75 .75 .75 389 274 @b 381 278.75 -.75 .75 389.75 282 .75 381 278 @a -.75 -.75 413.75 268.75 .75 .75 413 251 @b np 409 253 :M 417 253 :L 413 245 :L 409 253 :L eofill 409 253.75 -.75 .75 417.75 253 .75 409 253 @a 413 245.75 -.75 .75 417.75 253 .75 413 245 @a -.75 -.75 409.75 253.75 .75 .75 413 245 @b 376 245.75 -.75 .75 400.75 264 .75 376 245 @a np 401 260 :M 397 266 :L 405 268 :L 401 260 :L eofill -.75 -.75 397.75 266.75 .75 .75 401 260 @b 397 266.75 -.75 .75 405.75 268 .75 397 266 @a 401 260.75 -.75 .75 405.75 268 .75 401 260 @a 142 301 18 19 rC 142 316 :M f0_12 sf (X)S gR gS 98 300 18 19 rC 98 315 :M f0_12 sf (W)S gR gS 98 345 18 19 rC 98 360 :M f0_12 sf (Y)S gR gS 141 346 18 19 rC 141 361 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 117 311.75 -.75 .75 139.75 311 .75 117 311 @a np 119 315 :M 119 307 :L 111 311 :L 119 315 :L .75 lw eofill -.75 -.75 119.75 315.75 .75 .75 119 307 @b -.75 -.75 111.75 311.75 .75 .75 119 307 @b 111 311.75 -.75 .75 119.75 315 .75 111 311 @a 118 355.75 -.75 .75 136.75 355 .75 118 355 @a np 120 358 :M 120 351 :L 112 355 :L 120 358 :L eofill -.75 -.75 120.75 358.75 .75 .75 120 351 @b -.75 -.75 112.75 355.75 .75 .75 120 351 @b 112 355.75 -.75 .75 120.75 358 .75 112 355 @a -.75 -.75 144.75 339.75 .75 .75 144 322 @b np 148 337 :M 140 337 :L 144 345 :L 148 337 :L eofill 140 337.75 -.75 .75 148.75 337 .75 140 337 @a 140 337.75 -.75 .75 144.75 345 .75 140 337 @a -.75 -.75 144.75 345.75 .75 .75 148 337 @b 107 322.75 -.75 .75 131.75 341 .75 107 322 @a np 132 337 :M 128 343 :L 136 345 :L 132 337 :L eofill -.75 -.75 128.75 343.75 .75 .75 132 337 @b 128 343.75 -.75 .75 136.75 345 .75 128 343 @a 132 337.75 -.75 .75 136.75 345 .75 132 337 @a 233 304 18 19 rC 233 319 :M f0_12 sf (X)S gR gS 189 303 18 19 rC 189 318 :M f0_12 sf (W)S gR gS 189 348 18 19 rC 189 363 :M f0_12 sf (Y)S gR gS 232 349 18 19 rC 232 364 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 202 314.75 -.75 .75 224.75 314 .75 202 314 @a np 222 310 :M 222 318 :L 230 314 :L 222 310 :L .75 lw eofill -.75 -.75 222.75 318.75 .75 .75 222 310 @b -.75 -.75 222.75 318.75 .75 .75 230 314 @b 222 310.75 -.75 .75 230.75 314 .75 222 310 @a 203 358.75 -.75 .75 221.75 358 .75 203 358 @a np 219 354 :M 219 361 :L 227 358 :L 219 354 :L eofill -.75 -.75 219.75 361.75 .75 .75 219 354 @b -.75 -.75 219.75 361.75 .75 .75 227 358 @b 219 354.75 -.75 .75 227.75 358 .75 219 354 @a -.75 -.75 235.75 348.75 .75 .75 235 331 @b np 231 333 :M 239 333 :L 235 325 :L 231 333 :L eofill 231 333.75 -.75 .75 239.75 333 .75 231 333 @a 235 325.75 -.75 .75 239.75 333 .75 235 325 @a -.75 -.75 231.75 333.75 .75 .75 235 325 @b 203 329.75 -.75 .75 227.75 348 .75 203 329 @a np 202 333 :M 207 327 :L 198 325 :L 202 333 :L eofill -.75 -.75 202.75 333.75 .75 .75 207 327 @b 198 325.75 -.75 .75 207.75 327 .75 198 325 @a 198 325.75 -.75 .75 202.75 333 .75 198 325 @a 321 303 18 19 rC 321 318 :M f0_12 sf (X)S gR gS 277 302 18 19 rC 277 317 :M f0_12 sf (W)S gR gS 277 347 18 19 rC 277 362 :M f0_12 sf (Y)S gR gS 320 348 18 19 rC 320 363 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 296 313.75 -.75 .75 318.75 313 .75 296 313 @a np 298 317 :M 298 309 :L 290 313 :L 298 317 :L .75 lw eofill -.75 -.75 298.75 317.75 .75 .75 298 309 @b -.75 -.75 290.75 313.75 .75 .75 298 309 @b 290 313.75 -.75 .75 298.75 317 .75 290 313 @a 291 357.75 -.75 .75 309.75 357 .75 291 357 @a np 307 353 :M 307 360 :L 315 357 :L 307 353 :L eofill -.75 -.75 307.75 360.75 .75 .75 307 353 @b -.75 -.75 307.75 360.75 .75 .75 315 357 @b 307 353.75 -.75 .75 315.75 357 .75 307 353 @a -.75 -.75 323.75 347.75 .75 .75 323 330 @b np 319 332 :M 327 332 :L 323 324 :L 319 332 :L eofill 319 332.75 -.75 .75 327.75 332 .75 319 332 @a 323 324.75 -.75 .75 327.75 332 .75 323 324 @a -.75 -.75 319.75 332.75 .75 .75 323 324 @b 291 328.75 -.75 .75 315.75 347 .75 291 328 @a np 290 332 :M 295 326 :L 286 324 :L 290 332 :L eofill -.75 -.75 290.75 332.75 .75 .75 295 326 @b 286 324.75 -.75 .75 295.75 326 .75 286 324 @a 286 324.75 -.75 .75 290.75 332 .75 286 324 @a 407 303 18 19 rC 407 318 :M f0_12 sf (X)S gR gS 363 302 18 19 rC 363 317 :M f0_12 sf (W)S gR gS 363 347 18 19 rC 363 362 :M f0_12 sf (Y)S gR gS 406 348 18 19 rC 406 363 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 382 313.75 -.75 .75 404.75 313 .75 382 313 @a np 384 317 :M 384 309 :L 376 313 :L 384 317 :L .75 lw eofill -.75 -.75 384.75 317.75 .75 .75 384 309 @b -.75 -.75 376.75 313.75 .75 .75 384 309 @b 376 313.75 -.75 .75 384.75 317 .75 376 313 @a 383 357.75 -.75 .75 401.75 357 .75 383 357 @a np 385 360 :M 385 353 :L 377 357 :L 385 360 :L eofill -.75 -.75 385.75 360.75 .75 .75 385 353 @b -.75 -.75 377.75 357.75 .75 .75 385 353 @b 377 357.75 -.75 .75 385.75 360 .75 377 357 @a -.75 -.75 409.75 347.75 .75 .75 409 330 @b np 405 332 :M 413 332 :L 409 324 :L 405 332 :L eofill 405 332.75 -.75 .75 413.75 332 .75 405 332 @a 409 324.75 -.75 .75 413.75 332 .75 409 324 @a -.75 -.75 405.75 332.75 .75 .75 409 324 @b 377 328.75 -.75 .75 401.75 347 .75 377 328 @a np 376 332 :M 381 326 :L 372 324 :L 376 332 :L eofill -.75 -.75 376.75 332.75 .75 .75 381 326 @b 372 324.75 -.75 .75 381.75 326 .75 372 324 @a 372 324.75 -.75 .75 376.75 332 .75 372 324 @a 121 280 18 19 rC 121 295 :M f0_12 sf -.156(\(a\))A gR gS 215 280 18 19 rC 215 295 :M f0_12 sf (\(b\))S 215 313 :M <29>S gR gS 301 280 18 19 rC 301 295 :M f0_12 sf -.102(\(c\)\))A gR gS 388 280 18 19 rC 388 295 :M f0_12 sf (\(d\))S 388 313 :M <29>S gR gS 114 359 18 19 rC 114 374 :M f0_12 sf -.102(\(e\)\))A gR gS 208 359 18 19 rC 208 374 :M f0_12 sf (\(f\))S gR gS 294 359 18 19 rC 294 374 :M f0_12 sf (\(g\))S 294 392 :M <29>S gR gS 388 359 18 19 rC 388 374 :M f0_12 sf (\(h\))S 388 392 :M <292929>S gR endp %%Page: 5 5 %%BeginPageSetup initializepage (peter; page: 5 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (5)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf (is )S 71 56 :M (no )S 87 56 :M -.223(SEM )A 115 56 :M -.249(that )A 136 56 :M -.123(contains )A 179 56 :M -.165(path )A 203 56 :M -.188(diagram )A 245 56 :M -.167(in )A 258 56 :M -.054(Figure )A 293 56 :M (2 )S 303 56 :M -.104(\(a\) )A 321 56 :M -.083(with )A 347 56 :M -.206(marginal )A 393 56 :M -.196(covariance )A 448 56 :M -.22(matrix )A 483 56 :M f1_12 sf (S)S 59 74 :M f0_12 sf -.019(\(where T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf (, T)S f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.017(, and T)A f0_7 sf 0 3 rm (3)S 0 -3 rm f0_12 sf -.016( are latent variables\).)A 77 101 121 52 rC 198 153 :M psb currentpoint pse 77 101 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3872 div 1664 3 -1 roll exch div scale currentpoint translate 64 39 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (S) -1 889 sh 384 /Symbol f1 (=) 346 889 sh (\346) 662 355 sh (\350) 662 1468 sh (\347) 662 806 sh (\347) 662 1177 sh (\366) 3576 355 sh (\370) 3576 1468 sh (\367) 3576 806 sh (\367) 3576 1177 sh 384 /Times-Roman f1 (1) 925 313 sh (0) 1213 313 sh (0) 1880 313 sh (99) 2168 313 sh (0) 2911 313 sh (99) 3199 313 sh (0) 849 889 sh (99) 1137 889 sh (1) 1956 889 sh (0) 2244 889 sh (0) 2911 889 sh (99) 3199 889 sh (0) 849 1465 sh (99) 1137 1465 sh (0) 1880 1465 sh (99) 2168 1465 sh (1) 2987 1465 sh (0) 3275 1465 sh 384 /Times-Roman f1 (.) 1117 313 sh (.) 2072 313 sh (.) 3103 313 sh (.) 1041 889 sh (.) 2148 889 sh (.) 3103 889 sh (.) 1041 1465 sh (.) 2072 1465 sh (.) 3179 1465 sh end MTsave restore pse gR gS 0 0 552 730 rC 239 174 :M f0_12 sf -.052( \(a\))A 347 174 :M ( )S 383 174 :M ( \(b\))S 251 192 :M f2_12 sf 2.949 .295(Figure 2)J 77 216 :M f0_12 sf (In section 4, )S 140 216 :M (we )S 158 216 :M -.166(will )A 180 216 :M -.122(describe )A 223 216 :M .259 .026(how )J 248 216 :M -.167(to )A 261 216 :M -.209(efficiently )A 312 216 :M -.165(test )A 332 216 :M (when )S 362 216 :M (two )S 384 216 :M -.165(path )A 408 216 :M -.123(diagrams )A 455 216 :M -.111(without)A 59 234 :M -.133(correlated errors or directed cycles are covariance equivalent. We will )A 390 234 :M -.082(also )A 413 234 :M -.165(give )A 437 234 :M -.197(informative)A 59 252 :M -.07(necessary )A 111 252 :M -.1(conditions )A 166 252 :M (for )S 187 252 :M (two )S 212 252 :M -.165(path )A 239 252 :M -.123(diagrams )A 289 252 :M -.083(with )A 317 252 :M -.196(correlated )A 370 252 :M .604 .06(errors, )J 409 252 :M (cycles, )S 449 252 :M (or )S 466 252 :M -.331(latent)A 59 270 :M -.091(variables to be covariance equivalent over )A f2_12 sf -.178(O)A f0_12 sf -.092(. For related theorems see also Pearl \(1997\).)A 95 300 :M f4_12 sf (2)S 102 300 :M (.)S 107 300 :M (2)S 114 300 :M (.)S 119 300 :M 9 .9( )J 131 300 :M 3.119 .312(Features Common to a Covariance Equivalence Class)J 77 321 :M f0_12 sf -.123(A second )A 125 321 :M -.184(important )A 174 321 :M -.083(question )A 218 321 :M -.249(that )A 239 321 :M -.052(arises )A 270 321 :M -.083(with )A 295 321 :M -.139(respect )A 332 321 :M -.167(to )A 345 321 :M -.196(covariance )A 399 321 :M -.209(equivalence )A 458 321 :M -.052(classes)A 59 339 :M (is )S 71 339 :M -.092(whether )A 113 339 :M -.334(it )A 123 339 :M (is )S 135 339 :M -.041(possible )A 178 339 :M -.167(to )A 192 339 :M -.234(extract )A 228 339 :M -.22(the )A 247 339 :M -.121(features )A 289 339 :M -.249(that )A 311 339 :M -.22(the )A 330 339 :M -.109(set )A 348 339 :M (of )S 363 339 :M -.196(covariance )A 418 339 :M -.198(equivalent )A 471 339 :M -.22(path)A 59 357 :M -.123(diagrams )A 106 357 :M -.163(have )A 133 357 :M -.167(in )A 147 357 :M (common. )S 197 357 :M .258 .026(For )J 219 357 :M -.081(example, )A 267 357 :M -.129(every )A 298 357 :M -.165(path )A 323 357 :M -.188(diagram )A 366 357 :M -.167(in )A 380 357 :M -.054(Figure )A 416 357 :M (1 )S 427 357 :M (has )S 448 357 :M -.22(the )A 467 357 :M -.218(same)A 59 375 :M -.107(adjacencies, but the path diagrams do not have any edge )A 327 375 :M -.083(with )A 352 375 :M -.22(the )A 370 375 :M -.163(same )A 398 375 :M -.18(orientation )A 452 375 :M -.167(in )A 465 375 :M -.161(every)A 59 393 :M -.05(member of the equivalence class \(e.g. both W )A f1_12 sf -.121A f0_12 sf -.053( X, and W )A f1_12 sf -.121A f0_12 sf -.051( X occur in path )A 435 393 :M -.123(diagrams )A 482 393 :M -.334(in)A 59 411 :M .359 .036(Figure 1\).)J 77 429 :M -.039(However, there are other sets of )A 233 429 :M -.196(covariance )A 287 429 :M -.198(equivalent )A 339 429 :M -.165(path )A 363 429 :M -.123(diagrams )A 410 429 :M -.167(in )A 423 429 :M -.065(which )A 456 429 :M -.326(a )A 465 429 :M -.165(given)A 59 447 :M -.163(edge )A 85 447 :M -.053(always )A 122 447 :M -.052(occurs )A 157 447 :M -.083(with )A 182 447 :M -.22(the )A 200 447 :M -.163(same )A 228 447 :M -.18(orientation )A 282 447 :M -.167(in )A 296 447 :M -.129(every )A 327 447 :M -.219(member )A 370 447 :M (of )S 385 447 :M -.22(the )A 404 447 :M -.209(equivalence )A 464 447 :M .136(class.)A 59 465 :M -.087(For example, Figure 3 shows another simple covariance equivalence class of path )A 448 465 :M -.14(diagrams)A 59 483 :M -.098(in which the orientation X )A f1_12 sf -.243A f0_12 sf -.102( Z occurs in every member of the equivalence class.)A 251 555 :M f2_12 sf 2.949 .295(Figure 3)J 77 579 :M f0_12 sf -.083(This )A 104 579 :M (is )S 118 579 :M -.179(informative )A 178 579 :M -.139(because )A 221 579 :M -.163(even )A 249 579 :M -.056(though )A 288 579 :M -.22(the )A 308 579 :M -.247(data )A 334 579 :M (does )S 363 579 :M -.111(not )A 385 579 :M -.165(help )A 412 579 :M -.053(choose )A 452 579 :M -.163(between)A 59 597 :M -.081(members of the equivalence class, insofar as )A 273 597 :M -.22(the )A 291 597 :M -.247(data )A 314 597 :M (is )S 326 597 :M -.205(evidence )A 371 597 :M (for )S 389 597 :M -.22(the )A 407 597 :M -.121(disjunction )A 463 597 :M (of )S 477 597 :M -.33(the)A 59 615 :M -.096(members in the equivalence class, it is evidence for the orientation X )A f1_12 sf -.243A f0_12 sf -.137( Z.)A 77 633 :M (In )S 92 633 :M -.141(section )A 130 633 :M (4 )S 141 633 :M (we )S 160 633 :M -.166(will )A 183 633 :M .479 .048(show )J 214 633 :M .259 .026(how )J 240 633 :M -.167(to )A 254 633 :M -.234(extract )A 291 633 :M -.331(all )A 308 633 :M (of )S 324 633 :M -.22(the )A 344 633 :M -.121(features )A 387 633 :M -.166(common )A 434 633 :M -.167(to )A 449 633 :M -.326(a )A 460 633 :M -.199(simple)A 59 651 :M -.196(covariance )A 113 651 :M -.209(equivalence )A 173 651 :M -.064(class )A 201 651 :M (of )S 216 651 :M -.165(path )A 241 651 :M (diagrams, )S 293 651 :M -.109(and )A 315 651 :M -.14(briefly )A 351 651 :M -.248(indicate )A 392 651 :M -.249(that )A 414 651 :M -.334(it )A 425 651 :M (is )S 438 651 :M -.041(possible )A 482 651 :M -.334(to)A 359 109 18 19 rC 359 124 :M (X)S gR gS 225 112 18 19 rC 225 127 :M f0_12 sf (X)S gR gS 248 95 18 19 rC 248 110 :M f0_12 sf .115(T)A f0_7 sf 0 3 rm (1)S 0 -3 rm gR gS 404 108 18 19 rC 404 123 :M f0_12 sf (Y)S gR gS 212 136 18 19 rC 212 151 :M f0_12 sf .115(T)A f0_7 sf 0 3 rm (3)S 0 -3 rm gR gS 288 135 18 19 rC 288 150 :M f0_12 sf .115(T)A f0_7 sf 0 3 rm (2)S 0 -3 rm gR gS 277 114 18 19 rC 277 129 :M f0_12 sf (Y)S gR gS 253 148 18 19 rC 253 163 :M f0_12 sf (Z)S gR gS 403 143 18 19 rC 403 158 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 377 122.75 -.75 .75 394.75 122 .75 377 122 @a np 392 118 :M 392 125 :L 400 122 :L 392 118 :L .75 lw eofill -.75 -.75 392.75 125.75 .75 .75 392 118 @b -.75 -.75 392.75 125.75 .75 .75 400 122 @b 392 118.75 -.75 .75 400.75 122 .75 392 118 @a np 379 125 :M 379 118 :L 371 122 :L 379 125 :L eofill -.75 -.75 379.75 125.75 .75 .75 379 118 @b -.75 -.75 371.75 122.75 .75 .75 379 118 @b 371 122.75 -.75 .75 379.75 125 .75 371 122 @a -.75 -.75 406.75 142.75 .75 .75 406 131 @b np 410 140 :M 402 140 :L 406 148 :L 410 140 :L eofill 402 140.75 -.75 .75 410.75 140 .75 402 140 @a 402 140.75 -.75 .75 406.75 148 .75 402 140 @a -.75 -.75 406.75 148.75 .75 .75 410 140 @b np 402 133 :M 410 133 :L 406 125 :L 402 133 :L eofill 402 133.75 -.75 .75 410.75 133 .75 402 133 @a 406 125.75 -.75 .75 410.75 133 .75 406 125 @a -.75 -.75 402.75 133.75 .75 .75 406 125 @b 374 129.75 -.75 .75 393.75 144 .75 374 129 @a np 394 140 :M 390 146 :L 398 148 :L 394 140 :L eofill -.75 -.75 390.75 146.75 .75 .75 394 140 @b 390 146.75 -.75 .75 398.75 148 .75 390 146 @a 394 140.75 -.75 .75 398.75 148 .75 394 140 @a np 373 133 :M 378 127 :L 369 125 :L 373 133 :L eofill -.75 -.75 373.75 133.75 .75 .75 378 127 @b 369 125.75 -.75 .75 378.75 127 .75 369 125 @a 369 125.75 -.75 .75 373.75 133 .75 369 125 @a 223 491 18 19 rC 223 506 :M f0_12 sf (X)S gR gS 174 492 18 19 rC 174 507 :M f0_12 sf (W)S gR gS 180 530 18 19 rC 180 545 :M f0_12 sf (Y)S gR gS 223 531 18 19 rC 223 546 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 191 504.75 -.75 .75 214.75 504 .75 191 504 @a np 212 502 :M 212 506 :L 220 504 :L 212 502 :L .75 lw eofill -.75 -.75 212.75 506.75 .75 .75 212 502 @b -.75 -.75 212.75 506.75 .75 .75 220 504 @b 212 502.75 -.75 .75 220.75 504 .75 212 502 @a 194 540.75 -.75 .75 212.75 540 .75 194 540 @a np 210 538 :M 210 542 :L 218 540 :L 210 538 :L eofill -.75 -.75 210.75 542.75 .75 .75 210 538 @b -.75 -.75 210.75 542.75 .75 .75 218 540 @b 210 538.75 -.75 .75 218.75 540 .75 210 538 @a -.75 -.75 227.75 530.75 .75 .75 227 512 @b np 229 528 :M 225 528 :L 227 536 :L 229 528 :L eofill 225 528.75 -.75 .75 229.75 528 .75 225 528 @a 225 528.75 -.75 .75 227.75 536 .75 225 528 @a -.75 -.75 227.75 536.75 .75 .75 229 528 @b 359 491 18 19 rC 359 506 :M f0_12 sf (X)S gR gS 310 492 18 19 rC 310 507 :M f0_12 sf (W)S gR gS 316 530 18 19 rC 316 545 :M f0_12 sf (Y)S gR gS 359 531 18 19 rC 359 546 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 333 504.75 -.75 .75 356.75 504 .75 333 504 @a np 335 506 :M 335 502 :L 327 504 :L 335 506 :L .75 lw eofill -.75 -.75 335.75 506.75 .75 .75 335 502 @b -.75 -.75 327.75 504.75 .75 .75 335 502 @b 327 504.75 -.75 .75 335.75 506 .75 327 504 @a 330 540.75 -.75 .75 348.75 540 .75 330 540 @a np 346 538 :M 346 542 :L 354 540 :L 346 538 :L eofill -.75 -.75 346.75 542.75 .75 .75 346 538 @b -.75 -.75 346.75 542.75 .75 .75 354 540 @b 346 538.75 -.75 .75 354.75 540 .75 346 538 @a -.75 -.75 363.75 528.75 .75 .75 363 510 @b np 365 526 :M 361 526 :L 363 534 :L 365 526 :L eofill 361 526.75 -.75 .75 365.75 526 .75 361 526 @a 361 526.75 -.75 .75 363.75 534 .75 361 526 @a -.75 -.75 363.75 534.75 .75 .75 365 526 @b -.75 -.75 239.75 118.75 .75 .75 248 111 @b np 242 120 :M 238 114 :L 236 120 :L 242 120 :L eofill 238 114.75 -.75 .75 242.75 120 .75 238 114 @a -.75 -.75 236.75 120.75 .75 .75 238 114 @b 236 120.75 -.75 .75 242.75 120 .75 236 120 @a 263 111.75 -.75 .75 272.75 119 .75 263 111 @a np 273 115 :M 269 121 :L 275 121 :L 273 115 :L eofill -.75 -.75 269.75 121.75 .75 .75 273 115 @b 269 121.75 -.75 .75 275.75 121 .75 269 121 @a 273 115.75 -.75 .75 275.75 121 .75 273 115 @a -.75 -.75 218.75 141.75 .75 .75 225 132 @b np 221 131 :M 227 135 :L 227 129 :L 221 131 :L eofill 221 131.75 -.75 .75 227.75 135 .75 221 131 @a -.75 -.75 227.75 135.75 .75 .75 227 129 @b -.75 -.75 221.75 131.75 .75 .75 227 129 @b 285 132.75 -.75 .75 292.75 140 .75 285 132 @a np 284 135 :M 289 131 :L 283 129 :L 284 135 :L eofill -.75 -.75 284.75 135.75 .75 .75 289 131 @b 283 129.75 -.75 .75 289.75 131 .75 283 129 @a 283 129.75 -.75 .75 284.75 135 .75 283 129 @a 228 153.75 -.75 .75 244.75 158 .75 228 153 @a np 244 154 :M 242 161 :L 248 159 :L 244 154 :L eofill -.75 -.75 242.75 161.75 .75 .75 244 154 @b -.75 -.75 242.75 161.75 .75 .75 248 159 @b 244 154.75 -.75 .75 248.75 159 .75 244 154 @a -.75 -.75 269.75 158.75 .75 .75 289 151 @b np 271 161 :M 269 154 :L 265 159 :L 271 161 :L eofill 269 154.75 -.75 .75 271.75 161 .75 269 154 @a -.75 -.75 265.75 159.75 .75 .75 269 154 @b 265 159.75 -.75 .75 271.75 161 .75 265 159 @a endp %%Page: 6 6 %%BeginPageSetup initializepage (peter; page: 6 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (6)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.234(extract )A 95 56 :M -.082(some )A 125 56 :M -.121(features )A 167 56 :M -.166(common )A 213 56 :M -.167(to )A 227 56 :M -.326(a )A 237 56 :M -.196(covariance )A 292 56 :M -.209(equivalence )A 352 56 :M -.064(class )A 380 56 :M (of )S 395 56 :M -.165(path )A 421 56 :M -.123(diagrams )A 470 56 :M -.11(with)A 59 74 :M -.062(correlated errors, cycles, or latent variables.)A 95 104 :M f4_12 sf (2)S 102 104 :M (.)S 107 104 :M (3)S 114 104 :M (.)S 119 104 :M 9 .9( )J 131 104 :M 3.978 .398(Regression Coefficients and Structural Equation Coefficients)J 77 125 :M f0_12 sf -.104(It is common knowledge among practising social scientists )A 358 125 :M -.249(that )A 379 125 :M (for )S 397 125 :M -.22(the )A 415 125 :M -.208(coefficient )A 468 125 :M (of )S 482 125 :M (X)S 59 143 :M -.098(in the regression of Y upon X to be interpretable as the effect of X on Y )A 401 143 :M -.196(there )A 428 143 :M (should )S 464 143 :M -.163(be )A 479 143 :M (no)S 59 161 :M -.088("confounding" variable Z which is a cause of both X and Y:)A 251 204 10 12 rC 252 213 :M (X)S gR gS 308 186 9 12 rC 309 195 :M f0_12 sf (Z)S gR gS 278 243 10 12 rC 279 252 :M f0_12 sf (Y)S gR gS 250 179 68 77 rC np 278 243 :M 269 235 :L 271 233 :L 274 231 :L 278 243 :L eofill 260 217 -1 1 272 233 1 260 216 @a np 287 243 :M 291 231 :L 293 233 :L 296 234 :L 287 243 :L eofill -1 -1 294 234 1 1 314 198 @b np 260 207 :M 270 200 :L 271 203 :L 272 205 :L 260 207 :L eofill -1 -1 272 204 1 1 305 189 @b 276 180 9 17 rC 277 193 :M f1_12 sf (a)S gR gS 255 223 9 17 rC 256 236 :M f1_12 sf (b)S gR gS 304 214 9 17 rC 305 227 :M f1_12 sf (g)S gR gS 0 0 552 730 rC 260 271 :M f2_12 sf 2.949 .295(Figure 4)J 59 307 :M f0_12 sf -.11(Simple calculations confirm this conclusion \(using the notation in Figure 4\))A 414 304 :M f0_8 sf (4)S 418 307 :M f0_12 sf (:)S 203 320 147 14 rC 350 334 :M psb currentpoint pse 203 320 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4704 div 448 3 -1 roll exch div scale currentpoint translate 64 36 translate -12 284 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 629 284 moveto 384 /Times-Roman f1 (\() show 773 284 moveto 384 /Times-Roman f1 (X) show 1052 284 moveto 384 /Times-Roman f1 (,) show 1185 284 moveto 384 /Times-Roman f1 (Y) show 1475 284 moveto 384 /Times-Roman f1 (\)) show 1692 284 moveto 384 /Symbol f1 (=) show 1987 284 moveto 384 /Symbol f1 (b) show 2197 284 moveto 384 /Times-Roman f1 (V) show 2480 284 moveto 384 /Times-Roman f1 (\() show 2624 284 moveto 384 /Times-Roman f1 (X) show 2916 284 moveto 384 /Times-Roman f1 (\)) show 3118 284 moveto 384 /Symbol f1 (+) show 3405 284 moveto 384 /Symbol f1 (ag) show 3804 284 moveto 384 /Times-Roman f1 (V) show 4087 284 moveto 384 /Times-Roman f1 (\() show 4233 284 moveto 384 /Times-Roman f1 (Z) show 4483 284 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 59 355 :M f0_12 sf -.16(Hence)A 195 376 :M ( )S 198 358 170 31 rC 368 389 :M psb currentpoint pse 198 358 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5440 div 992 3 -1 roll exch div scale currentpoint translate 64 53 translate 20 284 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 661 284 moveto 384 /Times-Roman f1 (\() show 805 284 moveto 384 /Times-Roman f1 (X) show 1084 284 moveto 384 /Times-Roman f1 (,) show 1217 284 moveto 384 /Times-Roman f1 (Y) show 1507 284 moveto 384 /Times-Roman f1 (\)) show 407 816 moveto 384 /Times-Roman f1 (V) show 690 816 moveto 384 /Times-Roman f1 (\() show 834 816 moveto 384 /Times-Roman f1 (X) show 1126 816 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 424 moveto 1648 0 rlineto stroke 1756 523 moveto 384 /Symbol f1 (=) show 2083 284 moveto 384 /Symbol f1 (b) show 2293 284 moveto 384 /Times-Roman f1 (V) show 2576 284 moveto 384 /Times-Roman f1 (\() show 2720 284 moveto 384 /Times-Roman f1 (X) show 3012 284 moveto 384 /Times-Roman f1 (\)) show 3214 284 moveto 384 /Symbol f1 (+) show 3501 284 moveto 384 /Symbol f1 (ag) show 3900 284 moveto 384 /Times-Roman f1 (V) show 4183 284 moveto 384 /Times-Roman f1 (\() show 4329 284 moveto 384 /Times-Roman f1 (Z) show 4579 284 moveto 384 /Times-Roman f1 (\)) show 2980 816 moveto 384 /Times-Roman f1 (V) show 3263 816 moveto 384 /Times-Roman f1 (\() show 3407 816 moveto 384 /Times-Roman f1 (X) show 3699 816 moveto 384 /Times-Roman f1 (\)) show 2074 424 moveto 2646 0 rlineto stroke 4827 523 moveto 384 /Symbol f1 (\271) show 5123 523 moveto 384 /Symbol f1 (b) show end pse gR gS 0 0 552 730 rC 368 376 :M f0_12 sf (.)S 59 404 :M -.102(Thus the coefficient from the regression of Y on X alone will be a consistent )A 423 404 :M -.183(estimator )A 470 404 :M -.111(only)A 59 421 :M -.164(if )A 70 421 :M -.219(either )A 100 421 :M f1_12 sf .308(a)A f0_12 sf .122 .012( )J 112 421 :M (or )S 126 421 :M f1_12 sf (g)S f0_12 sf ( )S 135 421 :M (is )S 147 421 :M -.197(equal )A 176 421 :M -.167(to )A 189 421 :M .236 .024(zero. )J 217 421 :M .379 .038(Further, )J 260 421 :M -.045(observe )A 301 421 :M -.249(that )A 322 421 :M -.22(the )A 341 421 :M -.082(bias )A 365 421 :M -.247(term )A 391 421 :M f1_12 sf -.103(ag)A f0_12 sf -.102(V\(Z\)/V\(X\) )A 460 421 :M -.167(my )A 480 421 :M -.326(be)A 59 440 :M -.095(either positive or negative, and of arbitrary magnitude.)A 77 457 :M .571 .057(However, )J 128 447 97 14 rC 225 461 :M psb currentpoint pse 128 447 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3104 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate -12 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 629 261 moveto 384 /Times-Roman f1 (\() show 773 261 moveto 384 /Times-Roman f1 (X) show 1052 261 moveto 384 /Times-Roman f1 (,) show 1191 261 moveto 384 /Times-Roman f1 (Z) show 1441 261 moveto 384 /Times-Roman f1 (\)) show 1658 261 moveto 384 /Symbol f1 (=) show 1961 261 moveto 384 /Symbol f1 (a) show 2203 261 moveto 384 /Times-Roman f1 (V) show 2486 261 moveto 384 /Times-Roman f1 (\() show 2632 261 moveto 384 /Times-Roman f1 (Z) show 2882 261 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 225 457 :M f0_12 sf -.081( and )A 248 447 130 14 rC 378 461 :M psb currentpoint pse 248 447 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4160 div 448 3 -1 roll exch div scale currentpoint translate 64 36 translate -12 284 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 629 284 moveto 384 /Times-Roman f1 (\() show 769 284 moveto 384 /Times-Roman f1 (Y) show 1046 284 moveto 384 /Times-Roman f1 (,) show 1185 284 moveto 384 /Times-Roman f1 (Z) show 1435 284 moveto 384 /Times-Roman f1 (\)) show 1652 284 moveto 384 /Symbol f1 (=) show 1952 284 moveto 384 /Times-Roman f1 (\() show 2085 284 moveto 384 /Symbol f1 (ab) show 2617 284 moveto 384 /Symbol f1 (+) show 2918 284 moveto 384 /Symbol f1 (g) show 3124 284 moveto 384 /Times-Roman f1 (\)) show 3258 284 moveto 384 /Times-Roman f1 (V) show 3541 284 moveto 384 /Times-Roman f1 (\() show 3687 284 moveto 384 /Times-Roman f1 (Z) show 3937 284 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 378 457 :M f0_12 sf -.03(, and hence)A 159 479 249 68 rC 408 547 :M psb currentpoint pse 159 479 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7968 div 2176 3 -1 roll exch div scale currentpoint translate 64 -570 translate -12 1102 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 629 1102 moveto 384 /Times-Roman f1 (\() show 773 1102 moveto 384 /Times-Roman f1 (X) show 1052 1102 moveto 384 /Times-Roman f1 (,) show 1185 1102 moveto 384 /Times-Roman f1 (Y) show 1462 1102 moveto 384 /Times-Roman f1 ( ) show 1564 1102 moveto 384 /Times-Roman f1 (|) show 1678 1102 moveto 384 /Times-Roman f1 ( ) show 1774 1102 moveto 384 /Times-Roman f1 (Z) show 2024 1102 moveto 384 /Times-Roman f1 (\)) show 2240 1102 moveto 384 /Symbol f1 (\272) show 2546 1102 moveto 384 /Times-Roman f1 (Cov) show 3187 1102 moveto 384 /Times-Roman f1 (\() show 3331 1102 moveto 384 /Times-Roman f1 (X) show 3610 1102 moveto 384 /Times-Roman f1 (,) show 3743 1102 moveto 384 /Times-Roman f1 (Y) show 4033 1102 moveto 384 /Times-Roman f1 (\)) show 4234 1102 moveto 384 /Symbol f1 (-) show 4555 863 moveto 384 /Times-Roman f1 (Cov) show 5196 863 moveto 384 /Times-Roman f1 (\() show 5340 863 moveto 384 /Times-Roman f1 (X) show 5619 863 moveto 384 /Times-Roman f1 (,) show 5758 863 moveto 384 /Times-Roman f1 (Z) show 6008 863 moveto 384 /Times-Roman f1 (\)) show 6137 863 moveto 384 /Times-Roman f1 (Cov) show 6778 863 moveto 384 /Times-Roman f1 (\() show 6918 863 moveto 384 /Times-Roman f1 (Y) show 7195 863 moveto 384 /Times-Roman f1 (,) show 7334 863 moveto 384 /Times-Roman f1 (Z) show 7584 863 moveto 384 /Times-Roman f1 (\)) show 5733 1395 moveto 384 /Times-Roman f1 (V) show 6016 1395 moveto 384 /Times-Roman f1 (\() show 6162 1395 moveto 384 /Times-Roman f1 (Z) show 6412 1395 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 4535 1003 moveto 3190 0 rlineto stroke 2240 1978 moveto 384 /Symbol f1 (=) show 2535 1978 moveto 384 /Symbol f1 (b) show 2745 1978 moveto 384 /Times-Roman f1 (V) show 3028 1978 moveto 384 /Times-Roman f1 (\() show 3172 1978 moveto 384 /Times-Roman f1 (X) show 3464 1978 moveto 384 /Times-Roman f1 (\)) show 3666 1978 moveto 384 /Symbol f1 (+) show 3953 1978 moveto 384 /Symbol f1 (ag) show 4352 1978 moveto 384 /Times-Roman f1 (V) show 4635 1978 moveto 384 /Times-Roman f1 (\() show 4781 1978 moveto 384 /Times-Roman f1 (Z) show 5031 1978 moveto 384 /Times-Roman f1 (\)) show 5232 1978 moveto 384 /Symbol f1 (-) show 5518 1978 moveto 384 /Symbol f1 (a) show 5760 1978 moveto 384 /Times-Roman f1 (V) show 6043 1978 moveto 384 /Times-Roman f1 (\() show 6189 1978 moveto 384 /Times-Roman f1 (Z) show 6439 1978 moveto 384 /Times-Roman f1 (\)) show 6562 1978 moveto (\() show 6695 1978 moveto 384 /Symbol f1 (ab) show 7227 1978 moveto 384 /Symbol f1 (+) show 7528 1978 moveto 384 /Symbol f1 (g) show 7734 1978 moveto 384 /Times-Roman f1 (\)) show 2240 2619 moveto 384 /Symbol f1 (=) show 2535 2619 moveto 384 /Symbol f1 (b) show 2746 2619 moveto 384 /Times-Roman f1 (\() show 2887 2619 moveto 384 /Times-Roman f1 (V) show 3170 2619 moveto 384 /Times-Roman f1 (\() show 3314 2619 moveto 384 /Times-Roman f1 (X) show 3606 2619 moveto 384 /Times-Roman f1 (\)) show 3807 2619 moveto 384 /Symbol f1 (-) show 4093 2619 moveto 384 /Symbol f1 (a) show 4357 2448 moveto 224 /Times-Roman f1 (2) show 4488 2619 moveto 384 /Times-Roman f1 (V) show 4771 2619 moveto 384 /Times-Roman f1 (\() show 4917 2619 moveto 384 /Times-Roman f1 (Z) show 5167 2619 moveto 384 /Times-Roman f1 (\)) show 5297 2619 moveto (\)) show end pse gR gS 0 0 552 730 rC 59 562 :M f0_12 sf -.163(and)A 161 565 241 32 rC 402 597 :M psb currentpoint pse 161 565 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7712 div 1024 3 -1 roll exch div scale currentpoint translate -701 -2890 translate 758 3498 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (V) show 1041 3498 moveto 384 /Times-Roman f1 (\() show 1185 3498 moveto 384 /Times-Roman f1 (X) show 1462 3498 moveto 384 /Times-Roman f1 ( ) show 1564 3498 moveto 384 /Times-Roman f1 (|) show 1678 3498 moveto 384 /Times-Roman f1 ( ) show 1774 3498 moveto 384 /Times-Roman f1 (Z) show 2024 3498 moveto 384 /Times-Roman f1 (\)) show 2240 3498 moveto 384 /Symbol f1 (\272) show 2551 3498 moveto 384 /Times-Roman f1 (V) show 2834 3498 moveto 384 /Times-Roman f1 (\() show 2978 3498 moveto 384 /Times-Roman f1 (X) show 3270 3498 moveto 384 /Times-Roman f1 (\)) show 3471 3498 moveto 384 /Symbol f1 (-) show 3792 3259 moveto 384 /Times-Roman f1 (Cov) show 4433 3259 moveto 384 /Times-Roman f1 (\() show 4577 3259 moveto 384 /Times-Roman f1 (X) show 4856 3259 moveto 384 /Times-Roman f1 (,) show 4995 3259 moveto 384 /Times-Roman f1 (Z) show 5245 3259 moveto 384 /Times-Roman f1 (\)) show 5380 3088 moveto 224 /Times-Roman f1 (2) show 4264 3791 moveto 384 /Times-Roman f1 (V) show 4547 3791 moveto 384 /Times-Roman f1 (\() show 4693 3791 moveto 384 /Times-Roman f1 (Z) show 4943 3791 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3772 3399 moveto 1778 0 rlineto stroke 5658 3498 moveto 384 /Symbol f1 (=) show 5969 3498 moveto 384 /Times-Roman f1 (V) show 6252 3498 moveto 384 /Times-Roman f1 (\() show 6396 3498 moveto 384 /Times-Roman f1 (X) show 6688 3498 moveto 384 /Times-Roman f1 (\)) show 6889 3498 moveto 384 /Symbol f1 (-) show 7175 3498 moveto 384 /Symbol f1 (a) show 7439 3327 moveto 224 /Times-Roman f1 (2) show 7570 3498 moveto 384 /Times-Roman f1 (V) show 7853 3498 moveto 384 /Times-Roman f1 (\() show 7999 3498 moveto 384 /Times-Roman f1 (Z) show 8249 3498 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 402 584 :M f0_12 sf (,)S 59 629 :M -.082(so the coefficient of X in the regression of Y on X and Z is a consistent estimator of )A f1_12 sf -.116(b)A f0_12 sf -.097( since)A 59 647 :M .052 .005(Cov\(X,Y|Z\)/V\(X|Z\) = )J f1_12 sf (b)S f0_12 sf (.)S 59 674 :M ( )S 59 671.48 -.48 .48 203.48 671 .48 59 671 @a 59 684 :M f0_8 sf (4)S 63 687 :M f0_10 sf ( Section 6 contains a simple rule for calculating covariances from a path diagram.)S endp %%Page: 7 7 %%BeginPageSetup initializepage (peter; page: 7 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (7)S gR gS 0 0 552 730 rC 77 56 :M f0_12 sf -.108(The danger presented by failing to include confounding variables is )A 398 56 :M -.164(well )A 422 56 :M -.032(understood )A 479 56 :M (by)S 59 74 :M -.077(social scientists. Indeed, it is often used as the justification )A 339 74 :M (for )S 357 74 :M -.089(considering )A 416 74 :M -.326(a )A 425 74 :M -.083(long )A 450 74 :M -.14(\322laundry)A 59 92 :M -.089(list\323 of \322potential confounders\323 for inclusion in a given regression equation.)A 77 110 :M -.247(What )A 108 110 :M (is )S 122 110 :M -.045(perhaps )A 165 110 :M (less )S 189 110 :M -.164(well )A 215 110 :M -.032(understood )A 274 110 :M (is )S 288 110 :M -.249(that )A 311 110 :M -.148(including )A 361 110 :M -.326(a )A 372 110 :M -.205(variable )A 415 110 :M -.065(which )A 450 110 :M (is )S 464 110 :M -.111(not )A 486 110 :M (a)S 59 128 :M -.116(confounder can also lead to biased estimates of the )A 301 128 :M -.131(structural )A 349 128 :M -.108(coefficient. )A 406 128 :M -.326(We )A 426 128 :M .259 .026(now )J 451 128 :M -.092(consider)A 59 146 :M -.107(a number of simple cases demonstrating this:)A 225 183 10 12 rC 226 192 :M (X)S gR gS 279 183 10 12 rC 280 192 :M f0_12 sf (Y)S gR gS 333 183 9 12 rC 334 192 :M f0_12 sf (Z)S gR gS 224 167 119 29 rC np 279 188 :M 267 191 :L 267 188 :L 267 185 :L 279 188 :L eofill 234 189 -1 1 268 188 1 234 188 @a np 332 188 :M 320 191 :L 320 188 :L 320 185 :L 332 188 :L eofill 287 189 -1 1 321 188 1 287 188 @a 246 168 8 17 rC 247 181 :M f1_12 sf (b)S gR gS 298 168 9 17 rC 299 181 :M f1_12 sf (h)S gR gS 0 0 552 730 rC 260 211 :M f2_12 sf 2.949 .295(Figure 5)J 77 228 :M f0_12 sf -.129(In the SEM with the path diagram depicted in Figure )A 328 228 :M .833 .083(5, )J 342 228 :M .38 .038(Cov\(X,Y\) )J 395 228 :M .211 .021(= )J 406 228 :M f1_12 sf .191(b)A f0_12 sf .607 .061(V\(X\), )J 446 228 :M -.196(hence )A 477 228 :M -.33(the)A 59 246 :M -.208(coefficient )A 112 246 :M (of )S 126 246 :M .306 .031(X )J 139 246 :M -.167(in )A 152 246 :M -.22(the )A 170 246 :M -.031(regression )A 223 246 :M (of )S 237 246 :M -.663(Y )A 249 246 :M (upon )S 277 246 :M .306 .031(X )J 290 246 :M (is )S 302 246 :M -.326(a )A 312 246 :M -.099(consistent )A 364 246 :M -.183(estimator )A 412 246 :M (of )S 427 246 :M f1_12 sf .74(b)A f0_12 sf .613 .061(. )J 443 246 :M .147(However,)A 59 264 :M .063 .006(Cov\(Y,Z\) = )J f1_12 sf (h)S f0_12 sf .06 .006(V\(Y\), and Cov\(X,Z\) = )J f1_12 sf (bh)S f0_12 sf .064 .006(V\(X\), so that)J 140 286 287 36 rC 427 322 :M psb currentpoint pse 140 286 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9184 div 1152 3 -1 roll exch div scale currentpoint translate -73 -4037 translate 157 4470 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 798 4470 moveto 384 /Times-Roman f1 (\() show 942 4470 moveto 384 /Times-Roman f1 (X) show 1221 4470 moveto 384 /Times-Roman f1 (,) show 1354 4470 moveto 384 /Times-Roman f1 (Y) show 1629 4470 moveto 384 /Times-Roman f1 (|) show 1741 4470 moveto 384 /Times-Roman f1 (Z) show 1991 4470 moveto 384 /Times-Roman f1 (\)) show 543 5002 moveto 384 /Times-Roman f1 (V) show 826 5002 moveto 384 /Times-Roman f1 (\() show 970 5002 moveto 384 /Times-Roman f1 (X) show 1247 5002 moveto 384 /Times-Roman f1 (|) show 1359 5002 moveto 384 /Times-Roman f1 (Z) show 1609 5002 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 137 4610 moveto 1995 0 rlineto stroke 2240 4709 moveto 384 /Symbol f1 (=) show 2535 4709 moveto 384 /Symbol f1 (b) show 2993 4470 moveto 384 /Times-Roman f1 (V) show 3276 4470 moveto 384 /Times-Roman f1 (\() show 3422 4470 moveto 384 /Times-Roman f1 (Z) show 3672 4470 moveto 384 /Times-Roman f1 (\)) show 3873 4470 moveto 384 /Symbol f1 (-) show 4174 4470 moveto 384 /Symbol f1 (h) show 4402 4299 moveto 224 /Times-Roman f1 (2) show 4533 4470 moveto 384 /Times-Roman f1 (V) show 4816 4470 moveto 384 /Times-Roman f1 (\() show 4956 4470 moveto 384 /Times-Roman f1 (Y) show 5246 4470 moveto 384 /Times-Roman f1 (\)) show 2821 5002 moveto 384 /Times-Roman f1 (V) show 3104 5002 moveto 384 /Times-Roman f1 (\() show 3250 5002 moveto 384 /Times-Roman f1 (Z) show 3500 5002 moveto 384 /Times-Roman f1 (\)) show 3701 5002 moveto 384 /Symbol f1 (-) show 3979 5002 moveto 384 /Symbol f1 (b) show 4202 4831 moveto 224 /Times-Roman f1 (2) show 4340 5002 moveto 384 /Symbol f1 (h) show 4568 4831 moveto 224 /Times-Roman f1 (2) show 4699 5002 moveto 384 /Times-Roman f1 (V) show 4982 5002 moveto 384 /Times-Roman f1 (\() show 5126 5002 moveto 384 /Times-Roman f1 (X) show 5418 5002 moveto 384 /Times-Roman f1 (\)) show 2796 4610 moveto 2763 0 rlineto stroke 5667 4709 moveto 384 /Symbol f1 (=) show 5962 4709 moveto 384 /Symbol f1 (b) show 7244 4470 moveto 384 /Times-Roman f1 (V) show 7527 4470 moveto 384 /Times-Roman f1 (\() show 7667 4470 moveto 384 /Symbol f1 (e) show 7861 4566 moveto 224 /Times-Roman f1 (Z) show 8047 4470 moveto 384 /Times-Roman f1 (\)) show 6400 5002 moveto 384 /Times-Roman f1 (V) show 6683 5002 moveto 384 /Times-Roman f1 (\() show 6823 5002 moveto 384 /Symbol f1 (e) show 7017 5098 moveto 224 /Times-Roman f1 (Z) show 7203 5002 moveto 384 /Times-Roman f1 (\)) show 7405 5002 moveto 384 /Symbol f1 (+) show 7707 5002 moveto 384 /Symbol f1 (h) show 7935 4831 moveto 224 /Times-Roman f1 (2) show 8066 5002 moveto 384 /Times-Roman f1 (V) show 8349 5002 moveto 384 /Times-Roman f1 (\() show 8489 5002 moveto 384 /Symbol f1 (e) show 8680 5098 moveto 224 /Times-Roman f1 (Y) show 8890 5002 moveto 384 /Times-Roman f1 (\)) show 6375 4610 moveto 2656 0 rlineto stroke 6176 4445 moveto 384 /Symbol f1 (\346) show 6176 5018 moveto (\350) show 6176 4799 moveto (\347) show 9043 4445 moveto (\366) show 9043 5018 moveto (\370) show 9043 4799 moveto (\367) show end pse gR gS 0 0 552 730 rC 77 337 :M f0_12 sf -.128(Hence )A 112 337 :M -.22(the )A 131 337 :M -.208(coefficient )A 185 337 :M (of )S 200 337 :M .306 .031(X )J 215 337 :M -.167(in )A 230 337 :M -.22(the )A 250 337 :M -.031(regression )A 305 337 :M (of )S 321 337 :M -.663(Y )A 335 337 :M (on )S 353 337 :M .306 .031(X )J 368 337 :M -.109(and )A 391 337 :M -.33(Z )A 404 337 :M (is )S 418 337 :M -.163(an )A 435 337 :M -.12(inconsistent)A 59 354 :M -.183(estimator )A 106 354 :M (of )S 120 354 :M f1_12 sf .74(b)A f0_12 sf .613 .061(. )J 135 354 :M -.219(The )A 157 354 :M -.248(estimate )A 199 354 :M -.166(will )A 221 354 :M -.163(have )A 247 354 :M -.22(the )A 265 354 :M -.163(same )A 293 354 :M (sign )S 317 354 :M (as )S 331 354 :M f1_12 sf .74(b)A f0_12 sf .613 .061(, )J 346 354 :M -.111(but )A 365 354 :M -.166(will )A 387 354 :M -.163(have )A 413 354 :M -.188(smaller )A 452 354 :M -.141(absolute)A 59 372 :M -.044(magnitude. Note that Cov\(X,Y|Z\)/V\(X|Z\) = 0 if and only if )A f1_12 sf -.043(b )A f0_12 sf -.056(= 0.)A 77 391 :M -.164(It )A 88 391 :M -.2(might )A 119 391 :M -.163(be )A 134 391 :M -.206(objected )A 177 391 :M -.249(that )A 198 391 :M -.084(this )A 219 391 :M -.165(type )A 243 391 :M (of )S 258 391 :M -.061(error )A 286 391 :M (is )S 299 391 :M -.166(unlikely )A 342 391 :M -.167(to )A 356 391 :M -.129(arise )A 383 391 :M -.167(in )A 397 391 :M -.163(practise )A 438 391 :M -.131(since )A 467 391 :M -.163(often)A 59 409 :M -.099(information about time order would rule out Z as a potential unmeasured confounder. In the)A 59 427 :M -.122(next example this response is not available )A 263 427 :M -.131(since )A 291 427 :M -.33(Z )A 302 427 :M -.22(may )A 326 427 :M -.198(temporally )A 380 427 :M -.185(precede )A 420 427 :M -.083(both )A 445 427 :M .306 .031(X )J 458 427 :M -.109(and )A 479 427 :M .337(Y.)A 59 444 :M -.33(Let )A 78 444 :M f1_12 sf .219(e)A f0_7 sf 0 3 rm .21(X)A 0 -3 rm f0_12 sf .227 .023(, )J 96 444 :M f1_12 sf .219(e)A f0_7 sf 0 3 rm .21(Y)A 0 -3 rm f0_12 sf .227 .023(, )J 114 444 :M -.109(and )A 135 444 :M f1_12 sf -.228(e)A f0_7 sf 0 3 rm -.185(Z)A 0 -3 rm f0_12 sf ( )S 148 444 :M -.163(be )A 163 444 :M -.22(the )A 181 444 :M -.061(error )A 208 444 :M -.145(variables )A 254 444 :M -.167(in )A 267 444 :M -.054(Figure )A 302 444 :M .458 .046(6\(a\), )J 329 444 :M -.109(and )A 350 444 :M f1_12 sf .178(e)A f0_12 sf .135A f0_7 sf 0 3 rm .171(X)A 0 -3 rm f0_12 sf .184 .018(, )J 372 444 :M f1_12 sf .178(e)A f0_12 sf .135A f0_7 sf 0 3 rm .171(Y)A 0 -3 rm f0_12 sf .184 .018(, )J 394 444 :M -.109(and )A 415 444 :M f1_12 sf -.171(e)A f0_12 sf -.13A f0_7 sf 0 3 rm -.139(Z)A 0 -3 rm f0_12 sf ( )S 433 444 :M -.163(be )A 449 444 :M -.22(the )A 468 444 :M -.076(error)A 59 463 :M -.038(variables in Figure 6 \(b\).)A 312 527 10 12 rC 313 536 :M (X)S gR gS 365 577 10 12 rC 366 586 :M f0_12 sf (Y)S gR gS 363 527 9 12 rC 364 536 :M f0_12 sf (Z)S gR gS 162 484 244 126 rC np 368 574 :M 356 570 :L 358 567 :L 359 565 :L 368 574 :L eofill 316 542 -1 1 359 567 1 316 541 @a 327 555 8 17 rC 328 568 :M f1_12 sf (b)S gR gS 162 484 244 126 rC 180 270 36 12 344.5 521.5 @n -90 0 36 12 340.5 521.5 @n np 317 526 :M 326 517 :L 327 519 :L 329 522 :L 317 526 :L eofill -1 -1 326 522 1 1 327 519 @b np 367 525 :M 355 521 :L 357 519 :L 358 516 :L 367 525 :L eofill 357 520 -1 1 359 520 1 357 519 @a -90 0 26 36 382.5 549.5 @n 0 90 28 52 381.5 548.5 @n np 370 532 :M 382 529 :L 382 532 :L 382 535 :L 370 532 :L eofill 382 533 -1 1 386 532 1 382 532 @a np 370 573 :M 382 570 :L 382 573 :L 382 576 :L 370 573 :L eofill 382 574 -1 1 386 573 1 382 573 @a 338 495 8 17 rC 339 508 :M f1_12 sf (r)S gR gS 398 543 7 17 rC 399 556 :M f1_12 sf (t)S gR gS 199 597 15 12 rC 200 606 :M f0_12 sf -.156(\(a\))A gR gS 339 597 15 12 rC 340 606 :M f0_12 sf (\(b\))S gR gS 163 533 10 12 rC 164 542 :M f0_12 sf (X)S gR gS 222 580 10 12 rC 223 589 :M f0_12 sf (Y)S gR gS 214 533 9 12 rC 215 542 :M f0_12 sf (Z)S gR gS 162 484 244 126 rC np 219 580 :M 207 576 :L 209 573 :L 210 571 :L 219 580 :L eofill 167 548 -1 1 210 573 1 167 547 @a np 216 530 :M 206 522 :L 209 520 :L 211 518 :L 216 530 :L eofill 198 507 -1 1 210 520 1 198 506 @a 178 561 8 17 rC 179 574 :M f1_12 sf (b)S gR gS 162 484 244 126 rC np 169 531 :M 173 519 :L 176 521 :L 178 522 :L 169 531 :L eofill -1 -1 177 522 1 1 186 507 @b np 221 529 :M 226 517 :L 229 519 :L 231 521 :L 221 529 :L eofill -1 -1 230 520 1 1 238 509 @b np 230 577 :M 229 564 :L 232 565 :L 235 565 :L 230 577 :L eofill -1 -1 233 566 1 1 244 509 @b 1 G 31 21 193.5 496.5 @j 0 G 30 20 193.5 496.5 @f 1 G 31 21 246.5 498.5 @j 0 G 30 20 246.5 498.5 @f 186 487 15 17 rC 187 500 :M f1_12 sf (T)S gR gS 239 490 15 17 rC 240 503 :M f1_12 sf (T)S gR gS 207 502 8 17 rC 208 515 :M f1_12 sf (y)S gR gS 243 526 8 17 rC 244 539 :M f1_12 sf (f)S gR gS 226 502 7 17 rC 227 515 :M f1_12 sf (1)S gR gS 172 502 7 17 rC 173 515 :M f1_12 sf (1)S gR gS 192 493 6 12 rC 193 502 :M f0_10 sf (1)S gR gS 245 497 6 12 rC 246 506 :M f0_10 sf (2)S gR gS 0 0 552 730 rC 260 625 :M f2_12 sf 2.949 .295(Figure 6)J 77 642 :M f0_12 sf -.109(In the )A 108 642 :M -.165(path )A 132 642 :M -.188(diagram )A 174 642 :M -.206(depicted )A 217 642 :M -.167(in )A 230 642 :M -.054(Figure )A 265 642 :M -.078(6\(a\) )A 288 642 :M -.196(there )A 315 642 :M -.215(are )A 333 642 :M (two )S 355 642 :M -.097(unmeasured )A 416 642 :M -.028(confounders )A 479 642 :M -.196(T)A f0_10 sf 0 2 rm (1)S 0 -2 rm 59 660 :M f0_12 sf -.086(and T)A f0_10 sf 0 2 rm -.078(2)A 0 -2 rm f0_12 sf -.079(, which are uncorrelated with one another. Any SEM with )A 369 660 :M -.084(this )A 390 660 :M -.165(path )A 414 660 :M -.188(diagram )A 456 660 :M -.22(may )A 480 660 :M -.326(be)A 59 678 :M -.149(converted into a SEM with the path diagram depicted )A 312 678 :M -.167(in )A 325 678 :M -.054(Figure )A 360 678 :M .676 .068(6\(b\), )J 388 678 :M -.237(letting )A 421 678 :M f1_12 sf .285(r)A f0_12 sf .13 .013( )J 432 678 :M .211 .021(= )J 443 678 :M .145(Cov\(X,Z\))A endp %%Page: 8 8 %%BeginPageSetup initializepage (peter; page: 8 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (8)S gR gS 0 0 552 730 rC 59 55 :M f0_12 sf (= )S f1_12 sf (y)S f0_12 sf (V\(T)S f0_10 sf 0 2 rm (1)S 0 -2 rm f0_12 sf (\), )S f1_12 sf (t )S f0_12 sf (= )S 131 55 :M f1_12 sf (f)S f0_12 sf -.037(V\(T)A f0_10 sf 0 2 rm (2)S 0 -2 rm f0_12 sf -.028(\), )A 173 43 115 16 rC 288 59 :M psb currentpoint pse 173 43 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3680 div 512 3 -1 roll exch div scale currentpoint translate -1139 -5301 translate 1196 5685 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (V) show 1479 5685 moveto 384 /Times-Roman f1 (\() show 1619 5685 moveto 384 /Symbol f1 (e) show 1812 5781 moveto 224 /Times-Roman f1 (X) show 1800 5514 moveto 224 /Times-Roman f1 (*) show 2023 5685 moveto 384 ns (\)) show 2240 5685 moveto 384 /Symbol f1 (=) show 2551 5685 moveto 384 /Times-Roman f1 (V) show 2834 5685 moveto 384 /Times-Roman f1 (\() show 2974 5685 moveto 384 /Symbol f1 (e) show 3167 5781 moveto 224 /Times-Roman f1 (X) show 3378 5685 moveto 384 /Times-Roman f1 (\)) show 3580 5685 moveto 384 /Symbol f1 (+) show 3875 5685 moveto 384 /Times-Roman f1 (V) show 4158 5685 moveto 384 /Times-Roman f1 (\() show 4300 5685 moveto 384 /Times-Roman f1 (T) show 4536 5781 moveto 224 /Times-Roman f1 (1) show 4677 5685 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 288 55 :M f0_12 sf .909 .091(, )J 296 43 170 16 rC 466 59 :M psb currentpoint pse 296 43 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5440 div 512 3 -1 roll exch div scale currentpoint translate -1142 -5952 translate 1199 6336 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (V) show 1482 6336 moveto 384 /Times-Roman f1 (\() show 1622 6336 moveto 384 /Symbol f1 (e) show 1813 6432 moveto 224 /Times-Roman f1 (Y) show 1803 6165 moveto 224 /Times-Roman f1 (*) show 2023 6336 moveto 384 ns (\)) show 2240 6336 moveto 384 /Symbol f1 (=) show 2551 6336 moveto 384 /Times-Roman f1 (V) show 2834 6336 moveto 384 /Times-Roman f1 (\() show 2974 6336 moveto 384 /Symbol f1 (e) show 3165 6432 moveto 224 /Times-Roman f1 (Y) show 3375 6336 moveto 384 /Times-Roman f1 (\)) show 3577 6336 moveto 384 /Symbol f1 (+) show 3875 6336 moveto 384 /Symbol f1 (y) show 4170 6165 moveto 224 /Times-Roman f1 (2) show 4301 6336 moveto 384 /Times-Roman f1 (V) show 4584 6336 moveto 384 /Times-Roman f1 (\() show 4726 6336 moveto 384 /Times-Roman f1 (T) show 4962 6432 moveto 224 /Times-Roman f1 (1) show 5103 6336 moveto 384 /Times-Roman f1 (\)) show 5305 6336 moveto 384 /Symbol f1 (+) show 5600 6336 moveto 384 /Times-Roman f1 (V) show 5883 6336 moveto 384 /Times-Roman f1 (\() show 6025 6336 moveto 384 /Times-Roman f1 (T) show 6279 6432 moveto 224 /Times-Roman f1 (2) show 6438 6336 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 466 55 :M f0_12 sf .909 .091(, )J 474 55 :M -.163(and)A 59 61 126 16 rC 185 77 :M psb currentpoint pse 59 61 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4032 div 512 3 -1 roll exch div scale currentpoint translate -1163 -5952 translate 1220 6336 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (V) show 1503 6336 moveto 384 /Times-Roman f1 (\() show 1643 6336 moveto 384 /Symbol f1 (e) show 1837 6432 moveto 224 /Times-Roman f1 (Z) show 1824 6165 moveto 224 /Times-Roman f1 (*) show 2023 6336 moveto 384 ns (\)) show 2240 6336 moveto 384 /Symbol f1 (=) show 2551 6336 moveto 384 /Times-Roman f1 (V) show 2834 6336 moveto 384 /Times-Roman f1 (\() show 2974 6336 moveto 384 /Symbol f1 (e) show 3168 6432 moveto 224 /Times-Roman f1 (Z) show 3354 6336 moveto 384 /Times-Roman f1 (\)) show 3556 6336 moveto 384 /Symbol f1 (+) show 3848 6336 moveto 384 /Symbol f1 (f) show 4062 6165 moveto 224 /Times-Roman f1 (2) show 4193 6336 moveto 384 /Times-Roman f1 (V) show 4476 6336 moveto 384 /Times-Roman f1 (\() show 4618 6336 moveto 384 /Times-Roman f1 (T) show 4872 6432 moveto 224 /Times-Roman f1 (2) show 5031 6336 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 185 73 :M f0_12 sf (.)S 77 92 :M -.081(Note )A 105 92 :M .384 .038(however, )J 155 92 :M -.249(that )A 177 92 :M -.22(the )A 197 92 :M -.09(reverse )A 237 92 :M (is )S 251 92 :M -.111(not )A 272 92 :M -.167(in )A 287 92 :M -.187(general )A 327 92 :M -.197(true: )A 354 92 :M -.111(not )A 375 92 :M -.129(every )A 407 92 :M -.199(model )A 442 92 :M -.184(containing)A 59 109 :M -.196(correlated )A 109 109 :M (errors )S 141 109 :M .286 .029(\(X )J 158 109 :M f1_12 sf .451 .045J 175 109 :M f0_12 sf -.328(Y\) )A 191 109 :M -.217(can )A 211 109 :M -.163(be )A 226 109 :M -.145(converted )A 277 109 :M -.167(into )A 300 109 :M -.326(a )A 310 109 :M -.223(SEM )A 339 109 :M -.199(model )A 373 109 :M -.083(with )A 399 109 :M -.276(latent )A 429 109 :M -.145(variables )A 476 109 :M -.167(but)A 59 128 :M -.095(without )A 102 128 :M -.196(correlated )A 155 128 :M (errors )S 190 128 :M (by )S 209 128 :M -.12(introducing )A 271 128 :M -.326(a )A 284 128 :M -.276(latent )A 317 128 :M -.33(T )A 332 128 :M -.249(that )A 357 128 :M (is )S 373 128 :M -.326(a )A 386 128 :M -.163(parent )A 423 128 :M (of )S 441 128 :M .306 .031(X )J 458 128 :M -.109(and )A 483 128 :M (Y)S 59 145 :M <28>S 1 G 63 131 58 19 rC 13 14 92.5 142 @j 0 G 12 13 92.5 142 @f 64 132 56 17 rC 65 145 :M (X)S 73 145 :M f1_12 sf S 110 145 :M f0_12 sf (Y)S gR gS 0 0 552 730 rC 121 145 :M f0_12 sf .839 .084(\), )J 133 145 :M (as )S 147 145 :M -.142(pointed )A 186 145 :M -.111(out )A 205 145 :M -.167(in )A 218 145 :M -.141(section )A 255 145 :M 1.429 .143(2.1. )J 279 145 :M -.107(\(It )A 294 145 :M (is )S 306 145 :M -.044(however )A 351 145 :M -.053(always )A 388 145 :M -.041(possible )A 432 145 :M -.167(to )A 446 145 :M -.14(convert )A 486 145 :M (a)S 59 165 :M -.199(model )A 93 165 :M -.083(with )A 119 165 :M -.196(correlated )A 170 165 :M (errors )S 203 165 :M -.167(into )A 226 165 :M f5_12 sf .077(some)A f0_12 sf ( )S 256 165 :M -.276(latent )A 287 165 :M -.205(variable )A 330 165 :M -.199(model )A 365 165 :M -.095(without )A 407 165 :M -.196(correlated )A 459 165 :M .171(errors,)A 59 183 :M -.124(because every normal distribution is a linear )A 270 183 :M -.117(transformation )A 343 183 :M (of )S 357 183 :M -.326(a )A 366 183 :M -.109(set )A 383 183 :M (of )S 397 183 :M -.15(independent )A 458 183 :M -.197(normal)A 59 201 :M -.03(variables.\))A 77 219 :M -.102(Returning to the path diagram in Figure 6\(b\) note that the regression of Y on )A 441 219 :M .306 .031(X )J 454 219 :M -.11(yields )A 486 219 :M (a)S 59 236 :M -.002(consistent estimate of )A f1_12 sf (b)S f0_12 sf ( since Cov\(X,Y\) = )S f1_12 sf (b)S f0_12 sf (V\(X\). However,)S 151 258 266 100 rC 417 358 :M psb currentpoint pse 151 258 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8512 div 3200 3 -1 roll exch div scale currentpoint translate 64 49 translate 20 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 661 261 moveto 384 /Times-Roman f1 (\() show 805 261 moveto 384 /Times-Roman f1 (X) show 1084 261 moveto 384 /Times-Roman f1 (,) show 1217 261 moveto 384 /Times-Roman f1 (Y) show 1492 261 moveto 384 /Times-Roman f1 (|) show 1604 261 moveto 384 /Times-Roman f1 (Z) show 1854 261 moveto 384 /Times-Roman f1 (\)) show 406 793 moveto 384 /Times-Roman f1 (V) show 689 793 moveto 384 /Times-Roman f1 (\() show 833 793 moveto 384 /Times-Roman f1 (X) show 1110 793 moveto 384 /Times-Roman f1 (|) show 1222 793 moveto 384 /Times-Roman f1 (Z) show 1472 793 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 401 moveto 1995 0 rlineto stroke 2103 500 moveto 384 /Symbol f1 (=) show 2441 261 moveto 384 /Times-Roman f1 (Cov) show 3082 261 moveto 384 /Times-Roman f1 (\() show 3226 261 moveto 384 /Times-Roman f1 (X) show 3505 261 moveto 384 /Times-Roman f1 (,) show 3638 261 moveto 384 /Times-Roman f1 (Y) show 3928 261 moveto 384 /Times-Roman f1 (\)) show 4062 261 moveto 384 /Times-Roman f1 (V) show 4345 261 moveto 384 /Times-Roman f1 (\() show 4491 261 moveto 384 /Times-Roman f1 (Z) show 4741 261 moveto 384 /Times-Roman f1 (\)) show 4942 261 moveto 384 /Symbol f1 (-) show 5231 261 moveto 384 /Times-Roman f1 (Cov) show 5872 261 moveto 384 /Times-Roman f1 (\() show 6016 261 moveto 384 /Times-Roman f1 (X) show 6295 261 moveto 384 /Times-Roman f1 (,) show 6434 261 moveto 384 /Times-Roman f1 (Z) show 6684 261 moveto 384 /Times-Roman f1 (\)) show 6813 261 moveto 384 /Times-Roman f1 (Cov) show 7454 261 moveto 384 /Times-Roman f1 (\() show 7594 261 moveto 384 /Times-Roman f1 (Y) show 7871 261 moveto 384 /Times-Roman f1 (,) show 8010 261 moveto 384 /Times-Roman f1 (Z) show 8260 261 moveto 384 /Times-Roman f1 (\)) show 3534 793 moveto 384 /Times-Roman f1 (V) show 3817 793 moveto 384 /Times-Roman f1 (\() show 3961 793 moveto 384 /Times-Roman f1 (X) show 4253 793 moveto 384 /Times-Roman f1 (\)) show 4387 793 moveto 384 /Times-Roman f1 (V) show 4670 793 moveto 384 /Times-Roman f1 (\() show 4816 793 moveto 384 /Times-Roman f1 (Z) show 5066 793 moveto 384 /Times-Roman f1 (\)) show 5267 793 moveto 384 /Symbol f1 (-) show 5556 793 moveto 384 /Times-Roman f1 (Cov) show 6197 793 moveto 384 /Times-Roman f1 (\() show 6341 793 moveto 384 /Times-Roman f1 (X) show 6620 793 moveto 384 /Times-Roman f1 (,) show 6759 793 moveto 384 /Times-Roman f1 (Z) show 7009 793 moveto 384 /Times-Roman f1 (\)) show 7144 622 moveto 224 /Times-Roman f1 (2) show 2421 401 moveto 5980 0 rlineto stroke 2103 1615 moveto 384 /Symbol f1 (=) show 2430 1376 moveto 384 /Symbol f1 (b) show 2640 1376 moveto 384 /Times-Roman f1 (V) show 2923 1376 moveto 384 /Times-Roman f1 (\() show 3067 1376 moveto 384 /Times-Roman f1 (X) show 3359 1376 moveto 384 /Times-Roman f1 (\)) show 3493 1376 moveto 384 /Times-Roman f1 (V) show 3776 1376 moveto 384 /Times-Roman f1 (\() show 3922 1376 moveto 384 /Times-Roman f1 (Z) show 4172 1376 moveto 384 /Times-Roman f1 (\)) show 4373 1376 moveto 384 /Symbol f1 (-) show 4655 1376 moveto 384 /Symbol f1 (r) show 4857 1376 moveto 384 /Times-Roman f1 (\() show 4986 1376 moveto 384 /Symbol f1 (rb) show 5486 1376 moveto 384 /Symbol f1 (+) show 5785 1376 moveto 384 /Symbol f1 (t) show 5966 1376 moveto 384 /Times-Roman f1 (\)) show 3077 1908 moveto 384 /Times-Roman f1 (V) show 3360 1908 moveto 384 /Times-Roman f1 (\() show 3504 1908 moveto 384 /Times-Roman f1 (X) show 3796 1908 moveto 384 /Times-Roman f1 (\)) show 3930 1908 moveto 384 /Times-Roman f1 (V) show 4213 1908 moveto 384 /Times-Roman f1 (\() show 4359 1908 moveto 384 /Times-Roman f1 (Z) show 4609 1908 moveto 384 /Times-Roman f1 (\)) show 4810 1908 moveto 384 /Symbol f1 (-) show 5092 1908 moveto 384 /Symbol f1 (r) show 5306 1737 moveto 224 /Times-Roman f1 (2) show 2421 1516 moveto 3686 0 rlineto stroke 2103 2728 moveto 384 /Symbol f1 (=) show 2398 2728 moveto 384 /Symbol f1 (b) show 2687 2728 moveto 384 /Symbol f1 (-) show 4006 2489 moveto 384 /Symbol f1 (rt) show 3013 3021 moveto 384 /Times-Roman f1 (V) show 3296 3021 moveto 384 /Times-Roman f1 (\() show 3440 3021 moveto 384 /Times-Roman f1 (X) show 3732 3021 moveto 384 /Times-Roman f1 (\)) show 3866 3021 moveto 384 /Times-Roman f1 (V) show 4149 3021 moveto 384 /Times-Roman f1 (\() show 4295 3021 moveto 384 /Times-Roman f1 (Z) show 4545 3021 moveto 384 /Times-Roman f1 (\)) show 4746 3021 moveto 384 /Symbol f1 (-) show 5028 3021 moveto 384 /Symbol f1 (r) show 5242 2850 moveto 224 /Times-Roman f1 (2) show 2988 2629 moveto 2424 0 rlineto stroke end pse gR gS 0 0 552 730 rC 59 373 :M f0_12 sf -.088(Hence the coefficient of X in the regression of Y on X and Z is )A 360 373 :M -.111(not )A 379 373 :M -.326(a )A 388 373 :M -.099(consistent )A 439 373 :M -.248(estimate )A 481 373 :M (of)S 59 390 :M f1_12 sf .74(b)A f0_12 sf .613 .061(, )J 74 390 :M (\(unless )S 112 390 :M f1_12 sf .376 .038(r )J 123 390 :M f0_12 sf .211 .021(= )J 134 390 :M (0 )S 144 390 :M (or )S 158 390 :M f1_12 sf -.268(t )A 167 390 :M f0_12 sf .211 .021(= )J 178 390 :M .775 .077(0\), )J 196 390 :M -.109(and )A 218 390 :M -.22(may )A 243 390 :M -.163(even )A 270 390 :M -.163(have )A 297 390 :M -.326(a )A 307 390 :M -.231(completely )A 363 390 :M -.144(different )A 408 390 :M .666 .067(sign. )J 437 390 :M (In )S 452 390 :M -.22(the )A 471 390 :M -.215(case)A 59 408 :M -.062(where )A 93 408 :M f1_12 sf .376 .038(b )J 105 408 :M f0_12 sf .211 .021(= )J 117 408 :M .833 .083(0, )J 132 408 :M -.22(the )A 151 408 :M -.208(coefficient )A 205 408 :M (of )S 221 408 :M .306 .031(X )J 236 408 :M -.167(in )A 251 408 :M -.22(the )A 271 408 :M -.031(regression )A 326 408 :M (of )S 342 408 :M -.663(Y )A 356 408 :M (on )S 374 408 :M .306 .031(X )J 389 408 :M -.221(typically )A 435 408 :M -.166(will )A 459 408 :M -.111(not )A 480 408 :M -.326(be)A 59 427 :M -.091(significantly different from zero, but will become so once Z is included.)A 77 445 :M -.131(Analyses often appear to suggest that it is better to include rather than exclude a )A 454 445 :M -.234(variable)A 59 463 :M -.066(from a regression. This notion is perhaps given support by reference to \322controlling for )A 475 463 :M .172<5AD32C>A 59 481 :M -.123(the implication being that controlling for Z eliminates a source of bias. The conclusion to )A 480 481 :M -.326(be)A 59 499 :M (drawn )S 93 499 :M -.08(from )A 120 499 :M -.131(these )A 149 499 :M -.164(examples )A 198 499 :M (is )S 211 499 :M -.249(that )A 233 499 :M -.196(there )A 261 499 :M (is )S 274 499 :M (no )S 291 499 :M (sense )S 322 499 :M -.167(in )A 336 499 :M -.065(which )A 370 499 :M -.109(one )A 392 499 :M (is )S 405 499 :M -.165(\322playing )A 450 499 :M -.128(safe\323 )A 479 499 :M (by)S 59 517 :M -.1(including rather than excluding \322potential' confounders\323; if they turn out )A 404 517 :M -.111(not )A 423 517 :M -.167(to )A 436 517 :M -.163(be )A 451 517 :M -.249(potential)A 59 535 :M -.111(confounders then this could change a consistent estimate into an inconsistent estimate.)A 77 553 :M -.219(The )A 99 553 :M -.148(situation )A 144 553 :M (is )S 157 553 :M -.082(also )A 181 553 :M -.247(made )A 211 553 :M -.081(somewhat )A 264 553 :M .234 .023(worse )J 298 553 :M (by )S 315 553 :M -.22(the )A 334 553 :M (use )S 355 553 :M (of )S 370 553 :M -.166(misleading )A 426 553 :M -.12(definitions )A 481 553 :M (of)S 59 571 :M 1.178 .118('confounder': )J 128 571 :M -.147(sometimes )A 182 571 :M -.326(a )A 191 571 :M -.064(confounder )A 249 571 :M (is )S 261 571 :M -.082(said )A 284 571 :M -.167(to )A 298 571 :M -.163(be )A 314 571 :M -.326(a )A 324 571 :M -.205(variable )A 366 571 :M -.249(that )A 388 571 :M (is )S 401 571 :M -.041(strongly )A 445 571 :M -.218(correlated)A 59 589 :M -.046(with both X and Y, or even a )A 201 589 :M -.205(variable )A 242 589 :M .229 .023(whose )J 277 589 :M -.111(inclusion )A 324 589 :M -.092(changes )A 366 589 :M -.22(the )A 384 589 :M -.208(coefficient )A 437 589 :M (of )S 451 589 :M .306 .031(X )J 464 589 :M -.167(in )A 477 589 :M -.33(the)A 59 606 :M .33 .033(regression. )J 118 606 :M .214 .021(Since, )J 154 606 :M (for )S 174 606 :M -.137(sufficiently )A 233 606 :M -.196(large )A 262 606 :M f1_12 sf -.17(t)A f0_12 sf ( )S 273 606 :M -.109(and )A 296 606 :M f1_12 sf .74(r)A f0_12 sf .613 .061(, )J 313 606 :M -.33(Z )A 326 606 :M -.167(in )A 341 606 :M -.054(Figure )A 379 606 :M (6 )S 392 606 :M (would )S 429 606 :M -.141(qualify )A 469 606 :M (as )S 486 606 :M (a)S 59 625 :M -.101(confounder under either of these definitions, it follows that under either definition including)A 59 643 :M -.059(confounding )A 127 643 :M -.145(variables )A 177 643 :M -.167(in )A 195 643 :M -.326(a )A 209 643 :M -.031(regression )A 267 643 :M -.22(may )A 296 643 :M -.247(make )A 330 643 :M -.326(a )A 344 643 :M -.123(higherto )A 392 643 :M -.099(consistent )A 448 643 :M -.206(estimator)A 59 661 :M -.027(inconsistent.)A endp %%Page: 9 9 %%BeginPageSetup initializepage (peter; page: 9 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 485 695 6 30 rC 485 722 :M f0_12 sf (9)S gR gS 0 0 552 730 rC 77 56 :M f0_12 sf (Finally, )S 118 56 :M -.334(it )A 128 56 :M (is )S 140 56 :M (worth )S 172 56 :M -.209(reiterating )A 223 56 :M -.22(the )A 241 56 :M -.031(well-known )A 303 56 :M -.245(fact )A 325 56 :M -.249(that )A 347 56 :M -.167(in )A 361 56 :M -.234(certain )A 397 56 :M -.151(circumstances )A 468 56 :M -.245(there)A 59 74 :M -.22(may )A 83 74 :M -.163(be )A 98 74 :M (no )S 114 74 :M -.031(regression )A 167 74 :M -.065(which )A 200 74 :M -.166(will )A 222 74 :M -.248(estimate )A 264 74 :M -.22(the )A 282 74 :M -.218(parameter )A 332 74 :M (of )S 346 74 :M -.035(interest, )A 388 74 :M -.11(\(although )A 437 74 :M -.082(some )A 467 74 :M -.163(other)A 59 92 :M -.138(consistent estimator may exist\):)A 283 133 9 12 rC 284 142 :M (X)S gR gS 332 132 9 12 rC 333 141 :M f0_12 sf (Y)S gR gS 227 132 11 12 rC 228 141 :M f0_12 sf (W)S gR gS 226 95 116 63 rC np 282 138 :M 270 141 :L 270 138 :L 270 135 :L 282 138 :L eofill 243 139 -1 1 271 138 1 243 138 @a np 331 137 :M 319 140 :L 319 137 :L 319 134 :L 331 137 :L eofill 293 138 -1 1 320 137 1 293 137 @a np 334 132 :M 323 125 :L 325 123 :L 328 121 :L 334 132 :L eofill 316 114 -1 1 326 123 1 316 113 @a np 291 133 :M 297 122 :L 299 124 :L 302 126 :L 291 133 :L eofill -1 -1 300 125 1 1 311 113 @b 1 G 19 17 314.5 105.5 @j 0 G 18 16 314.5 105.5 @f 310 101 8 12 rC 311 110 :M f0_12 sf (T)S gR gS 306 138 8 17 rC 307 151 :M f1_12 sf (b)S gR gS 255 140 9 17 rC 256 153 :M f1_12 sf (a)S gR gS 332 105 8 17 rC 333 118 :M f1_12 sf (f)S gR gS 297 103 7 17 rC 298 116 :M f1_12 sf (1)S gR gS 0 0 552 730 rC 251 173 :M f2_12 sf 2.949 .295(Figure 7)J 77 196 :M f0_12 sf -.052(In the SEM shown in Figure 7, Cov\(X,Y\) = )A f1_12 sf -.066(b)A f0_12 sf -.055(V\(X\) + )A f1_12 sf -.063(f)A f0_12 sf -.05(V\(T\); hence the coefficient of X)A 59 214 :M -.053(in the regression of Y on X is not a consistent estimator of )A f1_12 sf -.076(b)A f0_12 sf -.06(. Further)A 389 214 :M ( )S 144 236 280 31 rC 424 267 :M psb currentpoint pse 144 236 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8960 div 992 3 -1 roll exch div scale currentpoint translate 64 -3938 translate 20 4243 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 661 4243 moveto 384 /Times-Roman f1 (\() show 805 4243 moveto 384 /Times-Roman f1 (X) show 1084 4243 moveto 384 /Times-Roman f1 (,) show 1217 4243 moveto 384 /Times-Roman f1 (Y) show 1492 4243 moveto 384 /Times-Roman f1 (|) show 1603 4243 moveto 384 /Times-Roman f1 (W) show 1983 4243 moveto 384 /Times-Roman f1 (\)) show 407 4775 moveto 384 /Times-Roman f1 (V) show 690 4775 moveto 384 /Times-Roman f1 (\() show 834 4775 moveto 384 /Times-Roman f1 (X) show 1111 4775 moveto 384 /Times-Roman f1 (|) show 1222 4775 moveto 384 /Times-Roman f1 (W) show 1602 4775 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 4383 moveto 2124 0 rlineto stroke 2232 4482 moveto 384 /Symbol f1 (=) show 2527 4482 moveto 384 /Symbol f1 (b) show 2817 4482 moveto 384 /Symbol f1 (+) show 3910 4243 moveto 384 /Symbol f1 (f) show 4110 4243 moveto 384 /Times-Roman f1 (V) show 4393 4243 moveto 384 /Times-Roman f1 (\() show 4535 4243 moveto 384 /Times-Roman f1 (T) show 4784 4243 moveto 384 /Times-Roman f1 (\)) show 3144 4775 moveto 384 /Times-Roman f1 (V) show 3427 4775 moveto 384 /Times-Roman f1 (\() show 3571 4775 moveto 384 /Times-Roman f1 (X) show 3863 4775 moveto 384 /Times-Roman f1 (\)) show 4064 4775 moveto 384 /Symbol f1 (-) show 4350 4775 moveto 384 /Symbol f1 (a) show 4614 4604 moveto 224 /Times-Roman f1 (2) show 4745 4775 moveto 384 /Times-Roman f1 (V) show 5028 4775 moveto 384 /Times-Roman f1 (\() show 5173 4775 moveto 384 /Times-Roman f1 (W) show 5553 4775 moveto 384 /Times-Roman f1 (\)) show 3119 4383 moveto 2575 0 rlineto stroke 5802 4482 moveto 384 /Symbol f1 (=) show 6097 4482 moveto 384 /Symbol f1 (b) show 6387 4482 moveto 384 /Symbol f1 (+) show 7275 4243 moveto 384 /Symbol f1 (f) show 7475 4243 moveto 384 /Times-Roman f1 (V) show 7758 4243 moveto 384 /Times-Roman f1 (\() show 7900 4243 moveto 384 /Times-Roman f1 (T) show 8149 4243 moveto 384 /Times-Roman f1 (\)) show 6714 4775 moveto 384 /Times-Roman f1 (V) show 6997 4775 moveto 384 /Times-Roman f1 (\() show 7139 4775 moveto 384 /Times-Roman f1 (T) show 7388 4775 moveto 384 /Times-Roman f1 (\)) show 7590 4775 moveto 384 /Symbol f1 (+) show 7885 4775 moveto 384 /Times-Roman f1 (V) show 8168 4775 moveto 384 /Times-Roman f1 (\() show 8308 4775 moveto 384 /Symbol f1 (e) show 8501 4871 moveto 224 /Times-Roman f1 (X) show 8712 4775 moveto 384 /Times-Roman f1 (\)) show 6689 4383 moveto 2164 0 rlineto stroke end pse gR gS 0 0 552 730 rC 59 282 :M f0_12 sf -.069(hence including W in the regression does not help matters. However, a consistent )A 448 282 :M -.206(estimator)A 59 300 :M -.136(exists, the so-called Instrumental Variable estimator:)A 217 321 134 31 rC 351 352 :M psb currentpoint pse 217 321 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4288 div 992 3 -1 roll exch div scale currentpoint translate -317 -5050 translate 404 5387 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 1045 5387 moveto 384 /Times-Roman f1 (\() show 1185 5387 moveto 384 /Times-Roman f1 (Y) show 1462 5387 moveto 384 /Times-Roman f1 (,) show 1600 5387 moveto 384 /Times-Roman f1 (W) show 1980 5387 moveto 384 /Times-Roman f1 (\)) show 401 5919 moveto 384 /Times-Roman f1 (Cov) show 1042 5919 moveto 384 /Times-Roman f1 (\() show 1186 5919 moveto 384 /Times-Roman f1 (X) show 1465 5919 moveto 384 /Times-Roman f1 (,) show 1603 5919 moveto 384 /Times-Roman f1 (W) show 1983 5919 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 381 5527 moveto 1743 0 rlineto stroke 2232 5626 moveto 384 /Symbol f1 (=) show 2567 5387 moveto 384 /Symbol f1 (ab) show 3019 5387 moveto 384 /Times-Roman f1 (V) show 3302 5387 moveto 384 /Times-Roman f1 (\() show 3447 5387 moveto 384 /Times-Roman f1 (W) show 3827 5387 moveto 384 /Times-Roman f1 (\)) show 2672 5919 moveto 384 /Symbol f1 (a) show 2914 5919 moveto 384 /Times-Roman f1 (V) show 3197 5919 moveto 384 /Times-Roman f1 (\() show 3342 5919 moveto 384 /Times-Roman f1 (W) show 3722 5919 moveto 384 /Times-Roman f1 (\)) show 2550 5527 moveto 1418 0 rlineto stroke 4076 5626 moveto 384 /Symbol f1 (=) show 4371 5626 moveto 384 /Symbol f1 (b) show end pse gR gS 0 0 552 730 rC 77 385 :M f0_12 sf -.078(In this discussion we have highlighted a number of problems that arise when )A 443 385 :M -.221(estimating)A 59 403 :M -.099(structural coefficients via regression. These examples raise the following general questions:)A 77 421 :M -.104(\(a\) )A 94 421 :M (If )S 106 421 :M -.663(Y )A 118 421 :M (is )S 130 421 :M -.033(regressed )A 179 421 :M (on )S 195 421 :M -.326(a )A 204 421 :M -.109(set )A 222 421 :M (of )S 237 421 :M -.145(variables )A 284 421 :M f2_12 sf .667(W)A f0_12 sf .303 .03(, )J 305 421 :M -.148(including )A 354 421 :M 1.114 .111(X, )J 372 421 :M -.167(in )A 386 421 :M -.065(which )A 420 421 :M -.084(SEMs )A 454 421 :M -.166(will )A 477 421 :M -.33(the)A 59 438 :M -.235(partial )A 92 438 :M -.031(regression )A 145 438 :M -.208(coefficient )A 198 438 :M (of )S 212 438 :M .306 .031(X )J 226 438 :M -.163(be )A 242 438 :M -.326(a )A 252 438 :M -.099(consistent )A 304 438 :M -.248(estimate )A 347 438 :M (of )S 362 438 :M -.22(the )A 381 438 :M -.131(structural )A 430 438 :M -.208(coefficient )A 484 438 :M f1_12 sf (b)S 59 456 :M f0_12 sf -.103(associated with the X )A f1_12 sf -.159A f0_12 sf -.142(Y edge?)A 77 475 :M (\(b\) If )S 106 475 :M -.663(Y )A 118 475 :M (is )S 130 475 :M -.033(regressed )A 179 475 :M (on )S 195 475 :M -.22(the )A 213 475 :M -.109(set )A 230 475 :M f2_12 sf .667(W)A f0_12 sf .303 .03(, )J 250 475 :M -.065(which )A 283 475 :M -.123(includes )A 326 475 :M 1.114 .111(X, )J 343 475 :M -.167(in )A 356 475 :M -.065(which )A 389 475 :M -.084(SEMs )A 422 475 :M -.166(will )A 444 475 :M -.22(the )A 462 475 :M -.275(partial)A 59 492 :M -.104(regression coefficient of )A 177 492 :M .306 .031(X )J 190 492 :M -.163(be )A 205 492 :M -.161(zero )A 229 492 :M -.164(if )A 240 492 :M -.22(the )A 258 492 :M -.131(structural )A 306 492 :M -.208(coefficient )A 359 492 :M -.131(associated )A 411 492 :M -.083(with )A 436 492 :M -.22(the )A 454 492 :M .306 .031(X )J 467 492 :M f1_12 sf .144 .014J 483 492 :M f0_12 sf (Y)S 59 511 :M -.135(edge is zero?)A 77 528 :M -.104(\(c\) )A 94 528 :M -.065(Given )A 127 528 :M -.326(a )A 136 528 :M -.197(particular )A 184 528 :M -.223(SEM )A 212 528 :M -.167(in )A 225 528 :M -.065(which )A 258 528 :M -.196(there )A 285 528 :M (is )S 297 528 :M -.163(an )A 312 528 :M -.163(edge )A 338 528 :M .306 .031(X )J 351 528 :M f1_12 sf .144 .014J 367 528 :M f0_12 sf -.663(Y )A 379 528 :M -.083(with )A 404 528 :M -.208(coefficient )A 457 528 :M f1_12 sf .74(b)A f0_12 sf .613 .061(, )J 472 528 :M (is )S 485 528 :M -.668(it)A 59 547 :M -.041(possible )A 103 547 :M -.167(to )A 117 547 :M -.082(find )A 141 547 :M -.326(a )A 151 547 :M (subset )S 186 547 :M f2_12 sf (W)S f0_12 sf ( )S 203 547 :M (of )S 218 547 :M -.039(observed )A 266 547 :M -.145(variables )A 313 547 :M -.132(\(including )A 366 547 :M 1.033 .103(X\), )J 388 547 :M (such )S 415 547 :M -.249(that )A 437 547 :M (when )S 469 547 :M -.663(Y )A 483 547 :M (is)S 59 564 :M -.079(regressed on the set )A f2_12 sf -.205(W)A f0_12 sf -.077(, the coefficient of X in the regression is a consistent estimate of )A f1_12 sf -.113(b)A f0_12 sf (?)S 77 582 :M (\(d\) )S 96 582 :M -.065(Given )A 130 582 :M -.326(a )A 140 582 :M -.197(particular )A 189 582 :M -.223(SEM )A 219 582 :M -.109(and )A 242 582 :M -.326(a )A 253 582 :M -.131(structural )A 303 582 :M -.208(coefficient )A 358 582 :M f1_12 sf .74(b)A f0_12 sf .613 .061(, )J 375 582 :M (is )S 389 582 :M -.334(it )A 401 582 :M -.041(possible )A 446 582 :M -.167(to )A 461 582 :M -.082(find )A 486 582 :M (a)S 59 600 :M -.082(function h\()A f2_12 sf -.113(S)A f0_12 sf -.082(\) \(where )A f2_12 sf -.113(S)A f0_12 sf -.078( is the sample covariance matrix\) that is a consistent estimator of )A f1_12 sf -.112(b)A f0_12 sf (?)S 77 619 :M -.326(We )A 98 619 :M -.132(shall )A 125 619 :M (answer )S 164 619 :M -.036(questions )A 214 619 :M .491 .049(\(a\), )J 236 619 :M (\(b\) )S 256 619 :M -.109(and )A 279 619 :M .491 .049(\(c\), )J 302 619 :M (by )S 320 619 :M -.124(applying )A 367 619 :M -.22(the )A 387 619 :M -.182(graphical )A 436 619 :M -.182(criterion )A 481 619 :M (of)S 59 637 :M -.023(d-separation. )A 126 637 :M -.33(One )A 150 637 :M -.182(advantage )A 202 637 :M (of )S 217 637 :M -.326(a )A 227 637 :M -.182(graphical )A 275 637 :M -.182(criterion )A 319 637 :M (is )S 332 637 :M -.249(that )A 354 637 :M -.334(it )A 365 637 :M -.217(can )A 386 637 :M -.163(be )A 402 637 :M -.189(applied )A 441 637 :M -.111(simply )A 479 637 :M (by)S 59 655 :M -.084(visual inspection of the path diagram, and does not )A 303 655 :M -.139(require )A 340 655 :M -.142(lengthy )A 379 655 :M -.218(algebraic )A 425 655 :M -.166(manipulations)A 59 673 :M -.065(which )A 92 673 :M -.219(become )A 132 673 :M -.137(increasingly )A 193 673 :M (arduous )S 235 673 :M (when )S 265 673 :M -.163(more )A 293 673 :M -.145(variables )A 339 673 :M -.215(are )A 358 673 :M -.124(involved )A 404 673 :M -.167(in )A 418 673 :M -.22(the )A 437 673 :M -.149(calculation.)A endp %%Page: 10 10 %%BeginPageSetup initializepage (peter; page: 10 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (10)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.326(We )A 79 56 :M (do )S 95 56 :M -.111(not )A 114 56 :M .24 .024(know )J 145 56 :M -.22(the )A 163 56 :M (answer )S 201 56 :M -.167(to )A 214 56 :M .724 .072(\(d\), )J 236 56 :M -.065(which )A 269 56 :M (is )S 281 56 :M -.109(one )A 302 56 :M -.08(form )A 329 56 :M (of )S 344 56 :M -.22(the )A 363 56 :M -.031(well-known )A 425 56 :M -.205("identification)A 59 74 :M -.135(problem"; )A 110 74 :M -.334(it )A 120 74 :M (is )S 132 74 :M -.041(possible )A 175 74 :M -.249(that )A 196 74 :M -.065(extensions )A 250 74 :M (of )S 264 74 :M -.22(the )A 283 74 :M -.182(graphical )A 331 74 :M -.246(criteria )A 368 74 :M (we )S 387 74 :M -.092(present )A 426 74 :M -.22(may )A 451 74 :M -.083(hold )A 477 74 :M -.33(the)A 59 92 :M -.014(key. For related theorems, see Pearl\(1997\).)A 95 122 :M f4_12 sf (2)S 102 122 :M (.)S 107 122 :M (4)S 114 122 :M (.)S 119 122 :M 9 .9( )J 131 122 :M 4.569 .457(Other Applications)J 77 149 :M f0_12 sf -.103(In addition to the uses described above, there are a number of other applications that )A 477 149 :M (we)S 59 167 :M (do )S 76 167 :M -.111(not )A 96 167 :M -.163(have )A 123 167 :M -.22(the )A 142 167 :M -.129(space )A 173 167 :M -.167(to )A 187 167 :M -.122(describe )A 231 167 :M .236 .024(here. )J 260 167 :M -.219(The )A 283 167 :M -.108(d-separation )A 346 167 :M -.206(relation )A 386 167 :M (has )S 408 167 :M -.053(proved )A 447 167 :M -.053(useful )A 482 167 :M -.334(in)A 59 185 :M -.22(automated )A 111 185 :M -.106(search )A 145 185 :M (for )S 163 185 :M -.163(causal )A 196 185 :M -.108(structure )A 241 185 :M -.08(from )A 268 185 :M -.247(data )A 292 185 :M -.109(and )A 314 185 :M -.065(background )A 375 185 :M -.072(knowledge )A 432 185 :M -.04(\(Spirtes )A 474 185 :M -.163(and)A 59 203 :M .493 .049(Glymour, 1991, )J 141 203 :M .375 .037(Spirtes, )J 182 203 :M -.046(Glymour )A 229 203 :M -.109(and )A 250 203 :M .184 .018(Scheines, )J 300 203 :M .667 .067(1993, )J 332 203 :M -.13(Pearl )A 360 203 :M -.109(and )A 381 203 :M -.107(Verma, )A 420 203 :M .667 .067(1991, )J 452 203 :M .113(Cooper,)A 59 221 :M -.105(1992\), in calculating the )A 177 221 :M -.138(effects )A 212 221 :M (of )S 226 221 :M -.127(interventions )A 291 221 :M (on )S 307 221 :M -.163(causal )A 340 221 :M (systems )S 382 221 :M .359 .036(\(Spirtes, )J 427 221 :M -.046(Glymour )A 474 221 :M -.163(and)A 59 239 :M .184 .018(Scheines, )J 109 239 :M .667 .067(1993, )J 141 239 :M -.109(and )A 162 239 :M .219 .022(Pearl, )J 194 239 :M .629 .063(1995\), )J 230 239 :M -.109(and )A 251 239 :M (has )S 271 239 :M (shed )S 297 239 :M -.2(light )A 322 239 :M (on )S 338 239 :M -.326(a )A 347 239 :M -.109(number )A 388 239 :M (of )S 403 239 :M .212 .021(issues )J 437 239 :M -.167(in )A 451 239 :M -.147(statistics)A 59 257 :M -.039(ranging from Simpson\325s Paradox to )A 233 257 :M -.219(experimental )A 297 257 :M -.055(design )A 332 257 :M .359 .036(\(Spirtes, )J 377 257 :M -.046(Glymour )A 424 257 :M -.109(and )A 445 257 :M .044(Scheines,)A 59 275 :M -.048(1993\). See also the applications in Pearl\(1997\).)A 59 305 :M f2_14 sf (3)S 67 305 :M (.)S 72 305 :M 19.5 1.95( )J 95 305 :M 3.04 .304(Linear Structural Equation Models and d-separation)J 77 326 :M f0_12 sf -.137(In a linear SEM the random variables are divided )A 311 326 :M -.167(into )A 333 326 :M (two )S 355 326 :M -.125(disjoint )A 394 326 :M .67 .067(sets, )J 420 326 :M -.22(the )A 438 326 :M -.099(substantive)A 59 344 :M -.093(variables and the error variables. Corresponding to each substantive random variable )A 462 344 :M -.663(V )A 474 344 :M (is )S 486 344 :M (a)S 59 362 :M -.11(unique )A 96 362 :M -.061(error )A 124 362 :M -.247(term )A 150 362 :M f1_12 sf .268(e)A f0_7 sf 0 3 rm .258(V)A 0 -3 rm f0_12 sf (.)S 164 359 :M f0_8 sf (5)S 168 362 :M f0_12 sf ( )S 173 362 :M -.663(A )A 186 362 :M -.219(linear )A 217 362 :M -.223(SEM )A 246 362 :M -.123(contains )A 291 362 :M -.326(a )A 302 362 :M -.109(set )A 321 362 :M (of )S 337 362 :M -.219(linear )A 369 362 :M -.11(equations )A 420 362 :M -.167(in )A 435 362 :M -.065(which )A 470 362 :M -.326(each)A 59 380 :M -.09(substantive )A 116 380 :M -.109(random )A 156 380 :M -.205(variable )A 197 380 :M -.663(V )A 209 380 :M (is )S 221 380 :M -.141(written )A 258 380 :M (as )S 272 380 :M -.326(a )A 281 380 :M -.219(linear )A 311 380 :M -.123(function )A 354 380 :M (of )S 368 380 :M -.131(other )A 397 380 :M -.09(substantive )A 455 380 :M -.131(random)A 59 398 :M -.099(variables together with )A f1_12 sf -.111(e)A f0_7 sf 0 3 rm -.106(V)A 0 -3 rm f0_12 sf -.098(, and a correlation matrix among the error terms. Initially, )A 455 398 :M (we )S 473 398 :M -.222(will)A 59 416 :M -.053(assume )A 98 416 :M -.249(that )A 119 416 :M -.22(the )A 137 416 :M -.061(error )A 164 416 :M -.145(variables )A 210 416 :M -.215(are )A 228 416 :M -.228(multi-variate )A 291 416 :M .535 .053(Gaussian. )J 343 416 :M .571 .057(However, )J 395 416 :M -.165(many )A 426 416 :M (of )S 441 416 :M -.22(the )A 460 416 :M -.054(results)A 59 434 :M -.099(that we will prove are about partial correlations, which do not )A 353 434 :M -.109(depend )A 391 434 :M (upon )S 419 434 :M -.22(the )A 437 434 :M -.121(distribution)A 59 452 :M (of )S 73 452 :M -.22(the )A 91 452 :M -.061(error )A 118 452 :M .216 .022(terms, )J 152 452 :M -.111(but )A 171 452 :M -.109(depend )A 209 452 :M -.083(only )A 234 452 :M (upon )S 262 452 :M -.22(the )A 280 452 :M -.219(linear )A 310 452 :M -.11(equations )A 359 452 :M -.109(and )A 380 452 :M -.22(the )A 399 452 :M -.136(correlations )A 459 452 :M -.165(among)A 59 470 :M -.041(the error terms.)A 77 488 :M -.061(Since we have no interest in first moments, without loss of )A 359 488 :M -.197(generality )A 409 488 :M -.245(each )A 434 488 :M -.205(variable )A 475 488 :M -.326(can)A 59 506 :M -.079(be expressed as a deviation from its mean.)A 77 530 :M .258 .026(For )J 98 530 :M -.081(example, )A 145 530 :M -.22(the )A 163 530 :M -.073(following )A 214 530 :M (is )S 227 530 :M -.326(a )A 237 530 :M -.219(linear )A 268 530 :M -.223(SEM )A 297 530 :M .136(M,)A f1_12 sf .06 .006( )J 316 530 :M .219(e)A f0_7 sf 0 3 rm .21(A)A 0 -3 rm f0_12 sf .227 .023(, )J 335 530 :M f1_12 sf .351(e)A f0_7 sf 0 3 rm .312(B)A 0 -3 rm f0_12 sf .364 .036(, )J 354 530 :M f1_12 sf .351(e)A f0_7 sf 0 3 rm .312(C)A 0 -3 rm f0_12 sf .364 .036(, )J 373 530 :M f1_12 sf .219(e)A f0_7 sf 0 3 rm .21(D)A 0 -3 rm f0_12 sf .227 .023(, )J 392 530 :M -.109(and )A 414 530 :M f1_12 sf -.228(e)A f0_7 sf 0 3 rm -.185(E)A 0 -3 rm f0_12 sf ( )S 428 530 :M -.215(are )A 447 530 :M (Gaussian)S 59 548 :M -.029("error terms", and A, B, C, D, and E are substantive random variables:)A 268 572 :M -.127(A = )A f1_12 sf -.125(e)A f0_7 sf 0 3 rm (A)S 0 -3 rm 268 590 :M f0_12 sf .082 .008(B = )J f1_12 sf .05(e)A f0_7 sf 0 3 rm (B)S 0 -3 rm 237 608 :M f0_12 sf .245 .024(C = .2B + .8D + )J f1_12 sf .121(e)A f0_7 sf 0 3 rm (C)S 0 -3 rm 236 626 :M f0_12 sf .16 .016(D = -.5C + .1E + )J f1_12 sf .077(e)A f0_7 sf 0 3 rm (D)S 0 -3 rm 269 644 :M f0_12 sf -.109(E = )A f1_12 sf -.114(e)A f0_7 sf 0 3 rm (E)S 0 -3 rm 59 662 :M f0_12 sf ( )S 59 659.48 -.48 .48 203.48 659 .48 59 659 @a 59 672 :M f0_8 sf (5)S 63 675 :M f0_10 sf .5 .05( )J 68 675 :M -.163(There )A 95 675 :M .694 .069(is )J 107 675 :M .051 .005(an )J 121 675 :M -.115(equivalent )A 166 675 :M -.038(definition )A 209 675 :M .144 .014(of )J 222 675 :M .056 .006(a )J 231 675 :M -.044(linear )A 259 675 :M .726 .073(SEM )J 286 675 :M .601 .06(in )J 300 675 :M .043 .004(which )J 330 675 :M -.073(parent-less )A 378 675 :M .144 .014(or )J 392 675 :M -.084(\324exogenous\325 )A 446 675 :M (substantive)S 59 687 :M -.081(variables have no associated error variables.)A endp %%Page: 11 11 %%BeginPageSetup initializepage (peter; page: 11 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (11)S gR gS 0 0 552 730 rC 167 74 :M f0_12 sf -.122( Correlation Matrix Among Error Terms)A 203 83 166 108 rC 369 191 :M psb currentpoint pse 203 83 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5312 div 3456 3 -1 roll exch div scale currentpoint translate 64 37 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 934 294 sh (e) 1686 294 sh (e) 2540 294 sh (e) 3415 294 sh (e) 4290 294 sh (e) 166 886 sh (e) 166 1478 sh (e) 166 2067 sh (e) 166 2659 sh (e) 166 3248 sh 224 /Times-Roman f1 (A) 1128 390 sh (B) 1879 390 sh (D) 2733 390 sh (D) 3608 390 sh (E) 4485 390 sh (A) 360 982 sh (B) 359 1574 sh (C) 356 2163 sh (D) 359 2755 sh (E) 361 3344 sh 384 /Times-Roman f1 (1) 892 886 sh (0) 1180 886 sh (0) 1737 886 sh (5) 2025 886 sh (0) 2568 886 sh (0) 2856 886 sh (0) 3413 886 sh (0) 3701 886 sh (0) 4258 886 sh (0) 4546 886 sh (0) 899 1478 sh (5) 1187 1478 sh (1) 1697 1478 sh (0) 1985 1478 sh (0) 2542 1478 sh (0) 2830 1478 sh (0) 3387 1478 sh (0) 3675 1478 sh (0) 4232 1478 sh (0) 4520 1478 sh (0) 905 2067 sh (0) 1193 2067 sh (0) 1750 2067 sh (0) 2038 2067 sh (1) 2562 2067 sh (0) 2850 2067 sh (0) 3407 2067 sh (0) 3695 2067 sh (0) 4252 2067 sh (0) 4540 2067 sh (0) 920 2659 sh (0) 1208 2659 sh (0) 1765 2659 sh (0) 2053 2659 sh (0) 2610 2659 sh (0) 2898 2659 sh (1) 3422 2659 sh (0) 3710 2659 sh (0) 4267 2659 sh (0) 4555 2659 sh (0) 901 3248 sh (0) 1189 3248 sh (0) 1746 3248 sh (0) 2034 3248 sh (0) 2591 3248 sh (0) 2879 3248 sh (0) 3436 3248 sh (0) 3724 3248 sh (1) 4248 3248 sh (0) 4536 3248 sh 384 /Times-Roman f1 (.) 1084 886 sh (.) 1929 886 sh (.) 2760 886 sh (.) 3605 886 sh (.) 4450 886 sh (.) 1091 1478 sh (.) 1889 1478 sh (.) 2734 1478 sh (.) 3579 1478 sh (.) 4424 1478 sh (.) 1097 2067 sh (.) 1942 2067 sh (.) 2754 2067 sh (.) 3599 2067 sh (.) 4444 2067 sh (.) 1112 2659 sh (.) 1957 2659 sh (.) 2802 2659 sh (.) 3614 2659 sh (.) 4459 2659 sh (.) 1093 3248 sh (.) 1938 3248 sh (.) 2783 3248 sh (.) 3628 3248 sh (.) 4440 3248 sh 384 /Symbol f1 (\346) -15 355 sh (\350) -15 3264 sh (\347) -15 806 sh (\347) -15 1177 sh (\347) -15 1548 sh (\347) -15 1919 sh (\347) -15 2290 sh (\347) -15 2661 sh (\347) -15 3032 sh (\366) 5044 355 sh (\370) 5044 3264 sh (\367) 5044 806 sh (\367) 5044 1177 sh (\367) 5044 1548 sh (\367) 5044 1919 sh (\367) 5044 2290 sh (\367) 5044 2661 sh (\367) 5044 3032 sh end MTsave restore pse gR gS 0 0 552 730 rC 77 212 :M f0_12 sf (If )S 89 212 :M -.22(the )A 107 212 :M -.163(coefficients )A 165 212 :M -.167(in )A 178 212 :M -.22(the )A 196 212 :M -.219(linear )A 226 212 :M -.11(equations )A 275 212 :M -.215(are )A 294 212 :M (such )S 321 212 :M -.249(that )A 343 212 :M -.22(the )A 362 212 :M -.09(substantive )A 420 212 :M -.145(variables )A 467 212 :M -.215(are )A 486 212 :M (a)S 59 230 :M -.11(unique )A 95 230 :M -.219(linear )A 125 230 :M -.123(function )A 168 230 :M (of )S 182 230 :M -.22(the )A 200 230 :M -.061(error )A 227 230 :M -.145(variables )A 273 230 :M (alone, )S 306 230 :M -.22(the )A 325 230 :M -.109(set )A 343 230 :M (of )S 358 230 :M -.11(equations )A 408 230 :M (is )S 421 230 :M -.082(said )A 445 230 :M -.167(to )A 459 230 :M -.163(have )A 486 230 :M (a)S 59 248 :M f2_12 sf 2.166 .217(reduced )J 108 248 :M .543(form)A f0_12 sf .468 .047(. )J 143 248 :M -.663(A )A 155 248 :M -.219(linear )A 186 248 :M -.223(SEM )A 215 248 :M -.083(with )A 241 248 :M -.326(a )A 251 248 :M -.139(reduced )A 293 248 :M -.08(form )A 321 248 :M -.082(also )A 345 248 :M -.164(determines )A 401 248 :M -.326(a )A 411 248 :M -.2(joint )A 437 248 :M -.121(distribution)A 59 266 :M -.092(over the substantive variables. We will )A 246 266 :M -.081(consider )A 290 266 :M -.083(only )A 315 266 :M -.219(linear )A 345 266 :M -.084(SEMs )A 378 266 :M -.065(which )A 411 266 :M -.163(have )A 437 266 :M -.178(coefficients)A 59 284 :M -.107(for which there is a reduced form, all variances and partial variances )A 384 284 :M -.132(among )A 420 284 :M -.22(the )A 438 284 :M -.099(substantive)A 59 302 :M -.131(variables are finite and positive, and all partial )A 279 302 :M -.136(correlations )A 338 302 :M -.132(among )A 374 302 :M -.22(the )A 392 302 :M -.09(substantive )A 449 302 :M -.163(variables)A 59 320 :M -.044(are well defined \(e.g. not infinite\).)A 77 338 :M -.219(The )A 101 338 :M -.165(path )A 128 338 :M -.188(diagram )A 173 338 :M (of )S 190 338 :M -.326(a )A 202 338 :M -.219(linear )A 235 338 :M -.223(SEM )A 266 338 :M -.083(with )A 294 338 :M -.163(uncorrelated )A 359 338 :M (errors )S 394 338 :M (is )S 409 338 :M -.141(written )A 449 338 :M -.083(with )A 477 338 :M -.33(the)A 59 356 :M -.09(conventions )A 121 356 :M -.249(that )A 143 356 :M -.334(it )A 154 356 :M -.123(contains )A 198 356 :M -.163(an )A 214 356 :M -.163(edge )A 241 356 :M -.663(A )A 254 356 :M f1_12 sf .126A f0_12 sf ( )S 271 356 :M (B )S 284 356 :M -.164(if )A 296 356 :M -.109(and )A 318 356 :M -.083(only )A 344 356 :M -.164(if )A 356 356 :M -.22(the )A 375 356 :M -.208(coefficient )A 429 356 :M (for )S 448 356 :M -.663(A )A 462 356 :M -.167(in )A 477 356 :M -.33(the)A 59 374 :M -.131(structural )A 108 374 :M -.165(equation )A 153 374 :M (for )S 172 374 :M (B )S 185 374 :M (is )S 198 374 :M .19 .019(non-zero, )J 249 374 :M -.109(and )A 271 374 :M -.196(there )A 299 374 :M (is )S 312 374 :M -.326(a )A 322 374 :M -.125(double-headed )A 396 374 :M (arrow )S 429 374 :M -.139(between )A 473 374 :M (two)S 59 392 :M -.145(variables )A 105 392 :M -.663(A )A 117 392 :M -.109(and )A 138 392 :M (B )S 150 392 :M -.164(if )A 161 392 :M -.109(and )A 182 392 :M -.083(only )A 207 392 :M -.164(if )A 218 392 :M -.22(the )A 236 392 :M -.179(correlation )A 290 392 :M -.139(between )A 333 392 :M f1_12 sf -.127(e)A f0_7 sf 0 3 rm -.122(A)A 0 -3 rm f0_12 sf ( )S 347 392 :M -.109(and )A 369 392 :M f1_12 sf (e)S f0_7 sf 0 3 rm (B)S 0 -3 rm f0_12 sf ( )S 384 392 :M (is )S 397 392 :M .19 .019(non-zero. )J 448 392 :M (Thus )S 477 392 :M -.33(the)A 59 410 :M -.041(path diagram for M is shown in Figure 8.)A 77 434 :M (In )S 91 434 :M -.062(order )A 120 434 :M -.167(to )A 133 434 :M -.163(define )A 166 434 :M -.22(the )A 185 434 :M -.108(d-separation )A 248 434 :M -.072(relation, )A 292 434 :M (we )S 311 434 :M -.163(need )A 338 434 :M -.167(to )A 352 434 :M -.146(introduce )A 401 434 :M -.22(the )A 420 434 :M -.073(following )A 471 434 :M -.22(path)A 59 452 :M -.118(diagram terminology. The concepts defined here are illustrated in Figure )A 403 452 :M .833 .083(8. )J 417 452 :M -.663(A )A 429 452 :M -.165(path )A 453 452 :M -.219(diagram)A 59 470 :M -.031(consists of two parts, a set of vertices )A 241 470 :M f2_12 sf .25(V)A f0_12 sf .087 .009( )J 254 470 :M -.109(and )A 275 470 :M -.326(a )A 284 470 :M -.109(set )A 301 470 :M (of )S 315 470 :M -.064(edges )A 346 470 :M f2_12 sf .569(E)A f0_12 sf .388 .039(. )J 362 470 :M -.246(Each )A 389 470 :M -.163(edge )A 415 470 :M -.167(in )A 428 470 :M f2_12 sf (E)S f0_12 sf ( )S 440 470 :M (is )S 452 470 :M -.163(between)A 59 488 :M -.074(two distinct vertices in )A f2_12 sf -.143(V)A f0_12 sf -.083(. There are two )A 253 488 :M (kinds )S 283 488 :M (of )S 297 488 :M -.064(edges )A 328 488 :M -.167(in )A 341 488 :M f2_12 sf .569(E)A f0_12 sf .388 .039(, )J 357 488 :M -.205(directed )A 398 488 :M -.064(edges )A 429 488 :M -.663(A )A 441 488 :M f1_12 sf .126A f0_12 sf ( )S 457 488 :M (B )S 469 488 :M (or )S 483 488 :M (A)S 59 506 :M f1_12 sf S f0_12 sf -.008( B, and double-headed edges A )A f1_12 sf S f0_12 sf -.007( B; in either case A and B are )A f2_12 sf -.008(endpoints)A f0_12 sf ( )S 434 506 :M (of )S 448 506 :M -.22(the )A 466 506 :M -.247(edge;)A 59 524 :M -.051(further, A and B are said to be )A f2_12 sf -.06(adjacent)A f0_12 sf -.055(. There may be multiple edges between vertices. In)A 59 542 :M -.004(Figure 8 the set of vertices is {A,B,C,D,E} and the set of edges is {A )A f1_12 sf S f0_12 sf ( B, B )S 435 542 :M f1_12 sf .126A f0_12 sf ( )S 451 542 :M .83 .083(C, )J 467 542 :M (C )S 479 542 :M f1_12 sf S 59 560 :M f0_12 sf .281 .028(D, )J 75 560 :M -.663(D )A 87 560 :M f1_12 sf .126A f0_12 sf ( )S 103 560 :M .83 .083(C, )J 119 560 :M -.33(E )A 130 560 :M f1_12 sf .126A f0_12 sf ( )S 146 560 :M .444 .044(D}. )J 168 560 :M .258 .026(For )J 189 560 :M -.326(a )A 198 560 :M -.205(directed )A 239 560 :M -.163(edge )A 265 560 :M -.663(A )A 277 560 :M f1_12 sf .126A f0_12 sf ( )S 293 560 :M .83 .083(B, )J 309 560 :M -.663(A )A 321 560 :M (is )S 333 560 :M -.22(the )A 351 560 :M f2_12 sf .284(tail)A f0_12 sf .204 .02( )J 373 560 :M (of )S 387 560 :M -.22(the )A 405 560 :M -.163(edge )A 431 560 :M -.109(and )A 452 560 :M (B )S 464 560 :M (is )S 477 560 :M -.33(the)A 59 578 :M f2_12 sf .095(head)A f0_12 sf .158 .016( of the edge, A is a )J f2_12 sf .087(parent)A f0_12 sf .141 .014( of B, and B is a )J f2_12 sf .078(child)A f0_12 sf .17 .017( of A.)J 77 596 :M .599 .06(An )J f2_12 sf 1.439 .144(undirected path)J f0_12 sf .294 .029( U )J 197 596 :M -.139(between )A 240 596 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 257 596 :M -.109(and )A 278 596 :M .478(X)A f0_7 sf 0 3 rm .193(n)A 0 -3 rm f0_12 sf .166 .017( )J 295 596 :M (is )S 307 596 :M -.326(a )A 316 596 :M -.121(sequence )A 363 596 :M (of )S 377 596 :M -.064(edges )A 408 596 :M .621( )J 469 596 :M (such)S 59 614 :M -.165(that one )A 100 614 :M -.124(endpoint )A 145 614 :M (of )S 159 614 :M .09(E)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S 174 614 :M (is )S 186 614 :M .876(X)A f0_7 sf 0 3 rm .354(1)A 0 -3 rm f0_12 sf .552 .055(, )J 207 614 :M -.109(one )A 228 614 :M -.124(endpoint )A 273 614 :M (of )S 287 614 :M -.36(E)A f0_7 sf 0 3 rm -.267(m)A 0 -3 rm f0_12 sf ( )S 303 614 :M (is )S 315 614 :M .876(X)A f0_7 sf 0 3 rm .354(n)A 0 -3 rm f0_12 sf .552 .055(, )J 336 614 :M -.109(and )A 357 614 :M (for )S 375 614 :M -.245(each )A 400 614 :M -.163(pair )A 422 614 :M (of )S 436 614 :M -.164(consecutive)A 59 632 :M -.036(edges E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.03(, E)A f0_7 sf 0 3 rm -.021(i+1)A 0 -3 rm f0_12 sf -.032( in the sequence, E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf -.034( E)A f0_7 sf 0 3 rm -.021(i+1)A 0 -3 rm f0_12 sf -.032(, and one endpoint of E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.033( equals one endpoint of )A 470 632 :M .475(E)A f0_7 sf 0 3 rm .203(i+1)A 0 -3 rm f0_12 sf (.)S 59 650 :M .077 .008(In Figure 8, A )J f1_12 sf .075A f0_12 sf ( )S 147 650 :M (B )S 159 650 :M f1_12 sf .126A f0_12 sf ( )S 175 650 :M (C )S 187 650 :M f1_12 sf .126A f0_12 sf ( )S 203 650 :M -.663(D )A 215 650 :M (is )S 227 650 :M -.163(an )A 242 650 :M -.235(example )A 285 650 :M (of )S 299 650 :M -.163(an )A 314 650 :M -.164(undirected )A 367 650 :M -.165(path )A 391 650 :M -.139(between )A 434 650 :M -.663(A )A 446 650 :M -.109(and )A 467 650 :M .281 .028(D. )J 483 650 :M (A)S 59 668 :M f2_12 sf .546 .055(directed path)J f0_12 sf .281 .028( P between X)J f0_7 sf 0 3 rm .064(1)A 0 -3 rm f0_12 sf .226 .023( and X)J f0_7 sf 0 3 rm .064(n)A 0 -3 rm f0_12 sf .294 .029( is a sequence of directed edges )J 448 668 :M (such )S 474 668 :M -.331(that)A 59 686 :M -.005(the tail of E)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( is X)S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.005(, the head of E)A f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf ( is X)S f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf (, )S 253 686 :M -.109(and )A 274 686 :M (for )S 292 686 :M -.245(each )A 317 686 :M -.163(pair )A 339 686 :M (of )S 353 686 :M -.064(edges )A 384 686 :M .348(E)A f0_7 sf 0 3 rm .092(i)A 0 -3 rm f0_12 sf .259 .026(, )J 401 686 :M .103(E)A f0_7 sf 0 3 rm .044(i+1)A 0 -3 rm f0_12 sf ( )S 422 686 :M -.247(adjacent )A 464 686 :M -.167(in )A 477 686 :M -.33(the)A endp %%Page: 12 12 %%BeginPageSetup initializepage (peter; page: 12 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (12)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf (sequence, )S 110 56 :M -.164(E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 123 56 :M f1_12 sf .285A f0_12 sf .13 .013( )J 134 56 :M .412(E)A f0_7 sf 0 3 rm .176(i+1)A 0 -3 rm f0_12 sf .307 .031(, )J 159 56 :M -.109(and )A 180 56 :M -.22(the )A 198 56 :M -.163(head )A 224 56 :M (of )S 238 56 :M -.164(E)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 251 56 :M (is )S 263 56 :M -.22(the )A 281 56 :M -.332(tail )A 299 56 :M (of )S 313 56 :M .412(E)A f0_7 sf 0 3 rm .176(i+1)A 0 -3 rm f0_12 sf .307 .031(. )J 338 56 :M .258 .026(For )J 359 56 :M -.081(example, )A 406 56 :M (B )S 418 56 :M f1_12 sf .126A f0_12 sf ( )S 434 56 :M (C )S 446 56 :M f1_12 sf .126A f0_12 sf ( )S 462 56 :M -.663(D )A 474 56 :M (is )S 486 56 :M (a)S 59 74 :M .904 .09(directed path. A )J f2_12 sf 1.006 .101(vertex occurs on a path)J f0_12 sf .159 .016( )J 273 74 :M -.164(if )A 284 74 :M -.334(it )A 294 74 :M (is )S 306 74 :M -.163(an )A 321 74 :M -.124(endpoint )A 366 74 :M (of )S 380 74 :M -.109(one )A 401 74 :M (of )S 415 74 :M -.22(the )A 433 74 :M -.064(edges )A 464 74 :M -.167(in )A 477 74 :M -.33(the)A 59 92 :M -.078(path. The set of vertices )A 176 92 :M (on )S 192 92 :M -.663(A )A 204 92 :M f1_12 sf .4A f0_12 sf .096 .01( )J 221 92 :M (B )S 233 92 :M f1_12 sf .126A f0_12 sf ( )S 249 92 :M (C )S 261 92 :M f1_12 sf .126A f0_12 sf ( )S 277 92 :M -.663(D )A 289 92 :M (is )S 301 92 :M .444 .044({A, )J 323 92 :M .83 .083(B, )J 339 92 :M .83 .083(C, )J 355 92 :M .444 .044(D}. )J 377 92 :M -.663(A )A 389 92 :M -.165(path )A 413 92 :M (is )S 425 92 :M f2_12 sf .572(acyclic)A f0_12 sf .347 .035( )J 468 92 :M -.164(if )A 479 92 :M (no)S 59 110 :M -.163(vertex )A 92 110 :M -.052(occurs )A 128 110 :M -.163(more )A 157 110 :M -.165(than )A 182 110 :M -.163(once )A 209 110 :M (on )S 226 110 :M -.22(the )A 245 110 :M .226 .023(path. )J 274 110 :M (C )S 287 110 :M f1_12 sf .126A f0_12 sf ( )S 304 110 :M -.663(D )A 317 110 :M f1_12 sf .126A f0_12 sf ( )S 334 110 :M (C )S 347 110 :M (is )S 360 110 :M -.326(a )A 370 110 :M -.274(cyclic )A 402 110 :M -.205(directed )A 444 110 :M .226 .023(path. )J 473 110 :M -.328(The)A 59 128 :M -.135(following is a list of all the acyclic directed paths in )A 304 128 :M -.054(Figure )A 339 128 :M -.167(8: )A 352 128 :M (B )S 364 128 :M f1_12 sf .126A f0_12 sf ( )S 380 128 :M .83 .083(C, )J 396 128 :M (C )S 408 128 :M f1_12 sf .126A f0_12 sf ( )S 424 128 :M .281 .028(D, )J 440 128 :M -.33(E )A 451 128 :M f1_12 sf .126A f0_12 sf ( )S 467 128 :M .281 .028(D, )J 483 128 :M (D)S 59 146 :M f1_12 sf .159A f0_12 sf .104 .01( C, B )J f1_12 sf .159A f0_12 sf .082 .008( C )J f1_12 sf .159A f0_12 sf .104 .01( D, E )J f1_12 sf .159A f0_12 sf .086 .009( D )J f1_12 sf .159A f0_12 sf .157 .016( C.)J 77 170 :M -.663(A )A 89 170 :M -.163(vertex )A 122 170 :M -.663(A )A 134 170 :M (is )S 146 170 :M -.163(an )A 161 170 :M f2_12 sf .431(ancestor)A f0_12 sf .239 .024( )J 212 170 :M (of )S 226 170 :M (B )S 238 170 :M -.08(\(and )A 263 170 :M (B )S 275 170 :M (is )S 287 170 :M -.326(a )A 296 170 :M f2_12 sf .445(descendant)A f0_12 sf .233 .023( )J 362 170 :M (of )S 376 170 :M -.328(A\) )A 393 170 :M -.164(if )A 405 170 :M -.109(and )A 427 170 :M -.083(only )A 453 170 :M -.164(if )A 465 170 :M -.263(either)A 59 188 :M -.155(there is a directed )A 145 188 :M -.165(path )A 169 188 :M -.08(from )A 196 188 :M -.663(A )A 208 188 :M -.167(to )A 221 188 :M (B )S 233 188 :M (or )S 247 188 :M -.663(A )A 259 188 :M .211 .021(= )J 270 188 :M .83 .083(B. )J 286 188 :M (Thus )S 314 188 :M -.22(the )A 332 188 :M -.122(ancestor )A 375 188 :M -.206(relation )A 414 188 :M (is )S 426 188 :M -.22(the )A 444 188 :M -.065(transitive,)A 59 206 :M -.181(reflexive )A 107 206 :M -.092(closure )A 148 206 :M (of )S 165 206 :M -.22(the )A 186 206 :M -.163(parent )A 222 206 :M -.072(relation. )A 268 206 :M -.219(The )A 293 206 :M -.073(following )A 346 206 :M -.264(table )A 375 206 :M -.067(lists )A 401 206 :M -.22(the )A 423 206 :M (child, )S 458 206 :M (parent,)S 59 224 :M -.08(descendant and ancestor relations in Figure 8.)A 82 267 :M -.329(Vertex)A 166 267 :M -.142(Children)A 257 267 :M -.053(Parents)A 334 267 :M -.164(Descendants)A 428 267 :M -.122(Ancestors)A 53 251 1 1 rF 53 251 1 1 rF 54 251 87 1 rF 141 251 1 1 rF 142 251 88 1 rF 230 251 1 1 rF 231 251 87 1 rF 318 251 1 1 rF 319 251 88 1 rF 407 251 1 1 rF 408 251 87 1 rF 495 251 1 1 rF 495 251 1 1 rF 53 252 1 18 rF 141 252 1 18 rF 230 252 1 18 rF 318 252 1 18 rF 407 252 1 18 rF 495 252 1 18 rF 94 286 :M (A)S 182 286 :M f1_12 sf S 270 286 :M S 354 286 :M f0_12 sf -.091({A})A 442 286 :M -.091({A})A 53 270 1 1 rF 54 270 87 1 rF 141 270 1 1 rF 142 270 88 1 rF 230 270 1 1 rF 231 270 87 1 rF 318 270 1 1 rF 319 270 88 1 rF 407 270 1 1 rF 408 270 87 1 rF 495 270 1 1 rF 53 271 1 18 rF 141 271 1 18 rF 230 271 1 18 rF 318 271 1 18 rF 407 271 1 18 rF 495 271 1 18 rF 94 305 :M (B)S 177 305 :M .238({C})A 270 305 :M f1_12 sf S 342 305 :M f0_12 sf .302({B,C,D})A 442 305 :M .238({B})A 53 289 1 1 rF 54 289 87 1 rF 141 289 1 1 rF 142 289 88 1 rF 230 289 1 1 rF 231 289 87 1 rF 318 289 1 1 rF 319 289 88 1 rF 407 289 1 1 rF 408 289 87 1 rF 495 289 1 1 rF 53 290 1 18 rF 141 290 1 18 rF 230 290 1 18 rF 318 290 1 18 rF 407 290 1 18 rF 495 290 1 18 rF 94 324 :M (C)S 177 324 :M -.091({D})A 259 324 :M .203({B,D})A 348 324 :M .203({C,D})A 424 324 :M .31({B,C.D,E})A 53 308 1 1 rF 54 308 87 1 rF 141 308 1 1 rF 142 308 88 1 rF 230 308 1 1 rF 231 308 87 1 rF 318 308 1 1 rF 319 308 88 1 rF 407 308 1 1 rF 408 308 87 1 rF 495 308 1 1 rF 53 309 1 18 rF 141 309 1 18 rF 230 309 1 18 rF 318 309 1 18 rF 407 309 1 18 rF 495 309 1 18 rF 94 343 :M (D)S 177 343 :M .238({C})A 259 343 :M .287({C,E})A 348 343 :M .203({C,D})A 424 343 :M .31({B,C,D,E})A 53 327 1 1 rF 54 327 87 1 rF 141 327 1 1 rF 142 327 88 1 rF 230 327 1 1 rF 231 327 87 1 rF 318 327 1 1 rF 319 327 88 1 rF 407 327 1 1 rF 408 327 87 1 rF 495 327 1 1 rF 53 328 1 18 rF 141 328 1 18 rF 230 328 1 18 rF 318 328 1 18 rF 407 328 1 18 rF 495 328 1 18 rF 94 362 :M (E)S 177 362 :M -.091({D})A 270 362 :M f1_12 sf S 342 362 :M f0_12 sf .247({C,D,E})A 442 362 :M .075({E})A 53 346 1 1 rF 54 346 87 1 rF 141 346 1 1 rF 142 346 88 1 rF 230 346 1 1 rF 231 346 87 1 rF 318 346 1 1 rF 319 346 88 1 rF 407 346 1 1 rF 408 346 87 1 rF 495 346 1 1 rF 53 347 1 18 rF 53 365 1 1 rF 53 365 1 1 rF 54 365 87 1 rF 141 347 1 18 rF 141 365 1 1 rF 142 365 88 1 rF 230 347 1 18 rF 230 365 1 1 rF 231 365 87 1 rF 318 347 1 18 rF 318 365 1 1 rF 319 365 88 1 rF 407 347 1 18 rF 407 365 1 1 rF 408 365 87 1 rF 495 347 1 18 rF 495 365 1 1 rF 495 365 1 1 rF 77 387 :M -.023(A vertex X is a )A f2_12 sf -.023(collider)A f0_12 sf -.023( on undirected path U if and only if U contains a subpath )A 466 387 :M -.663(Y )A 478 387 :M f1_12 sf S 59 405 :M f0_12 sf .154 .015(X )J f1_12 sf .199A f0_12 sf .133 .013( Z, or Y )J f1_12 sf .188A f0_12 sf .101 .01( X )J f1_12 sf .199A f0_12 sf .133 .013( Z, or Y )J f1_12 sf .188A f0_12 sf .101 .01( X )J f1_12 sf .188A f0_12 sf .113 .011( Z, )J 263 405 :M (or )S 277 405 :M -.663(Y )A 289 405 :M f1_12 sf .4A f0_12 sf .096 .01( )J 306 405 :M .306 .031(X )J 319 405 :M f1_12 sf .126A f0_12 sf ( )S 335 405 :M -.332(Z; )A 349 405 :M -.072(otherwise )A 399 405 :M -.164(if )A 410 405 :M .306 .031(X )J 423 405 :M (is )S 435 405 :M (on )S 451 405 :M .306 .031(U )J 464 405 :M -.334(it )A 474 405 :M (is )S 486 405 :M (a)S 59 423 :M f2_12 sf .118(non-collider)A f0_12 sf .295 .03( on U. For example, C is a collider on B )J f1_12 sf .271A f0_12 sf .069 .007( )J 338 423 :M (C )S 350 423 :M f1_12 sf .126A f0_12 sf ( )S 366 423 :M -.663(D )A 378 423 :M -.111(but )A 397 423 :M -.326(a )A 406 423 :M -.137(non-collider )A 467 423 :M (on )S 483 423 :M (B)S 59 441 :M f1_12 sf .69A f0_12 sf .355 .035( C )J f1_12 sf .69A f0_12 sf .513 .051( D. X is an )J f2_12 sf .925 .093(ancestor of a )J 230 441 :M .553(set)A f0_12 sf .356 .036( )J 250 441 :M (of )S 264 441 :M -.163(vertices )A 304 441 :M f2_12 sf (Z)S f0_12 sf ( )S 316 441 :M -.164(if )A 327 441 :M .306 .031(X )J 340 441 :M (is )S 352 441 :M -.163(an )A 367 441 :M -.122(ancestor )A 410 441 :M (of )S 424 441 :M -.082(some )A 453 441 :M -.263(member)A 59 459 :M .418 .042(of )J f2_12 sf .335(Z)A f0_12 sf (.)S 77 483 :M .414 .041(For disjoint sets of vertices, )J f2_12 sf .213(X)A f0_12 sf .123 .012(, )J f2_12 sf .213(Y)A f0_12 sf .249 .025(, and )J f2_12 sf .197(Z)A f0_12 sf .123 .012(, )J f2_12 sf .213(X)A f0_12 sf .144 .014( is )J f2_12 sf .139(d-connected)A f0_12 sf .157 .016( to )J f2_12 sf .213(Y)A f0_12 sf .298 .03( given )J f2_12 sf .197(Z)A f0_12 sf .224 .022( if and )J 470 483 :M -.111(only)A 59 501 :M -.155(if there is an acyclic undirected path )A 231 501 :M .306 .031(U )J 244 501 :M -.139(between )A 287 501 :M -.082(some )A 316 501 :M -.219(member )A 358 501 :M .306 .031(X )J 371 501 :M (of )S 385 501 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 403 501 :M -.109(and )A 424 501 :M -.082(some )A 453 501 :M -.263(member)A 59 519 :M -.041(Y of )A f2_12 sf -.072(Y)A f0_12 sf -.04(, such that every )A 174 519 :M -.206(collider )A 213 519 :M (on )S 229 519 :M .306 .031(U )J 242 519 :M (is )S 254 519 :M -.163(an )A 269 519 :M -.122(ancestor )A 312 519 :M (of )S 326 519 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 342 519 :M -.109(and )A 363 519 :M -.129(every )A 393 519 :M -.137(non-collider )A 454 519 :M (on )S 470 519 :M .306 .031(U )J 483 519 :M (is)S 59 537 :M .224 .022(not in )J f2_12 sf .158(Z)A f0_12 sf .301 .03(. For disjoint sets of vertices, )J f2_12 sf .171(X)A f0_12 sf .099 .01(, )J f2_12 sf .171(Y)A f0_12 sf .2 .02(, and )J f2_12 sf .158(Z)A f0_12 sf .099 .01(, )J f2_12 sf .171(X)A f0_12 sf .059 .006( )J 321 537 :M (is )S 333 537 :M f2_12 sf .377(d-separated)A f0_12 sf .205 .021( )J 402 537 :M -.08(from )A 429 537 :M f2_12 sf .25(Y)A f0_12 sf .087 .009( )J 442 537 :M -.132(given )A 472 537 :M f2_12 sf (Z)S f0_12 sf ( )S 484 537 :M -.327(if)A 59 555 :M -.026(and only if )A f2_12 sf (X)S f0_12 sf -.027( is not d-connected to )A f2_12 sf (Y)S f0_12 sf -.027( given )A f2_12 sf (Z)S f0_12 sf (.)S endp %%Page: 13 13 %%BeginPageSetup initializepage (peter; page: 13 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (13)S gR gS 0 0 552 730 rC 251 152 :M f2_12 sf 2.949 .295(Figure 8)J 77 176 :M f0_12 sf -.085(For example, the path E )A 194 176 :M f1_12 sf .126A f0_12 sf ( )S 210 176 :M -.663(D )A 222 176 :M f1_12 sf .126A f0_12 sf ( )S 238 176 :M (C )S 250 176 :M -.097(d-connects )A 305 176 :M -.33(E )A 316 176 :M -.109(and )A 337 176 :M (C )S 349 176 :M -.132(given )A 379 176 :M f1_12 sf -.128A f0_12 sf -.082(; )A 396 176 :M -.334(it )A 406 176 :M -.082(also )A 429 176 :M -.097(d-connects )A 484 176 :M (E)S 59 194 :M -.109(and )A 80 194 :M (C )S 92 194 :M -.132(given )A 122 194 :M .584 .058({A}, )J 150 194 :M 1.055 .105({B}, )J 178 194 :M (or )S 192 194 :M 1.133 .113({A,B}. )J 232 194 :M -.33(E )A 243 194 :M f1_12 sf .126A f0_12 sf ( )S 259 194 :M -.663(D )A 272 194 :M f1_12 sf .126A f0_12 sf ( )S 289 194 :M (C )S 302 194 :M -.097(d-connects )A 358 194 :M -.33(E )A 370 194 :M -.109(and )A 392 194 :M (C )S 405 194 :M -.132(given )A 436 194 :M .584 .058({D}, )J 465 194 :M -.165(given)A 59 212 :M 1.133 .113({D,B}, )J 101 212 :M .721 .072({D,A}, )J 143 212 :M (or )S 159 212 :M 1.195 .119({D,A,B}. )J 213 212 :M -.219(The )A 237 212 :M -.073(following )A 289 212 :M (is )S 303 212 :M -.326(a )A 314 212 :M -.167(list )A 334 212 :M (of )S 350 212 :M -.331(all )A 367 212 :M -.22(the )A 387 212 :M -.08(pairwise )A 433 212 :M -.118(d-separation)A 59 230 :M -.095(relations in Figure 8 \(where each pair is )A 250 230 :M -.081(followed )A 296 230 :M (by )S 312 230 :M -.326(a )A 321 230 :M -.167(list )A 339 230 :M (of )S 353 230 :M -.331(all )A 368 230 :M (of )S 382 230 :M -.22(the )A 400 230 :M (sets )S 422 230 :M -.249(that )A 443 230 :M -.144(d-separate)A 59 248 :M -.264(them\):)A 77 266 :M .264 .026({A} and {C} are d-separated given: {B}, {B,D}, {B,E}, {B,D,E})J 77 284 :M .308 .031({A} and {D} are d-separated given: {B}, {B,C}, {B,E}, {B,C,E})J 77 302 :M .359 .036({A} and {E} are d-separated given: )J f1_12 sf .193A f0_12 sf .43 .043(, {B}, {B,C}, {B,D}, {B,C,D}, {C,D})J 77 320 :M -.016({B} and {E} are d-separated given: )A f1_12 sf S f0_12 sf -.019(, {C,D})A 77 356 :M -.219(The )A 99 356 :M -.064(first )A 122 356 :M -.188(theorem )A 164 356 :M -.109(states )A 194 356 :M -.249(that )A 215 356 :M -.108(d-separation )A 277 356 :M -.167(in )A 290 356 :M -.326(a )A 299 356 :M -.165(path )A 323 356 :M -.188(diagram )A 365 356 :M .306 .031(G )J 378 356 :M (is )S 390 356 :M -.326(a )A 399 356 :M -.131(sufficient )A 447 356 :M -.166(condition)A 59 374 :M (for )S 77 374 :M .306 .031(G )J 91 374 :M -.167(to )A 105 374 :M -.276(entail )A 135 374 :M -.249(that )A 157 374 :M f1_12 sf .261(r\()A f0_12 sf .287(X,Y.)A f2_12 sf .394(Z)A f0_12 sf .197<29>A f0_7 sf 0 3 rm .086 .009( )J 0 -3 rm 209 374 :M f0_12 sf .211 .021(= )J 221 374 :M (0 )S 232 374 :M .898 .09(\(i.e. )J 257 374 :M -.167(in )A 271 374 :M -.129(every )A 302 374 :M -.223(SEM )A 331 374 :M -.083(with )A 357 374 :M -.165(path )A 382 374 :M -.188(diagram )A 425 374 :M 1.114 .111(G, )J 443 374 :M -.22(the )A 462 374 :M -.275(partial)A 59 392 :M -.072(correlation of X and Y given )A f2_12 sf -.118(Z)A f0_12 sf -.073( equals 0.\))A 77 422 :M f2_12 sf .693 .069(Theorem 1: )J f0_12 sf .4 .04(If M is a SEM, )J 220 422 :M -.109(and )A 241 422 :M .629 .063({X} )J 266 422 :M -.109(and )A 287 422 :M -.061({Y} )A 311 422 :M -.215(are )A 329 422 :M -.117(d-separated )A 387 422 :M -.132(given )A 417 422 :M f2_12 sf (Z)S f0_12 sf ( )S 429 422 :M -.167(in )A 442 422 :M .171(G\(M\),)A 59 439 :M .307 .031(then )J f1_12 sf .103(r\()A f0_12 sf .114(X,Y.)A f2_12 sf .156(Z)A f0_12 sf .151 .015(\) = 0 in )J f1_12 sf .138(S)A f0_12 sf .141(\(M\).)A 77 470 :M -.107(The second theorem states that d-separation is a necessary )A 354 470 :M -.148(condition )A 402 470 :M (for )S 420 470 :M -.326(a )A 429 470 :M -.165(path )A 453 470 :M -.219(diagram)A 59 488 :M -.155(to entail a zero partial correlation.)A 77 518 :M f2_12 sf .046 .005(Theorem 2:)J f0_12 sf ( If {X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .024 .002(} and {X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .026 .003(} are not d-separated given )J f2_12 sf (Z)S f0_12 sf .022 .002( in path diagram )J 436 518 :M 1.114 .111(G, )J 453 518 :M -.22(then)A 59 535 :M .079 .008(there is a SEM M such that G\(M\) = G, and )J f1_12 sf .03(r\()A f0_12 sf (X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .033(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (.)S f2_12 sf (Z)S f0_12 sf (\) )S f1_12 sf S f0_12 sf .038 .004( 0 in )J f1_12 sf (S)S f0_12 sf .041(\(M\).)A 77 566 :M -.123(Theorem 2 does )A f5_12 sf -.117(not)A f0_12 sf -.11( say that there might not be an )A 316 566 :M -.166(individual )A 367 566 :M -.223(SEM )A 395 566 :M -.667(M )A 409 566 :M -.083(with )A 434 566 :M -.233(\322extra\323 )A 471 566 :M -.215(zero)A 59 584 :M -.235(partial )A 93 584 :M -.136(correlations )A 153 584 :M -.132(among )A 190 584 :M -.145(variables )A 237 584 :M -.249(that )A 259 584 :M -.215(are )A 278 584 :M -.111(not )A 298 584 :M -.117(d-separated )A 357 584 :M -.167(in )A 372 584 :M .456 .046(G\(M\), )J 409 584 :M (as )S 425 584 :M -.22(the )A 445 584 :M -.082(following)A 59 602 :M .155 .016(example shows.)J 234 674 :M .157 .016(X = .3 Y + .6 Z + )J f1_12 sf .093(e)A f0_7 sf 0 3 rm (X)S 0 -3 rm 264 628 30 20 rC 282 643 :M f0_12 sf (Y)S gR gS 335 629 30 20 rC 353 644 :M f0_12 sf (Z)S gR gS 195 629 30 20 rC 213 644 :M f0_12 sf (X)S gR gS 0 0 552 730 rC 232 640.75 -.75 .75 278.75 640 .75 232 640 @a np 234 644 :M 234 637 :L 226 640 :L 234 644 :L .75 lw eofill -.75 -.75 234.75 644.75 .75 .75 234 637 @b -.75 -.75 226.75 640.75 .75 .75 234 637 @b 226 640.75 -.75 .75 234.75 644 .75 226 640 @a 300 640.75 -.75 .75 347.75 640 .75 300 640 @a np 302 644 :M 302 637 :L 294 640 :L 302 644 :L eofill -.75 -.75 302.75 644.75 .75 .75 302 637 @b -.75 -.75 294.75 640.75 .75 .75 302 637 @b 294 640.75 -.75 .75 302.75 644 .75 294 640 @a 186 57 33 20 rC 204 72 :M f5_12 sf (A)S gR gS 0 0 552 730 rC -.75 -.75 217.75 619.75 .75 .75 217 615 @b np 221 617 :M 213 617 :L 217 625 :L 221 617 :L .75 lw eofill 213 617.75 -.75 .75 221.75 617 .75 213 617 @a 213 617.75 -.75 .75 217.75 625 .75 213 617 @a -.75 -.75 217.75 625.75 .75 .75 221 617 @b 324 56 33 20 rC 342 71 :M f5_12 sf (C)S gR .75 lw gS 0 0 552 730 rC -180 -90 142 28 288.5 616.5 @n 253 56 33 20 rC 271 71 :M f5_12 sf (B)S gR gS 0 0 552 730 rC -90 0 148 46 287.5 625.5 @n 330 113 33 20 rC 348 128 :M f5_12 sf (D)S gR gS 259 113 33 20 rC 277 128 :M f5_12 sf (E)S gR gS 0 0 552 730 rC 226 68.75 -.75 .75 265.75 68 .75 226 68 @a np 265 66 :M 265 70 :L 268 68 :L 265 66 :L eofill -.75 -.75 265.75 70.75 .75 .75 265 66 @b -.75 -.75 265.75 70.75 .75 .75 268 68 @b 265 66.75 -.75 .75 268.75 68 .75 265 66 @a np 227 70 :M 227 66 :L 223 68 :L 227 70 :L eofill -.75 -.75 227.75 70.75 .75 .75 227 66 @b -.75 -.75 223.75 68.75 .75 .75 227 66 @b 223 68.75 -.75 .75 227.75 70 .75 223 68 @a 288 68.75 -.75 .75 330.75 68 .75 288 68 @a np 330 66 :M 330 70 :L 333 68 :L 330 66 :L eofill -.75 -.75 330.75 70.75 .75 .75 330 66 @b -.75 -.75 330.75 70.75 .75 .75 333 68 @b 330 66.75 -.75 .75 333.75 68 .75 330 66 @a 295 125.75 -.75 .75 337.75 125 .75 295 125 @a np 337 123 :M 337 127 :L 340 125 :L 337 123 :L eofill -.75 -.75 337.75 127.75 .75 .75 337 123 @b -.75 -.75 337.75 127.75 .75 .75 340 125 @b 337 123.75 -.75 .75 340.75 125 .75 337 123 @a -.75 -.75 342.75 113.75 .75 .75 342 74 @b np 344 113 :M 340 113 :L 342 116 :L 344 113 :L eofill 340 113.75 -.75 .75 344.75 113 .75 340 113 @a 340 113.75 -.75 .75 342.75 116 .75 340 113 @a -.75 -.75 342.75 116.75 .75 .75 344 113 @b -.75 -.75 352.75 111.75 .75 .75 352 77 @b np 350 78 :M 354 78 :L 352 74 :L 350 78 :L eofill 350 78.75 -.75 .75 354.75 78 .75 350 78 @a 352 74.75 -.75 .75 354.75 78 .75 352 74 @a -.75 -.75 350.75 78.75 .75 .75 352 74 @b endp %%Page: 14 14 %%BeginPageSetup initializepage (peter; page: 14 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (14)S gR gS 0 0 552 730 rC 252 56 :M f0_12 sf -.067(Y = -2 Z +)A f1_12 sf -.054( e)A f0_7 sf 0 3 rm (Y)S 0 -3 rm 269 74 :M f0_12 sf -.109(Z = )A f1_12 sf -.114(e)A f0_7 sf 0 3 rm (Z)S 0 -3 rm 251 92 :M f2_12 sf 2.949 .295(Figure 9)J 59 116 :M f0_12 sf -.13( \(The )A 90 116 :M (errors )S 124 116 :M -.215(are )A 144 116 :M -.163(uncorrelated )A 208 116 :M -.139(because )A 251 116 :M -.196(there )A 280 116 :M -.215(are )A 300 116 :M (no )S 318 116 :M -.125(double-headed )A 394 116 :M .223 .022(arrows )J 434 116 :M -.167(in )A 450 116 :M -.22(the )A 471 116 :M -.22(path)A 59 134 :M -.034(diagram.\) )A 109 134 :M (In )S 123 134 :M -.084(this )A 144 134 :M -.161(case )A 168 134 :M .306 .031(X )J 181 134 :M -.109(and )A 202 134 :M -.663(Y )A 214 134 :M -.215(are )A 232 134 :M -.054(independent, )A 297 134 :M .957 .096(i.e. )J 317 134 :M f1_12 sf .154(r\()A f0_12 sf .567 .057(X,Y\) )J 357 134 :M .211 .021(= )J 368 134 :M .833 .083(0, )J 383 134 :M -.163(even )A 410 134 :M -.056(though )A 448 134 :M .629 .063({X} )J 474 134 :M -.163(and)A 59 152 :M -.061({Y} )A 84 152 :M -.215(are )A 103 152 :M -.111(not )A 123 152 :M -.117(d-separated )A 182 152 :M -.132(given )A 213 152 :M f1_12 sf .699A f0_12 sf .386 .039(. )J 232 152 :M .571 .057(However, )J 285 152 :M -.084(this )A 307 152 :M -.161(zero )A 332 152 :M -.179(correlation )A 387 152 :M (holds )S 418 152 :M -.139(because )A 461 152 :M (of )S 477 152 :M -.33(the)A 59 170 :M -.197(particular )A 107 170 :M -.219(linear )A 137 170 :M -.074(coefficients. )A 199 170 :M .668 .067(Thus, )J 231 170 :M -.145(according )A 281 170 :M -.167(to )A 294 170 :M -.187(Theorem )A 341 170 :M (2 )S 352 170 :M -.196(there )A 380 170 :M (is )S 393 170 :M -.082(some )A 423 170 :M -.131(other )A 452 170 :M -.223(SEM )A 481 170 :M (M)S 59 188 :M -.076(such with the )A 126 188 :M -.163(same )A 154 188 :M -.165(path )A 178 188 :M -.188(diagram )A 220 188 :M -.167(in )A 233 188 :M -.065(which )A 266 188 :M f1_12 sf .154(r\()A f0_12 sf .567 .057(X,Y\) )J 306 188 :M f1_12 sf 1.347A f0_12 sf 2.359 .236(\3120. )J 333 188 :M -.164(It )A 344 188 :M (has )S 364 188 :M -.163(been )A 390 188 :M .447 .045(shown )J 426 188 :M -.04(\(Spirtes )A 467 188 :M .261 .026(et. )J 483 188 :M -.66(al)A 59 206 :M (1993\) )S 91 206 :M -.249(that )A 112 206 :M -.22(the )A 130 206 :M -.109(set )A 147 206 :M (of )S 161 206 :M -.163(parameters )A 216 206 :M -.065(which )A 249 206 :M -.092(produce )A 291 206 :M -.181(conditional )A 347 206 :M -.164(independence )A 415 206 :M -.146(relations )A 459 206 :M -.165(among)A 59 224 :M -.145(variables )A 105 224 :M -.065(which )A 138 224 :M -.215(are )A 156 224 :M -.111(not )A 175 224 :M -.117(d-separated )A 233 224 :M -.167(in )A 246 224 :M .306 .031(G )J 260 224 :M (has )S 281 224 :M -.161(zero )A 306 224 :M -.122(Lebesgue )A 356 224 :M -.139(measure )A 400 224 :M -.08(over )A 426 224 :M -.22(the )A 445 224 :M -.245(parameter)A 59 242 :M .071(space.)A 59 272 :M f2_14 sf (4)S 67 272 :M (.)S 72 272 :M 19.5 1.95( )J 95 272 :M .687(Applications)A 95 305 :M f4_12 sf (4)S 102 305 :M (.)S 107 305 :M (1)S 114 305 :M (.)S 119 305 :M 9 .9( )J 131 305 :M 3.673 .367(Covariance )J 199 305 :M 3.658 .366(Equivalence )J 272 305 :M 2.569 .257(for )J 294 305 :M 1.904 .19(Path )J 324 305 :M 3.332 .333(diagrams )J 381 305 :M 3.523 .352(Without )J 432 305 :M .705(Correlated)A 131 323 :M 3.463 .346(Errors or Directed Cycles)J 77 344 :M f0_12 sf (If )S 89 344 :M (for )S 107 344 :M -.223(SEM )A 135 344 :M .277 .028(M, )J 153 344 :M -.196(there )A 180 344 :M (is )S 192 344 :M -.14(another )A 231 344 :M -.223(SEM )A 260 344 :M -.33<4DD520>A 279 344 :M -.083(with )A 305 344 :M -.326(a )A 315 344 :M -.144(different )A 360 344 :M -.165(path )A 385 344 :M -.188(diagram )A 428 344 :M -.111(but )A 448 344 :M -.22(the )A 467 344 :M -.218(same)A 59 362 :M -.109(number )A 100 362 :M (of )S 115 362 :M -.091(degrees )A 156 362 :M (of )S 171 362 :M (freedom, )S 219 362 :M -.109(and )A 242 362 :M -.22(the )A 262 362 :M -.163(same )A 292 362 :M -.206(marginal )A 339 362 :M -.111(distribution )A 399 362 :M -.08(over )A 426 362 :M -.22(the )A 446 362 :M -.139(measured)A 59 380 :M -.145(variables )A 105 380 :M -.167(in )A 118 380 :M .277 .028(M, )J 136 380 :M -.165(then )A 160 380 :M -.22(the )A 178 380 :M .171(p\()A f1_12 sf .226(c)A f0_7 sf 0 -5 rm .12(2)A 0 5 rm f0_12 sf .218 .022(\) )J 207 380 :M (for )S 225 380 :M -.33<4DD520>A 244 380 :M -.109(equals )A 279 380 :M .146(p\()A f1_12 sf .253(C)A f0_7 sf 0 -5 rm .102(2)A 0 5 rm f0_12 sf .185 .019(\) )J 311 380 :M (for )S 330 380 :M .277 .028(M, )J 349 380 :M -.109(and )A 371 380 :M -.165(they )A 396 380 :M -.163(have )A 423 380 :M -.22(the )A 442 380 :M -.163(same )A 471 380 :M (BIC)S 59 398 :M -.058(scores and AIC scores. Such SEMs are guaranteed to )A 315 398 :M -.132(exist )A 341 398 :M -.164(if )A 352 398 :M -.196(there )A 379 398 :M -.215(are )A 397 398 :M -.084(SEMs )A 430 398 :M -.249(that )A 451 398 :M -.163(have )A 477 398 :M -.33(the)A 59 416 :M -.163(same )A 88 416 :M -.109(number )A 129 416 :M (of )S 145 416 :M -.091(degrees )A 187 416 :M (of )S 203 416 :M -.139(freedom )A 248 416 :M -.109(and )A 271 416 :M -.189(contain )A 311 416 :M -.165(path )A 337 416 :M -.123(diagrams )A 386 416 :M -.065(which )A 421 416 :M -.215(are )A 441 416 :M -.218(covariance)A 59 434 :M -.198(equivalent )A 111 434 :M -.167(to )A 124 434 :M -.245(each )A 149 434 :M .217 .022(other. )J 182 434 :M -.109(Stelzl\(1986\) )A 245 434 :M -.109(and )A 267 434 :M -.327(Lee )A 289 434 :M -.109(and )A 311 434 :M -.016(Hershberger\(1990\) )A 407 434 :M .199 .02(discuss )J 447 434 :M -.145(sufficient)A 59 452 :M -.1(conditions )A 113 452 :M (for )S 132 452 :M -.196(covariance )A 187 452 :M -.209(equivalence )A 247 452 :M -.053(\(which )A 285 452 :M -.165(they )A 310 452 :M -.33(call )A 331 452 :M -.111(simply )A 368 452 :M -.099(equivalence\). )A 437 452 :M -.187(Theorem )A 485 452 :M (3)S 59 470 :M -.109(states )A 91 470 :M .184 .018(necessary, )J 147 470 :M (as )S 164 470 :M -.164(well )A 191 470 :M (as )S 208 470 :M -.131(sufficient )A 259 470 :M -.1(conditions )A 315 470 :M (for )S 336 470 :M -.196(covariance )A 393 470 :M -.209(equivalence )A 455 470 :M -.167(in )A 471 470 :M -.22(path)A 59 488 :M -.096(diagrams without corrrelated errors or directed cycles.)A 77 506 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 94 506 :M -.109(and )A 115 506 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 132 506 :M -.215(are )A 150 506 :M f2_12 sf 2.885 .289(d-separation )J 225 506 :M .442(equivalent)A f0_12 sf .249 .025( )J 287 506 :M -.164(if )A 298 506 :M (for )S 316 506 :M -.245(each )A 341 506 :M -.125(disjoint )A 380 506 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 398 506 :M f2_12 sf .79(Y)A f0_12 sf .497 .05(, )J 415 506 :M -.109(and )A 436 506 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 453 506 :M f2_12 sf .993(X)A f0_12 sf .344 .034( )J 468 506 :M (is )S 481 506 :M (d-)S 59 524 :M -.023(separated from )A f2_12 sf (Y)S f0_12 sf -.021( given )A f2_12 sf (Z)S f0_12 sf -.022( in G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.019( if and only if )A f2_12 sf (X)S f0_12 sf -.021( is d-separated from )A f2_12 sf (Y)S f0_12 sf -.021( given )A f2_12 sf (Z)S f0_12 sf -.022( in G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf (.)S 77 542 :M f2_12 sf 1.973 .197(Theorem )J 131 542 :M .387(3:)A f0_12 sf .232 .023( )J 146 542 :M (If )S 158 542 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 175 542 :M -.109(and )A 196 542 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 213 542 :M -.215(are )A 231 542 :M -.205(directed )A 272 542 :M -.282(acyclic )A 308 542 :M .596 .06(graphs, )J 348 542 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 365 542 :M -.109(and )A 386 542 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 404 542 :M -.215(are )A 423 542 :M -.218(covariance)A 59 560 :M -.076(equivalent if and only if G)A f0_7 sf 0 3 rm -.057(1)A 0 -3 rm f0_12 sf -.086( and G)A f0_7 sf 0 3 rm -.057(2)A 0 -3 rm f0_12 sf -.078( are d-separation equivalent.)A 77 578 :M -.219(The )A 101 578 :M -.165(test )A 123 578 :M (for )S 144 578 :M -.196(covariance )A 201 578 :M -.209(equivalence )A 263 578 :M (of )S 280 578 :M (two )S 305 578 :M -.165(path )A 332 578 :M -.123(diagrams )A 382 578 :M -.108(described )A 434 578 :M -.167(in )A 450 578 :M -.327(Lee )A 474 578 :M -.163(and)A 59 596 :M -.035(Hershberger\(1990\) requires )A 195 596 :M -.18(determining )A 255 596 :M -.092(whether )A 297 596 :M -.196(there )A 324 596 :M (is )S 336 596 :M -.326(a )A 345 596 :M -.052(series )A 376 596 :M (of )S 390 596 :M -.163(edge )A 416 596 :M -.191(replacements )A 481 596 :M (or)S 59 614 :M -.113(reversals preserving equivalence that lead from one path diagram to the other. )A 428 614 :M -.139(Because )A 471 614 :M -.22(they)A 59 632 :M (do )S 75 632 :M -.111(not )A 94 632 :M -.092(specify )A 132 632 :M -.163(an )A 147 632 :M -.081(ordering )A 191 632 :M -.167(in )A 204 632 :M -.065(which )A 237 632 :M -.22(the )A 256 632 :M -.066(tests )A 282 632 :M -.215(are )A 301 632 :M -.167(to )A 315 632 :M -.163(be )A 331 632 :M .449 .045(done, )J 363 632 :M -.084(this )A 385 632 :M -.132(could )A 416 632 :M -.163(be )A 432 632 :M -.326(a )A 442 632 :M -.08(very )A 468 632 :M .112(slow)A 59 650 :M -.013(process. The following )A 173 650 :M -.039(theorem, )A 219 650 :M -.109(due )A 240 650 :M -.167(to )A 253 650 :M -.13(Pearl )A 281 650 :M -.109(and )A 302 650 :M -.328(Verma )A 337 650 :M (\(1991\) )S 373 650 :M .669 .067(shows )J 408 650 :M .259 .026(how )J 433 650 :M -.118(d-separation)A 59 668 :M -.209(equivalence )A 118 668 :M -.217(can )A 139 668 :M -.163(be )A 155 668 :M -.263(calculated )A 206 668 :M -.167(in )A 220 668 :M -.105(O\(E)A f0_7 sf 0 -5 rm -.055(2)A 0 5 rm f0_12 sf -.11(\) )A 252 668 :M -.066(time, )A 281 668 :M -.062(where )A 315 668 :M -.33(E )A 327 668 :M (is )S 340 668 :M -.22(the )A 359 668 :M -.109(number )A 400 668 :M (of )S 415 668 :M -.064(edges )A 447 668 :M -.167(in )A 461 668 :M -.326(a )A 471 668 :M -.22(path)A endp %%Page: 15 15 %%BeginPageSetup initializepage (peter; page: 15 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (15)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.039(diagram. )A 106 56 :M .306 .031(X )J 120 56 :M (is )S 133 56 :M -.163(an )A 149 56 :M f2_12 sf 3.329 .333(unshielded )J 216 56 :M .504(collider)A f0_12 sf .313 .031( )J 264 56 :M -.167(in )A 278 56 :M -.205(directed )A 320 56 :M -.282(acyclic )A 358 56 :M -.064(graph )A 391 56 :M .306 .031(G )J 406 56 :M -.164(if )A 419 56 :M -.109(and )A 442 56 :M -.083(only )A 469 56 :M -.164(if )A 482 56 :M (G)S 59 74 :M -.051(contains edges A )A f1_12 sf -.12A f0_12 sf -.049( X )A f1_12 sf -.12A f0_12 sf -.049( B, and A is not adjacent to B in G.)A 77 92 :M f2_12 sf -.056(Theorem 4:)A f0_12 sf -.046( Two directed acyclic graphs are d-separation equivalent )A 409 92 :M -.164(if )A 420 92 :M -.109(and )A 441 92 :M -.083(only )A 466 92 :M -.327(if)A 59 110 :M -.105(they contain the same vertices, the same adjacencies, and the same unshielded colliders.)A 77 128 :M -.164(It )A 89 128 :M (is )S 102 128 :M -.163(apparent )A 148 128 :M -.08(from )A 177 128 :M -.187(Theorem )A 225 128 :M (4 )S 237 128 :M -.249(that )A 260 128 :M -.109(any )A 283 128 :M (two )S 307 128 :M -.084(SEMs )A 342 128 :M -.083(with )A 369 128 :M -.196(covariance )A 425 128 :M -.22(equivalent)A 59 146 :M -.11(directed acyclic graphs have the same degrees of freedom.)A 95 176 :M f4_12 sf (4)S 102 176 :M (.)S 107 176 :M (2)S 114 176 :M (.)S 119 176 :M 9 .9( )J 131 176 :M 3.673 .367(Covariance )J 202 176 :M 3.658 .366(Equivalence )J 278 176 :M 2.569 .257(for )J 303 176 :M 1.904 .19(Path )J 336 176 :M 2.964 .296(Diagrams )J 398 176 :M 3.326 .333(with )J 432 176 :M .705(Correlated)A 131 194 :M 3.463 .346(Errors or Directed Cycles)J 77 215 :M f0_12 sf -.033(Necessary )A 131 215 :M -.1(conditions )A 186 215 :M (for )S 206 215 :M -.196(covariance )A 262 215 :M -.209(equivalence )A 323 215 :M (for )S 343 215 :M -.165(path )A 369 215 :M -.123(diagrams )A 418 215 :M -.083(with )A 445 215 :M -.218(correlated)A 59 233 :M (errors )S 91 233 :M (or )S 105 233 :M (cycles, )S 142 233 :M -.109(and )A 163 233 :M (for )S 181 233 :M -.165(path )A 205 233 :M -.123(diagrams )A 252 233 :M -.083(with )A 277 233 :M -.276(latent )A 306 233 :M -.145(variables )A 352 233 :M -.054(follow )A 388 233 :M -.08(from )A 416 233 :M -.187(Theorem )A 463 233 :M (1 )S 474 233 :M -.163(and)A 59 251 :M -.068(Theorem 2. If )A f2_12 sf -.128(O)A f0_12 sf -.061( is a subset of the vertices in G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.064( and a subset of the vertices )A 420 251 :M -.167(in )A 433 251 :M .876(G)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 454 251 :M -.165(then )A 478 251 :M .596(G)A f0_7 sf 0 3 rm (1)S 0 -3 rm 59 269 :M f0_12 sf 1.339 .134(and G)J f0_7 sf 0 3 rm .242(2)A 0 -3 rm f0_12 sf .572 .057( are )J f2_12 sf 1.835 .184(d-separation equivalent over O)J f0_12 sf .208 .021( )J 290 269 :M -.164(if )A 301 269 :M (for )S 319 269 :M -.245(each )A 344 269 :M -.125(disjoint )A 383 269 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 401 269 :M f2_12 sf .79(Y)A f0_12 sf .497 .05(, )J 418 269 :M -.109(and )A 439 269 :M f2_12 sf (Z)S f0_12 sf ( )S 451 269 :M -.189(included)A 59 287 :M (in )S f2_12 sf (O)S f0_12 sf (, )S f2_12 sf (X)S f0_12 sf -.008( is d-separated from )A f2_12 sf (Y)S f0_12 sf -.008( given )A f2_12 sf (Z)S f0_12 sf ( in G)S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.008( if and )A 305 287 :M -.083(only )A 330 287 :M -.164(if )A 341 287 :M f2_12 sf .993(X)A f0_12 sf .344 .034( )J 355 287 :M (is )S 367 287 :M -.117(d-separated )A 425 287 :M -.08(from )A 452 287 :M f2_12 sf .25(Y)A f0_12 sf .087 .009( )J 465 287 :M -.165(given)A 59 305 :M f2_12 sf .312(Z)A f0_12 sf .374 .037( in G)J f0_7 sf 0 3 rm .136(2)A 0 -3 rm f0_12 sf (.)S 77 335 :M f2_12 sf -.022(Theorem 5: )A f0_12 sf -.019(If G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.02( and G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.019( are path diagrams that are covariance equivalent over )A 460 335 :M f2_12 sf .504(O)A f0_12 sf (,)S 59 353 :M -.064(then G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.063( and G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.056( are d-separation equivalent over )A f2_12 sf -.111(O)A f0_12 sf (.)S 77 383 :M -.219(The )A 99 383 :M -.08(converse )A 145 383 :M (is )S 157 383 :M -.111(not )A 176 383 :M -.182(generally )A 223 383 :M -.163(true )A 245 383 :M -.139(because )A 287 383 :M -.131(while )A 318 383 :M -.108(d-separation )A 381 383 :M -.209(equivalence )A 441 383 :M -.144(guarantees)A 59 401 :M -.17(that the conditional independence )A 219 401 :M -.089(constraints )A 274 401 :M -.094(imposed )A 318 401 :M (by )S 334 401 :M (two )S 356 401 :M -.165(path )A 380 401 :M -.123(diagrams )A 427 401 :M -.215(are )A 445 401 :M -.22(the )A 463 401 :M .087(same,)A 59 419 :M -.102(there are other, non-conditional independence constraints, that can )A 375 419 :M -.163(be )A 390 419 :M -.094(imposed )A 434 419 :M (by )S 450 419 :M -.109(one )A 471 419 :M -.22(path)A 59 437 :M -.188(diagram )A 101 437 :M -.111(but )A 120 437 :M -.111(not )A 139 437 :M -.22(the )A 157 437 :M .217 .022(other. )J 189 437 :M -.219(The )A 211 437 :M -.165(path )A 235 437 :M -.123(diagrams )A 282 437 :M -.167(in )A 295 437 :M -.054(Figure )A 330 437 :M (2 )S 340 437 :M -.215(are )A 359 437 :M -.164(examples )A 408 437 :M (of )S 423 437 :M -.165(path )A 448 437 :M -.14(diagrams)A 59 455 :M .059 .006(that are d-separation equivalent, but not covariance equivalent over )J f2_12 sf .026 .003(O = )J f0_12 sf ({)S f2_12 sf .018(X,Y,Z)A f0_12 sf (}.)S 77 473 :M -.119(If V is the )A 128 473 :M -.237(maximum )A 179 473 :M (of )S 193 473 :M -.22(the )A 211 473 :M -.109(number )A 251 473 :M (of )S 265 473 :M -.145(variables )A 311 473 :M -.167(in )A 324 473 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 341 473 :M (or )S 355 473 :M .876(G)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 376 473 :M -.109(and )A 397 473 :M -.667(M )A 411 473 :M (is )S 423 473 :M -.22(the )A 441 473 :M -.109(number )A 481 473 :M (of)S 59 491 :M -.145(variables )A 107 491 :M -.167(in )A 122 491 :M f2_12 sf .405(O)A f0_12 sf .237 .024(, )J 141 491 :M -.046(Spirtes )A 180 491 :M -.109(and )A 203 491 :M -.065(Richardson )A 263 491 :M (1996 )S 293 491 :M -.039(presents )A 338 491 :M -.163(an )A 355 491 :M -.215(O\(M)A f0_7 sf 0 -5 rm -.097(3)A 0 5 rm f0_12 sf ( )S 388 491 :M f1_12 sf .285A f0_12 sf .13 .013( )J 402 491 :M -.071(V)A f0_7 sf 0 -5 rm (2)S 0 5 rm f0_12 sf -.057(\) )A 425 491 :M -.184(algorithm )A 477 491 :M (for)S 59 509 :M -.081(checking whether two acyclic path diagrams G)A f0_7 sf 0 3 rm -.056(1)A 0 -3 rm f0_12 sf -.085( and G)A f0_7 sf 0 3 rm -.056(2)A 0 -3 rm f0_12 sf -.091( \(which may )A 382 509 :M -.189(contain )A 420 509 :M -.276(latent )A 449 509 :M -.163(variables)A 59 527 :M -.109(and )A 80 527 :M -.196(correlated )A 130 527 :M (errors\) )S 166 527 :M -.215(are )A 184 527 :M -.108(d-separation )A 246 527 :M -.198(equivalent )A 298 527 :M -.08(over )A 323 527 :M f2_12 sf .405(O)A f0_12 sf .237 .024(. )J 340 527 :M -.065(Richardson )A 399 527 :M (\(1996\) )S 436 527 :M -.039(presents )A 480 527 :M -.326(an)A 59 545 :M -.187(O\(V)A f0_7 sf 0 -5 rm -.092(7)A 0 5 rm f0_12 sf -.13(\) algorithm for determining when two cyclic path diagrams without latent )A 431 545 :M -.145(variables )A 477 545 :M -.323(are)A 59 563 :M -.099(d-separation equivalent.)A 95 593 :M f4_12 sf (4)S 102 593 :M (.)S 107 593 :M (3)S 114 593 :M (.)S 119 593 :M 9 .9( )J 131 593 :M 3.165 .316(Extracting )J 198 593 :M 2.78 .278(Features )J 255 593 :M 3.536 .354(Common )J 314 593 :M 2.222 .222(to )J 334 593 :M .909 .091(a )J 350 593 :M 3.673 .367(Covariance )J 423 593 :M .668(Equivalence)A 131 611 :M 1.082(Class)A 77 632 :M f0_12 sf -.187(Theorem )A 123 632 :M (4 )S 133 632 :M (is )S 145 632 :M -.082(also )A 168 632 :M -.22(the )A 186 632 :M (basis )S 214 632 :M (of )S 228 632 :M -.326(a )A 237 632 :M -.166(simple )A 272 632 :M -.14(representation )A 342 632 :M -.234(\(called )A 377 632 :M -.326(a )A 387 632 :M -.188(pattern )A 424 632 :M -.167(in )A 438 632 :M -.328(Verma )A 474 632 :M -.163(and)A 59 650 :M -.1(Pearl 1990\) of the entire set of path diagrams without correlated errors )A 394 650 :M (or )S 408 650 :M -.163(cycles )A 441 650 :M -.218(covariance)A 59 668 :M -.131(equivalent to a given path diagram without correlated errors )A 343 668 :M (or )S 357 668 :M (cycles. )S 394 668 :M -.219(The )A 416 668 :M -.188(pattern )A 452 668 :M (for )S 470 668 :M -.326(each)A endp %%Page: 16 16 %%BeginPageSetup initializepage (peter; page: 16 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (16)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.069(path diagram in Figure 1 is shown in Figure 10\(a\), and the pattern for each )A 416 56 :M -.165(path )A 440 56 :M -.188(diagram )A 482 56 :M -.334(in)A 59 74 :M .075 .008(Figure 3 is shown in Figure 10\(b\).)J 203 164 :M -.156(\(a\))A 311 164 :M (\(b\))S 247 188 :M f2_12 sf 3.34 .334(Figure 10)J 77 212 :M f0_12 sf -.663(A )A 89 212 :M -.188(pattern )A 125 212 :M (has )S 145 212 :M -.22(the )A 163 212 :M -.163(same )A 191 212 :M -.208(adjacencies )A 248 212 :M (as )S 262 212 :M -.22(the )A 280 212 :M -.165(path )A 304 212 :M -.123(diagrams )A 351 212 :M -.167(in )A 364 212 :M -.22(the )A 382 212 :M -.196(covariance )A 436 212 :M -.23(equivalence)A 59 230 :M -.062(class that it represents. In addition, an edge is oriented as X )A 344 230 :M f1_12 sf .126A f0_12 sf ( )S 360 230 :M -.33(Z )A 371 230 :M -.167(in )A 384 230 :M -.22(the )A 402 230 :M -.188(pattern )A 438 230 :M -.164(if )A 449 230 :M -.109(and )A 470 230 :M -.111(only)A 59 248 :M -.095(if it is oriented as X )A f1_12 sf -.264A f0_12 sf -.11( Z in every path diagram in )A 298 248 :M -.22(the )A 316 248 :M -.166(simple )A 351 248 :M -.196(covariance )A 405 248 :M -.209(equivalence )A 464 248 :M .136(class.)A 59 266 :M -.032(Meek 1995, Andersson )A 174 266 :M -.33(et )A 186 266 :M .261 .026(al. )J 202 266 :M .667 .067(1995, )J 234 266 :M -.109(and )A 255 266 :M -.132(Chickering )A 311 266 :M (1995 )S 339 266 :M .479 .048(show )J 369 266 :M .259 .026(how )J 394 266 :M -.167(to )A 407 266 :M -.204(generate )A 450 266 :M -.326(a )A 459 266 :M -.219(pattern)A 59 284 :M -.102(from an acyclic graph in O\(E\) time \(where E is the number of edges.\))A 77 302 :M (In )S 91 302 :M -.22(the )A 109 302 :M -.161(case )A 133 302 :M (of )S 147 302 :M -.282(acyclic )A 183 302 :M -.165(path )A 207 302 :M -.123(diagrams )A 254 302 :M -.065(which )A 287 302 :M -.22(may )A 311 302 :M -.082(also )A 335 302 :M -.189(contain )A 374 302 :M -.276(latent )A 404 302 :M -.031(variables, )A 455 302 :M -.109(and )A 477 302 :M -.33(the)A 59 320 :M -.161(case )A 84 320 :M (of )S 99 320 :M -.274(cyclic )A 131 320 :M -.165(path )A 157 320 :M -.123(diagrams )A 206 320 :M -.065(which )A 241 320 :M (do )S 259 320 :M -.111(not )A 280 320 :M -.189(contain )A 320 320 :M -.276(latent )A 351 320 :M -.031(variables, )A 403 320 :M -.196(there )A 432 320 :M (is )S 446 320 :M -.163(an )A 463 320 :M -.264(object)A 59 338 :M -.073(analogous )A 111 338 :M -.167(to )A 124 338 :M -.326(a )A 133 338 :M -.188(pattern )A 169 338 :M -.274(called )A 200 338 :M -.326(a )A 209 338 :M -.188(Partial )A 243 338 :M -.219(Ancestral )A 291 338 :M (Graph )S 325 338 :M .635 .064(\(PAG\), )J 365 338 :M -.065(which )A 398 338 :M -.063(represents )A 451 338 :M -.082(some )A 481 338 :M (of)S 59 356 :M -.148(the features common to the members of a covariance equivalence )A 368 356 :M -.064(class )A 395 356 :M -.08(over )A 420 356 :M f2_12 sf .405(O)A f0_12 sf .237 .024(. )J 437 356 :M -.046(Spirtes )A 474 356 :M -.163(and)A 59 374 :M -.328(Verma )A 94 374 :M (\(1992\) )S 130 374 :M .669 .067(shows )J 165 374 :M .259 .026(how )J 190 374 :M -.167(to )A 203 374 :M -.272(create )A 234 374 :M -.326(a )A 243 374 :M (PAG)S 267 371 :M f0_8 sf (6)S 271 374 :M f0_12 sf ( )S 275 374 :M -.08(from )A 303 374 :M -.163(an )A 319 374 :M -.282(acyclic )A 356 374 :M -.165(path )A 381 374 :M -.188(diagram )A 424 374 :M -.167(in )A 438 374 :M -.182(O\(V)A f0_7 sf 0 -5 rm -.089(5)A 0 5 rm f0_12 sf -.179(\) )A 471 374 :M -.443(time)A 59 392 :M -.05(\(where )A 96 392 :M -.663(V )A 108 392 :M (is )S 120 392 :M -.22(the )A 138 392 :M -.109(number )A 178 392 :M (of )S 192 392 :M -.163(vertices )A 232 392 :M -.167(in )A 245 392 :M -.22(the )A 264 392 :M -.165(path )A 289 392 :M -.034(diagram\). )A 340 392 :M -.057(Richardson\(1996c\) )A 436 392 :M -.039(presents )A 480 392 :M -.326(an)A 59 410 :M -.118(O\(V)A f0_7 sf 0 -5 rm -.058(7)A 0 5 rm f0_12 sf -.081(\) algorithm for constructing a PAG from a \(possibly cyclic\) graph.)A 95 440 :M f4_12 sf (4)S 102 440 :M (.)S 107 440 :M (4)S 114 440 :M (.)S 119 440 :M 9 .9( )J 131 440 :M 3.594 .359(Solutions to the questions on regression)J 77 461 :M f0_12 sf -.078(In this section we apply d-separation in order to answer three questions about the use of)A 59 479 :M -.031(regression )A 114 479 :M -.167(to )A 129 479 :M -.248(estimate )A 173 479 :M -.131(structural )A 223 479 :M -.163(coefficients )A 283 479 :M -.249(that )A 307 479 :M (we )S 328 479 :M -.108(raised )A 363 479 :M -.079(earlier. )A 403 479 :M -.326(We )A 426 479 :M -.146(introduce )A 477 479 :M -.33(the)A 59 496 :M .083 .008(following notation first: Given an SEM with path diagram G, we define )J f2_12 sf .03(G\\{X)A f3_12 sf .054A f2_12 sf .03(Y})A f0_12 sf ( )S 463 496 :M (as )S 477 496 :M -.33(the)A 59 514 :M -.082(path diagram in which the X )A f1_12 sf -.122A f0_12 sf -.091(Y edge is removed.)A 77 533 :M -.104(\(a\) )A 94 533 :M (If )S 106 533 :M -.663(Y )A 118 533 :M (is )S 130 533 :M -.033(regressed )A 179 533 :M (on )S 195 533 :M -.326(a )A 205 533 :M -.109(set )A 223 533 :M (of )S 238 533 :M -.145(variables )A 285 533 :M .562 .056(W, )J 305 533 :M -.148(including )A 354 533 :M 1.114 .111(X, )J 372 533 :M -.167(in )A 386 533 :M -.065(which )A 420 533 :M -.084(SEMs )A 454 533 :M -.166(will )A 477 533 :M -.33(the)A 59 551 :M -.235(partial )A 92 551 :M -.031(regression )A 145 551 :M -.208(coefficient )A 198 551 :M (of )S 212 551 :M .306 .031(X )J 226 551 :M -.163(be )A 242 551 :M -.326(a )A 252 551 :M -.099(consistent )A 304 551 :M -.248(estimate )A 347 551 :M (of )S 362 551 :M -.22(the )A 381 551 :M -.131(structural )A 430 551 :M -.208(coefficient )A 484 551 :M f1_12 sf (b)S 59 569 :M f0_12 sf -.103(associated with the X )A f1_12 sf -.159A f0_12 sf -.142(Y edge?)A 77 586 :M -.094(The coefficient of X is a consistent estimator of )A f1_12 sf -.133(b)A f0_12 sf -.067( if )A f2_12 sf -.243(W)A f0_12 sf -.101( does not contain any )A 439 586 :M -.145(descendant)A 59 604 :M -.026(of Y in G, and X is d-separated from Y given )A f2_12 sf -.063(W)A f0_12 sf -.027( in G\\{X)A f1_12 sf -.062A f0_12 sf -.046(Y}.)A 361 601 :M f0_8 sf (7)S 365 604 :M f0_12 sf ( )S 369 604 :M (If )S 381 604 :M -.084(this )A 402 604 :M -.148(condition )A 450 604 :M (does )S 476 604 :M -.167(not)A 59 626 :M ( )S 59 623.48 -.48 .48 203.48 623 .48 59 623 @a 59 636 :M f0_8 sf (6)S 63 639 :M f0_10 sf .5 .05( )J 67 639 :M (The )S 86 639 :M .327 .033(algorithm )J 129 639 :M .189 .019(given )J 155 639 :M .417 .042(by )J 169 639 :M .429 .043(Spirtes )J 201 639 :M -.313(and )A 218 639 :M -.14(Verma )A 248 639 :M (was )S 267 639 :M -.256(designed )A 304 639 :M .601 .06(to )J 316 639 :M .59 .059(output )J 346 639 :M .051 .005(an )J 359 639 :M .042 .004(object )J 387 639 :M -.229(called )A 413 639 :M .056 .006(a )J 422 639 :M .096 .01(partially )J 460 639 :M -.252(oriented)A 59 651 :M -.022(inducing path graph \(POIPG\); however, it has subsequently been shown that the output can be re-interpreted)A 59 663 :M .263 .026(as a PAG.)J 59 672 :M f0_8 sf (7)S 63 675 :M f0_10 sf .027 .003(Note this criterion is similar to Pearl's back door criterion \(Pearl, 1993\), except that the back-door )J 458 675 :M -.108(criterion)A 59 687 :M .023 .002(was proposed as a means of estimating the )J f5_10 sf (total)S f0_10 sf .017 .002( effect of X on Y.)J 228 82 18 19 rC 228 97 :M f0_12 sf (X)S gR gS 179 83 18 19 rC 179 98 :M f0_12 sf (W)S gR gS 185 121 18 19 rC 185 136 :M f0_12 sf (Y)S gR gS 228 122 18 19 rC 228 137 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 196 96 -1 1 226 95 1 196 95 @a 199 132 -1 1 224 131 1 199 131 @a -1 -1 232 122 1 1 231 98 @b 194 99 -1 1 224 121 1 194 98 @a 343 82 18 19 rC 343 97 :M f0_12 sf (X)S gR gS 294 83 18 19 rC 294 98 :M f0_12 sf (W)S gR gS 300 121 18 19 rC 300 136 :M f0_12 sf (Y)S gR gS 343 122 18 19 rC 343 137 :M f0_12 sf (Z)S gR gS 0 0 552 730 rC 311 96 -1 1 341 95 1 311 95 @a 314 131.75 -.75 .75 332.75 131 .75 314 131 @a np 330 129 :M 330 133 :L 338 131 :L 330 129 :L .75 lw eofill -.75 -.75 330.75 133.75 .75 .75 330 129 @b -.75 -.75 330.75 133.75 .75 .75 338 131 @b 330 129.75 -.75 .75 338.75 131 .75 330 129 @a -.75 -.75 346.75 115.75 .75 .75 346 98 @b np 348 113 :M 344 113 :L 346 121 :L 348 113 :L eofill 344 113.75 -.75 .75 348.75 113 .75 344 113 @a 344 113.75 -.75 .75 346.75 121 .75 344 113 @a -.75 -.75 346.75 121.75 .75 .75 348 113 @b endp %%Page: 17 17 %%BeginPageSetup initializepage (peter; page: 17 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (17)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf .444 .044(hold, )J 88 56 :M -.165(then )A 112 56 :M (for )S 130 56 :M -.166(almost )A 165 56 :M -.331(all )A 180 56 :M -.142(instantiations )A 246 56 :M (of )S 260 56 :M -.22(the )A 278 56 :M -.163(parameters )A 333 56 :M -.167(in )A 346 56 :M -.22(the )A 364 56 :M .237 .024(SEM, )J 396 56 :M -.22(the )A 414 56 :M -.208(coefficient )A 467 56 :M (of )S 482 56 :M (X)S 59 73 :M -.087(will fail to be a consistent estimator of )A f1_12 sf -.129(b)A f0_12 sf (.)S 77 91 :M -.09(It follows directly from this )A 211 91 :M -.249(that )A 232 91 :M -.141(\(almost )A 271 91 :M -.045(surely\) )A 308 91 :M f1_12 sf .285(b)A f0_12 sf .13 .013( )J 319 91 :M -.164(cannot )A 354 91 :M -.163(be )A 369 91 :M -.22(estimated )A 417 91 :M -.11(consistently )A 477 91 :M -.33(via)A 59 109 :M -.082(any regression equation if either there is an edge X )A f1_12 sf -.137A f0_12 sf -.073(Y \(i.e. )A f1_12 sf -.093(e)A f0_10 sf 0 2 rm -.127(X)A 0 -2 rm f0_12 sf -.082( and )A f1_12 sf -.093(e)A f0_10 sf 0 2 rm -.086(Y )A 0 -2 rm f0_12 sf -.082(are correlated\) or if)A 59 128 :M .14 .014(X is )J 83 128 :M -.326(a )A 92 128 :M -.131(descendant )A 148 128 :M (of )S 162 128 :M -.663(Y )A 174 128 :M .261 .026(\(so )J 193 128 :M -.249(that )A 214 128 :M -.22(the )A 232 128 :M -.165(path )A 256 128 :M -.188(diagram )A 298 128 :M (is )S 310 128 :M -.08(cyclic\). )A 349 128 :M -.219(The )A 371 128 :M -.109(result )A 401 128 :M -.165(itself )A 428 128 :M (follows )S 468 128 :M -.107(from)A 59 146 :M -.112(the fact that under the conditions stated,)A 173 181 :M .059(Cov\(X,)A f1_12 sf .052(e)A f0_10 sf 0 2 rm .071(Y)A 0 -2 rm f0_12 sf .036 .004( | )J f2_12 sf .119(W)A f0_12 sf .159 .016(\\{X}\) = Cov\(X,Q | )J f2_12 sf .119(W)A f0_12 sf .164 .016(\\{X}\) = 0)J 59 217 :M (for )S 78 217 :M -.245(each )A 104 217 :M -.127(Q)A f0_10 sf 0 2 rm ( )S 0 -2 rm 117 217 :M f1_12 sf .405 .04J 131 217 :M f2_12 sf .281(Parents)A f0_12 sf 1.446 .145(\(Y,G\)\\{X}. )J 235 217 :M .181<28>A f2_12 sf .254(Parents)A f0_12 sf .946 .095(\(Y,G\) )J 315 217 :M (is )S 328 217 :M -.22(the )A 347 217 :M -.109(set )A 365 217 :M (of )S 380 217 :M -.092(parents )A 419 217 :M (of )S 434 217 :M -.663(Y )A 447 217 :M -.167(in )A 461 217 :M 1.033 .103(G.\) )J 484 217 :M -.327(It)A 59 236 :M -.09(follows that)A 101 257 365 27 rC 466 284 :M psb currentpoint pse 101 257 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11680 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate -12 346 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 629 346 moveto (\() show 773 346 moveto (X) show 1052 346 moveto (,) show 1185 346 moveto (Y) show 1460 346 moveto (|) show 1567 346 moveto 384 /Times-Bold f1 (W) show 2069 346 moveto 384 /Times-Roman f1 (\\) show 2261 346 moveto ({) show 2419 346 moveto (X) show 2669 346 moveto (}) show 2832 346 moveto (\)) show 3049 346 moveto 384 /Symbol f1 (=) show 3355 346 moveto 384 /Times-Roman f1 (Cov) show 3996 346 moveto (\() show 4140 346 moveto (X) show 4419 346 moveto (,) show 4537 346 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (b) show 4766 346 moveto 384 /Times-Roman f1 (X) show 5130 346 moveto 384 /Symbol f1 (+) show 5874 346 moveto 384 /Times-Roman f1 (a) show 6071 442 moveto 224 ns (i) show 6155 346 moveto 384 ns (T) show 6411 442 moveto 224 ns (i) show 5509 719 moveto (T) show 5671 776 moveto 160 ns (i) show 5424 433 moveto 576 /Symbol f1 (\345) show 6594 346 moveto 384 ns (+) show 6888 346 moveto 384 /Symbol f2 (e) show 7084 442 moveto 224 /Times-Roman f1 (Y) show 7279 346 moveto 384 ns (|) show 7386 346 moveto 384 /Times-Bold f1 (W) show 7888 346 moveto 384 /Times-Roman f1 (\\) show 8080 346 moveto ({) show 8238 346 moveto (X) show 8488 346 moveto (}) show 8651 346 moveto (\)) show 8868 346 moveto 384 /Symbol f1 (=) show 9163 346 moveto 384 /Symbol f2 (b) show 9392 346 moveto 384 /Times-Roman f1 (V) show 9675 346 moveto (\() show 9819 346 moveto (X|) show 10203 346 moveto 384 /Times-Bold f1 (W) show 10705 346 moveto 384 /Times-Roman f1 (\\) show 10897 346 moveto ({) show 11055 346 moveto (X) show 11305 346 moveto (}) show 11468 346 moveto (\)) show end pse gR gS 0 0 552 730 rC 59 319 :M f0_12 sf -.145(and hence )A 109 302 116 30 rC 225 332 :M psb currentpoint pse 109 302 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3712 div 960 3 -1 roll exch div scale currentpoint translate 64 44 translate 20 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 661 261 moveto 384 /Times-Roman f1 (\() show 805 261 moveto 384 /Times-Roman f1 (X) show 1084 261 moveto 384 /Times-Roman f1 (,) show 1217 261 moveto 384 /Times-Roman f1 (Y) show 1492 261 moveto 384 /Times-Roman f1 (|) show 1599 261 moveto 384 /Times-Bold f1 (W) show 2101 261 moveto 384 /Times-Roman f1 (\\) show 2293 261 moveto ({) show 2451 261 moveto 384 /Times-Roman f1 (X) show 2701 261 moveto 384 /Times-Roman f1 (}) show 2864 261 moveto (\)) show 406 793 moveto 384 /Times-Roman f1 (V) show 689 793 moveto 384 /Times-Roman f1 (\() show 833 793 moveto 384 /Times-Roman f1 (X) show 1110 793 moveto 384 /Times-Roman f1 (|) show 1217 793 moveto 384 /Times-Bold f1 (W) show 1719 793 moveto 384 /Times-Roman f1 (\\) show 1911 793 moveto ({) show 2069 793 moveto 384 /Times-Roman f1 (X) show 2319 793 moveto 384 /Times-Roman f1 (}) show 2482 793 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 401 moveto 3005 0 rlineto stroke 3113 500 moveto 384 /Symbol f1 (=) show 3408 500 moveto 384 /Symbol f1 (b) show end pse gR gS 0 0 552 730 rC 225 319 :M f0_12 sf (.)S 77 347 :M ( \(b\) )S 99 347 :M (If )S 112 347 :M -.663(Y )A 125 347 :M (is )S 138 347 :M -.033(regressed )A 188 347 :M (on )S 205 347 :M -.22(the )A 224 347 :M -.109(set )A 242 347 :M f2_12 sf .667(W)A f0_12 sf .303 .03(, )J 264 347 :M -.148(including )A 314 347 :M 1.114 .111(X, )J 333 347 :M -.167(in )A 348 347 :M -.065(which )A 383 347 :M -.084(SEMs )A 418 347 :M -.166(will )A 442 347 :M -.22(the )A 462 347 :M -.275(partial)A 59 365 :M -.101(regression coefficient of X be zero if there is no edge between X and Y?)A 77 383 :M -.219(The )A 100 383 :M -.208(coefficient )A 154 383 :M (of )S 169 383 :M .306 .031(X )J 183 383 :M -.166(will )A 206 383 :M -.163(be )A 222 383 :M -.161(zero )A 247 383 :M -.164(if )A 260 383 :M .306 .031(X )J 275 383 :M -.109(and )A 298 383 :M -.663(Y )A 312 383 :M -.215(are )A 332 383 :M -.117(d-separated )A 392 383 :M -.132(given )A 424 383 :M f2_12 sf .429(W)A f0_12 sf .703 .07(\\{X}. )J 470 383 :M -.105(\(See)A 59 401 :M -.009(Scheines 1994 and Glymour 1994\). )A 233 401 :M -.083(This )A 258 401 :M (follows )S 298 401 :M -.206(directly )A 337 401 :M -.08(from )A 364 401 :M -.22(the )A 382 401 :M -.245(fact )A 403 401 :M -.249(that )A 424 401 :M -.22(the )A 442 401 :M -.229(coefficient)A 59 418 :M (of )S 73 418 :M .306 .031(X )J 86 418 :M -.167(in )A 99 418 :M -.22(the )A 117 418 :M -.031(regression )A 170 418 :M -.165(equation )A 214 418 :M (is )S 226 418 :M -.109(proportional )A 288 418 :M -.167(to )A 301 418 :M f1_12 sf .297(r)A f0_12 sf .247(\(X,Y,)A f2_12 sf .542(W)A f0_12 sf .945 .095(\\{X}\), )J 385 418 :M -.065(which )A 419 418 :M -.167(in )A 433 418 :M -.082(turn )A 457 418 :M -.166(will )A 480 418 :M -.326(be)A 59 437 :M -.056(zero if {X} is d-separated from {Y} )A 234 437 :M -.132(given )A 264 437 :M f2_12 sf .429(W)A f0_12 sf .703 .07(\\{X}. )J 308 437 :M -.165(As )A 325 437 :M .212 .021(before, )J 363 437 :M -.164(if )A 374 437 :M .629 .063({X} )J 399 437 :M -.109(and )A 420 437 :M -.061({Y} )A 444 437 :M -.215(are )A 462 437 :M -.111(not )A 481 437 :M (d-)S 59 455 :M -.144(separated )A 107 455 :M -.132(given )A 137 455 :M f2_12 sf .429(W)A f0_12 sf .703 .07(\\{X}, )J 181 455 :M .226 .023(then, )J 209 455 :M -.163(even )A 235 455 :M -.164(if )A 246 455 :M -.196(there )A 273 455 :M (is )S 285 455 :M (no )S 301 455 :M -.163(edge )A 328 455 :M -.139(between )A 372 455 :M .306 .031(X )J 386 455 :M -.109(and )A 408 455 :M .281 .028(Y, )J 425 455 :M (for )S 444 455 :M -.166(almost )A 480 455 :M -.497(all)A 59 473 :M -.098(assignments of values to the model parameters the coefficient of X will be non-zero.)A 77 490 :M -.104(\(c\) )A 94 490 :M -.065(Given )A 127 490 :M -.326(a )A 136 490 :M -.197(particular )A 184 490 :M .237 .024(SEM, )J 216 490 :M -.083(with )A 241 490 :M -.165(path )A 265 490 :M -.188(diagram )A 307 490 :M 1.114 .111(G, )J 324 490 :M -.167(in )A 337 490 :M -.065(which )A 370 490 :M -.196(there )A 397 490 :M (is )S 409 490 :M -.163(an )A 424 490 :M -.163(edge )A 450 490 :M .306 .031(X )J 463 490 :M f1_12 sf .144 .014J 479 490 :M f0_12 sf .337(Y,)A 59 508 :M -.164(with coefficient )A 136 508 :M f1_12 sf .74(b)A f0_12 sf .613 .061(, )J 151 508 :M (is )S 163 508 :M -.334(it )A 173 508 :M -.041(possible )A 216 508 :M -.167(to )A 229 508 :M -.082(find )A 252 508 :M -.326(a )A 261 508 :M (subset )S 295 508 :M f2_12 sf (W)S f0_12 sf ( )S 311 508 :M (of )S 325 508 :M -.039(observed )A 372 508 :M -.031(variables, )A 422 508 :M -.132(\(including )A 474 508 :M .672(X\),)A 59 527 :M (such )S 85 527 :M -.249(that )A 106 527 :M (when )S 136 527 :M -.663(Y )A 148 527 :M (is )S 160 527 :M -.033(regressed )A 209 527 :M (on )S 226 527 :M -.22(the )A 245 527 :M -.109(set )A 263 527 :M f2_12 sf .667(W)A f0_12 sf .303 .03(, )J 284 527 :M -.22(the )A 303 527 :M -.208(coefficient )A 357 527 :M (of )S 372 527 :M .306 .031(X )J 386 527 :M -.167(in )A 400 527 :M -.22(the )A 419 527 :M -.031(regression )A 473 527 :M (is )S 486 527 :M (a)S 59 544 :M -.112(consistent estimate of )A f1_12 sf -.159(b)A f0_12 sf (?)S 77 563 :M -.04(From \(a\), we know that if there is a subset )A f2_12 sf -.103(W)A f0_12 sf -.043( of the observed )A 373 563 :M -.145(variables )A 419 563 :M -.065(which )A 452 563 :M -.141(contains)A 59 580 :M (no )S 75 580 :M -.131(descendant )A 131 580 :M (of )S 145 580 :M .281 .028(Y, )J 161 580 :M -.111(but )A 180 580 :M -.065(which )A 213 580 :M -.087(d-separates )A 270 580 :M .306 .031(X )J 283 580 :M -.08(from )A 310 580 :M -.663(Y )A 322 580 :M -.167(in )A 336 580 :M .148(G\\{X)A f1_12 sf .265A f0_12 sf .352 .035(Y}, )J 398 580 :M -.165(then )A 423 580 :M -.22(the )A 442 580 :M -.034(regression)A 59 598 :M -.084(coefficient of X in the regression of Y on )A f2_12 sf -.216(W)A f0_12 sf -.082( will be a consistent estimate of )A f1_12 sf -.118(b)A f0_12 sf (.)S 59 629 :M f2_14 sf (5)S 67 629 :M (.)S 72 629 :M 19.5 1.95( )J 95 629 :M .679(Conclusion)A 77 650 :M f0_12 sf -.113(D-separation is a widely studied graphical relation, )A 320 650 :M -.065(which )A 353 650 :M (has )S 373 650 :M -.053(proved )A 410 650 :M -.053(useful )A 443 650 :M -.167(in )A 456 650 :M -.056(solving)A 59 668 :M -.165(many )A 89 668 :M .182 .018(problems. )J 141 668 :M -.326(We )A 161 668 :M -.163(have )A 187 668 :M -.18(illustrated )A 237 668 :M -.082(some )A 266 668 :M (of )S 280 668 :M -.112(its )A 295 668 :M -.075(applications, )A 360 668 :M (such )S 387 668 :M (as )S 402 668 :M -.094(finding )A 441 668 :M -.218(covariance)A 59 686 :M -.209(equivalence )A 119 686 :M .381 .038(classes, )J 161 686 :M -.094(finding )A 200 686 :M -.121(features )A 242 686 :M -.166(common )A 288 686 :M -.167(to )A 302 686 :M -.331(all )A 318 686 :M -.14(members )A 366 686 :M (of )S 381 686 :M -.196(covariance )A 436 686 :M -.23(equivalence)A endp %%Page: 18 18 %%BeginPageSetup initializepage (peter; page: 18 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (18)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf .381 .038(classes, )J 104 56 :M -.109(and )A 129 56 :M -.165(explaining )A 186 56 :M -.101(relationships )A 254 56 :M -.139(between )A 301 56 :M -.031(regression )A 358 56 :M -.163(coefficients )A 421 56 :M -.109(and )A 447 56 :M -.145(structural)A 59 74 :M -.165(equation )A 103 74 :M -.074(coefficients. )A 165 74 :M -.263(Other )A 195 74 :M -.165(applications )A 255 74 :M -.167(to )A 268 74 :M -.082(problems )A 317 74 :M -.167(in )A 331 74 :M -.131(structural )A 380 74 :M -.165(equation )A 425 74 :M -.185(modelling )A 477 74 :M -.323(are)A 59 92 :M -.108(described )A 109 92 :M -.167(in )A 124 92 :M -.046(Spirtes )A 163 92 :M -.33(et )A 177 92 :M .261 .026(al. )J 195 92 :M (\(1993\) )S 233 92 :M -.109(and )A 256 92 :M .165 .016(Pearl\(1997\). )J 322 92 :M -.054(Instructions )A 384 92 :M (for )S 404 92 :M (using )S 436 92 :M -.326(a )A 447 92 :M .051(program,)A 59 110 :M -.387(TETRAD )A 108 110 :M .78 .078(II, )J 124 110 :M -.249(that )A 145 110 :M .243 .024(uses )J 170 110 :M -.108(d-separation )A 232 110 :M (as )S 247 110 :M -.326(a )A 257 110 :M (basis )S 286 110 :M (for )S 305 110 :M -.108(searching )A 355 110 :M (for )S 374 110 :M -.22(automated )A 427 110 :M -.108(searching )A 477 110 :M (for)S 59 128 :M -.11(models )A 97 128 :M -.132(given )A 127 128 :M -.247(data )A 151 128 :M -.109(and )A 173 128 :M -.065(background )A 234 128 :M -.072(knowledge )A 291 128 :M -.217(can )A 312 128 :M -.163(be )A 328 128 :M (found )S 361 128 :M (on )S 378 128 :M -.22(the )A 397 128 :M (world )S 430 128 :M -.081(wide )A 458 128 :M (web )S 483 128 :M -.66(at)A 59 146 :M -.11(http://hss.cmu.edu/html/departments/philosophy/TETRAD/tetrad.html. )A 401 146 :M (By )S 423 146 :M -.147(extending )A 477 146 :M -.33(the)A 59 164 :M -.165(relation to path )A 133 164 :M (diagrams, )S 184 164 :M (as )S 198 164 :M -.164(well )A 222 164 :M (as )S 236 164 :M .596 .06(graphs, )J 276 164 :M (we )S 294 164 :M -.163(have )A 320 164 :M .447 .045(shown )J 356 164 :M .259 .026(how )J 381 164 :M -.123(theorems )A 428 164 :M -.053(proved )A 465 164 :M -.165(about)A 59 182 :M -.097(d-separation can be applied to a wider class of structural equation models.)A 59 212 :M f2_14 sf (6)S 67 212 :M (.)S 72 212 :M 19.5 1.95( )J 95 212 :M 3.244 .324(Proofs of Main Results)J 77 239 :M f0_12 sf -.077(We will prove Theorem 1 and Theorem 2 in two steps. First we will prove them )A 459 239 :M (for )S 477 239 :M -.33(the)A 59 257 :M -.161(case )A 83 257 :M -.062(where )A 116 257 :M -.079(G\(M\) )A 147 257 :M -.123(contains )A 191 257 :M (no )S 208 257 :M -.125(double-headed )A 282 257 :M (arrows; )S 323 257 :M -.165(then )A 348 257 :M (we )S 367 257 :M -.166(will )A 390 257 :M -.064(prove )A 422 257 :M -.334(it )A 433 257 :M (for )S 452 257 :M -.22(the )A 471 257 :M -.215(case)A 59 275 :M -.05(where G\(M\) does contain double-headed arrows.)A 77 299 :M .275 .027(A probability measure P over )J f2_12 sf .13(V)A f0_12 sf .208 .021( satisfies the )J f2_12 sf .341 .034(global directed )J 377 299 :M 1.673 .167(Markov )J 425 299 :M .315(property)A f0_12 sf .167 .017( )J 477 299 :M (for)S 59 317 :M -.101(path diagram G if and only if for any three disjoint )A 301 317 :M (sets )S 323 317 :M (of )S 337 317 :M -.145(variables )A 383 317 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(, )J 401 317 :M f2_12 sf .79(Y)A f0_12 sf .497 .05(, )J 418 317 :M -.109(and )A 439 317 :M f2_12 sf (Z)S f0_12 sf ( )S 451 317 :M -.189(included)A 59 335 :M (in )S f2_12 sf (V)S f0_12 sf (, if )S f2_12 sf (X)S f0_12 sf .032 .003( is d-separated from )J f2_12 sf (Y)S f0_12 sf .024 .002( given )J f2_12 sf (Z)S f0_12 sf .022 .002(, then )J f2_12 sf (X)S f0_12 sf .029 .003( is independent of )J f2_12 sf (Y)S f0_12 sf .024 .002( given )J f2_12 sf (Z)S f0_12 sf ( in P.)S 77 359 :M -.158(The following lemma relates )A 215 359 :M -.22(the )A 233 359 :M -.166(global )A 266 359 :M -.205(directed )A 307 359 :M -.164(Markov )A 348 359 :M -.081(property )A 392 359 :M -.167(to )A 405 359 :M -.164(factorizations )A 472 359 :M (of )S 486 359 :M (a)S 59 377 :M -.048(density function. Denote a density function over )A f2_12 sf -.087(V)A f0_12 sf -.043( by f\()A f2_12 sf -.087(V)A f0_12 sf -.053(\), where )A 377 377 :M (for )S 395 377 :M -.109(any )A 416 377 :M (subset )S 450 377 :M f2_12 sf .993(X)A f0_12 sf .344 .034( )J 464 377 :M (of )S 478 377 :M f2_12 sf .993(V)A f0_12 sf (,)S 59 395 :M (f\()S f2_12 sf (X)S f0_12 sf .043 .004(\) denotes the marginal of f\()J f2_12 sf (V)S f0_12 sf .027 .003(\). Let )J f2_12 sf .016(An\(X\))A f0_12 sf .034 .003( be the set of ancestors )J 390 395 :M (of )S 404 395 :M -.14(members )A 451 395 :M (of )S 465 395 :M f2_12 sf 1.381(X)A f0_12 sf .869 .087(. )J 483 395 :M (If)S 59 413 :M .06(f\()A f2_12 sf .131(V)A f0_12 sf .096 .01(\) )J 84 413 :M (is )S 96 413 :M -.22(the )A 114 413 :M -.094(density )A 152 413 :M -.123(function )A 195 413 :M (for )S 213 413 :M -.326(a )A 222 413 :M -.15(probability )A 277 413 :M -.139(measure )A 320 413 :M -.08(over )A 345 413 :M -.326(a )A 355 413 :M -.109(set )A 373 413 :M (of )S 388 413 :M -.145(variables )A 435 413 :M f2_12 sf .79(V)A f0_12 sf .497 .05(, )J 453 413 :M (say )S 474 413 :M -.331(that)A 59 431 :M .385(f\()A f2_12 sf .836(V)A f0_12 sf .562 .056(\) )J f2_12 sf 2.025 .202(factors according to )J 201 431 :M 2.237 .224(directed )J 251 431 :M 1.554 .155(graph )J 288 431 :M f0_12 sf .306 .031(G )J 301 431 :M -.083(with )A 326 431 :M -.163(vertices )A 366 431 :M f2_12 sf .25(V)A f0_12 sf .087 .009( )J 379 431 :M -.164(if )A 390 431 :M -.109(and )A 411 431 :M -.083(only )A 436 431 :M -.164(if )A 447 431 :M (for )S 465 431 :M -.161(every)A 59 449 :M .752 .075(subset )J f2_12 sf .336(X)A f0_12 sf .258 .026( of )J f2_12 sf .336(V)A f0_12 sf (,)S 188 482 174 26 rC 362 508 :M psb currentpoint pse 188 482 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5568 div 832 3 -1 roll exch div scale currentpoint translate 64 40 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (f\() -7 344 sh (\() 747 344 sh (\)\)) 1170 344 sh (g) 2585 344 sh (\(V,) 2994 344 sh (\() 4785 344 sh (\)) 5209 344 sh 224 ns (V) 2793 440 sh (V) 1860 714 sh (n\(X\)) 2341 714 sh 384 /Times-Bold f1 (An) 258 344 sh (X) 885 344 sh (Parents) 3530 344 sh (V) 4923 344 sh 224 ns (A) 2180 714 sh 384 /Symbol f1 (=) 1530 344 sh 224 ns (\316) 2039 714 sh 576 ns (\325) 2069 432 sh 384 /Times-Roman f1 (\)) 5337 344 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 535 :M f0_12 sf -.077(where g)A f0_10 sf 0 2 rm -.102(V)A 0 -2 rm f0_12 sf -.068( is a non-negative function.)A 77 559 :M -.33(Lemma )A 117 559 :M (1 )S 128 559 :M .264 .026(was )J 152 559 :M -.053(proved )A 190 559 :M -.167(in )A 204 559 :M -.219(Lauritzen )A 253 559 :M -.33(et )A 266 559 :M .261 .026(al. )J 283 559 :M (\(1990\) )S 320 559 :M (for )S 339 559 :M -.22(the )A 358 559 :M -.282(acyclic )A 395 559 :M .236 .024(case, )J 424 559 :M -.109(and )A 446 559 :M -.22(the )A 465 559 :M (proof)S 59 577 :M -.108(carries over essentially unchanged for the cyclic case.)A 77 601 :M f2_12 sf 1.122 .112(Lemma )J 122 601 :M .387(1:)A f0_12 sf .232 .023( )J 137 601 :M (If )S 149 601 :M f2_12 sf .25(V)A f0_12 sf .087 .009( )J 162 601 :M (is )S 174 601 :M -.326(a )A 183 601 :M -.109(set )A 200 601 :M (of )S 214 601 :M -.109(random )A 254 601 :M -.145(variables )A 300 601 :M -.083(with )A 325 601 :M -.326(a )A 334 601 :M -.15(probability )A 389 601 :M -.139(measure )A 432 601 :M .299 .03(P )J 443 601 :M -.249(that )A 465 601 :M (has )S 486 601 :M (a)S 59 619 :M -.094(density )A 97 619 :M -.123(function )A 140 619 :M .06(f\()A f2_12 sf .131(V)A f0_12 sf .096 .01(\) )J 165 619 :M -.109(and )A 186 619 :M .06(f\()A f2_12 sf .131(V)A f0_12 sf .096 .01(\) )J 211 619 :M -.091(factors )A 247 619 :M -.145(according )A 297 619 :M -.167(to )A 310 619 :M -.205(directed )A 351 619 :M -.064(graph )A 382 619 :M 1.114 .111(G, )J 399 619 :M -.165(then )A 423 619 :M .299 .03(P )J 434 619 :M -.072(satisfies )A 477 619 :M -.33(the)A 59 637 :M -.079(global directed Markov property for G.)A 77 661 :M -.03(Lemma 2 was proved in Spirtes\(1995\) and Koster\(1995\).)A endp %%Page: 19 19 %%BeginPageSetup initializepage (peter; page: 19 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (19)S gR gS 0 0 552 730 rC 77 56 :M f2_12 sf -.017(Lemma 2:)A f0_12 sf -.013( If M is a SEM, and {X} and {Y} are d-separated given )A f2_12 sf (Z)S f0_12 sf ( in )S 423 56 :M -.205(directed )A 464 56 :M -.08(graph)A 59 74 :M .362 .036(G\(M\), then )J f1_12 sf .13(r)A f0_12 sf .108(\(X,Y.)A f2_12 sf .158(Z)A f0_12 sf .153 .015(\) = 0 in )J f1_12 sf .14(S)A f0_12 sf .142(\(M\).)A 77 98 :M -.036(Lemma 3 was proved in Spirtes\(1995\).)A 77 122 :M f2_12 sf .225 .023(Lemma 3:)J f0_12 sf .115 .012( For any directed )J 216 122 :M -.064(graph )A 247 122 :M 1.114 .111(G, )J 264 122 :M -.164(if )A 275 122 :M .629 .063({X} )J 300 122 :M -.109(and )A 321 122 :M -.061({Y} )A 345 122 :M -.215(are )A 363 122 :M -.111(not )A 382 122 :M -.117(d-separated )A 440 122 :M -.132(given )A 470 122 :M f2_12 sf (Z)S f0_12 sf ( )S 482 122 :M -.334(in)A 59 140 :M .112 .011(G\(M\), there is a SEM M, G = G\(M\) and )J f1_12 sf .052(r)A f0_12 sf .043(\(X,Y.)A f2_12 sf .063(Z)A f0_12 sf .061 .006(\) )J cF f1_12 sf .006A sf .061 .006( 0 in )J f1_12 sf .056(S)A f0_12 sf .057(\(M\).)A 77 164 :M -.092(We will now show that Theorem 1 and Theorem 2 hold )A 343 164 :M -.163(even )A 369 164 :M (when )S 399 164 :M .306 .031(G )J 412 164 :M -.123(contains )A 455 164 :M -.109(double-)A 59 182 :M -.056(headed arrows. Let the set of vertices in G be )A f2_12 sf -.104(V)A f0_12 sf -.055(. For a given triple )A 377 182 :M 1.114 .111(X, )J 394 182 :M .281 .028(Y, )J 410 182 :M -.109(and )A 431 182 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 447 182 :M -.164(if )A 458 182 :M .629 .063({X} )J 483 182 :M (is)S 59 200 :M -.117(d-separated )A 118 200 :M -.08(from )A 146 200 :M -.061({Y} )A 171 200 :M -.132(given )A 202 200 :M f2_12 sf (Z)S f0_12 sf ( )S 215 200 :M -.167(in )A 229 200 :M -.079(G\(M\) )A 261 200 :M -.109(and )A 283 200 :M -.079(G\(M\) )A 315 200 :M -.123(contains )A 359 200 :M -.125(double-headed )A 434 200 :M .798 .08(arrows, )J 477 200 :M -.33(the)A 59 218 :M -.094(strategy is to )A 123 218 :M -.14(convert )A 162 218 :M -.667(M )A 176 218 :M -.167(into )A 198 218 :M -.14(another )A 237 218 :M -.223(SEM )A 265 218 :M .119(M\325\(M,X,Y,)A f2_12 sf .154(Z)A f0_12 sf .122 .012(\) )J 338 218 :M (such )S 364 218 :M -.249(that )A 385 218 :M .121(G\(M\325\(M,X,Y,)A f2_12 sf .157(Z)A f0_12 sf .179 .018(\)\) )J 475 218 :M (has)S 59 236 :M -.199(additional )A 112 236 :M -.276(latent )A 145 236 :M -.031(variables, )A 199 236 :M -.111(but )A 222 236 :M (no )S 242 236 :M -.125(double-headed )A 319 236 :M .798 .08(arrows, )J 364 236 :M -.22(the )A 386 236 :M -.206(marginal )A 435 236 :M -.08(over )A 464 236 :M f2_12 sf .25(V)A f0_12 sf .087 .009( )J 481 236 :M (of)S 59 254 :M f1_12 sf .188(S)A f0_12 sf .158(\(M\325\(M,X,Y,)A f2_12 sf .212(Z)A f0_12 sf .242 .024(\)\) )J 150 254 :M (is )S 164 254 :M -.197(equal )A 196 254 :M -.167(to )A 212 254 :M f1_12 sf .278(S)A f0_12 sf .69 .069(\(M\), )J 249 254 :M -.109(and )A 273 254 :M .629 .063({X} )J 301 254 :M -.109(and )A 325 254 :M -.061({Y} )A 352 254 :M -.215(are )A 373 254 :M -.117(d-separated )A 434 254 :M -.132(given )A 467 254 :M f2_12 sf (Z)S f0_12 sf ( )S 482 254 :M -.334(in)A 59 272 :M .186(G\(M\325\(M,X,Y,)A f2_12 sf .24(Z)A f0_12 sf .323 .032(\)\). )J 155 272 :M -.215(\(We )A 181 272 :M -.13(write )A 211 272 :M .119(M\325\(M,X,Y,)A f2_12 sf .154(Z)A f0_12 sf .122 .012(\) )J 286 272 :M -.167(in )A 301 272 :M -.062(order )A 332 272 :M -.167(to )A 347 272 :M -.182(emphasize )A 402 272 :M -.249(that )A 425 272 :M -.22(the )A 446 272 :M -.223(SEM )A 477 272 :M -.66<4DD5>A 59 290 :M -.088(constructed from M is a function of the path diagram of M, and the vertices X, )A 433 290 :M .281 .028(Y, )J 449 290 :M -.109(and )A 470 290 :M f2_12 sf (Z)S f0_12 sf ( )S 482 290 :M -.334(in)A 59 308 :M -.22(the )A 79 308 :M -.108(d-separation )A 144 308 :M -.206(relation )A 186 308 :M -.132(being )A 219 308 :M (considered.\) )S 285 308 :M -.164(It )A 299 308 :M -.166(will )A 324 308 :M -.165(then )A 351 308 :M -.054(follow )A 389 308 :M -.08(from )A 419 308 :M -.33(Lemma )A 461 308 :M (2 )S 474 308 :M -.331(that)A 59 326 :M f1_12 sf .196(r)A f0_12 sf .163(\(X,Y.)A f2_12 sf .239(Z)A f0_12 sf .232 .023(\) = 0 in )J f1_12 sf .212(S)A f0_12 sf .215(\(M\).)A 77 350 :M (If )S 90 350 :M .629 .063({X} )J 116 350 :M (is )S 129 350 :M -.117(d-separated )A 188 350 :M -.08(from )A 216 350 :M -.061({Y} )A 241 350 :M -.132(given )A 272 350 :M f2_12 sf (Z)S f0_12 sf ( )S 286 350 :M -.167(in )A 301 350 :M .456 .046(G\(M\), )J 338 350 :M -.22(the )A 358 350 :M -.064(graph )A 391 350 :M .121(G\(M\325\(M,X,Y,)A f2_12 sf .157(Z)A f0_12 sf .179 .018(\)\) )J 483 350 :M (is)S 59 368 :M -.093(constructed by )A 132 368 :M -.22(the )A 150 368 :M -.073(following )A 200 368 :M -.066(algorithm, )A 253 368 :M -.062(where )A 286 368 :M -.326(a )A 295 368 :M f2_12 sf .15(trek)A f0_12 sf .084 .008( )J 321 368 :M -.139(between )A 364 368 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 379 368 :M -.109(and )A 400 368 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 415 368 :M (is )S 427 368 :M -.163(an )A 442 368 :M -.182(undirected)A 59 386 :M -.086(path between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.086( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.077( that )A 193 386 :M -.123(contains )A 236 386 :M (no )S 252 386 :M -.031(colliders. )A 300 386 :M -.215(\(We )A 324 386 :M -.166(will )A 346 386 :M -.198(illustrate )A 390 386 :M -.22(the )A 408 386 :M -.21(application )A 463 386 :M (of )S 477 386 :M -.33(the)A 59 404 :M -.099(algorithm to the path diagram in Figure 11.\))A 159 446 :M f2_12 sf 2.028 .203(Algorithm: Construct Latent Directed Graph)J 412 446 :M f0_12 sf ( )S 77 464 :M f2_12 sf (Inputs)S f0_12 sf -.004( - Path Diagram G with vertex set )A f2_12 sf (V)S f0_12 sf -.004(, Vertices X, Y, )A f2_12 sf (Z)S f0_12 sf (;)S 77 482 :M f2_12 sf .052(Output)A f0_12 sf .136 .014( - Directed Graph G)J f0_7 sf 0 3 rm .025(Construct)A 0 -3 rm f0_12 sf .047(\(G,X,Y,)A f2_12 sf .067(Z)A f0_12 sf .118 .012(\), with vertex set )J f2_12 sf .072(V)A f1_12 sf .077A f2_12 sf .067(T)A f0_12 sf (;)S 77 500 :M -.071(1. Order the variables so that X )A 229 500 :M (is )S 241 500 :M .424 .042(first, )J 268 500 :M -.663(Y )A 280 500 :M (is )S 292 500 :M .401 .04(second, )J 333 500 :M -.081(followed )A 379 500 :M (by )S 395 500 :M -.245(each )A 420 500 :M -.205(variable )A 461 500 :M -.083(with )A 486 500 :M (a)S 59 518 :M -.049(descendant in )A f2_12 sf -.081(Z)A f0_12 sf -.052(, followed )A 187 518 :M (by )S 203 518 :M -.109(any )A 224 518 :M -.183(remaining )A 275 518 :M -.145(variables )A 321 518 :M -.249(that )A 342 518 :M -.163(have )A 368 518 :M .306 .031(X )J 381 518 :M (or )S 395 518 :M -.663(Y )A 407 518 :M (as )S 421 518 :M -.088(descendants )A 482 518 :M -.334(in)A 59 536 :M -.048(G\(M\), followed by the rest of the )A 221 536 :M -.031(variables. )A 271 536 :M -.065(Given )A 304 536 :M -.084(this )A 325 536 :M .186 .019(ordering, )J 373 536 :M (we )S 391 536 :M -.166(will )A 413 536 :M .259 .026(now )J 438 536 :M -.126(refer )A 464 536 :M -.167(to )A 477 536 :M -.33(the)A 59 554 :M .197 .02(variables as X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .039(,...,X)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf .124 .012(, where for all i, X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .088 .009( is the i)J f0_7 sf 0 -5 rm .027(th)A 0 5 rm f0_12 sf .179 .018( variable in the ordering.)J 77 572 :M -.041(2. For each variable X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.039(, add to the existing graph G, a variable T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.043(, and edges )A 442 572 :M -.08(from )A 469 572 :M -.164(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 482 572 :M -.334(to)A 59 590 :M .726(X)A f0_7 sf 0 3 rm .163(j)A 0 -3 rm f0_12 sf .457 .046(, )J 78 590 :M (for )S 96 590 :M -.245(each )A 122 590 :M -.334(j )A 130 590 :M cF f1_12 sf .038A sf .376 .038( )J 142 590 :M .555 .055(i. )J 154 590 :M -.249(Call )A 178 590 :M -.22(the )A 197 590 :M -.11(resulting )A 243 590 :M .425 .043(graph, )J 279 590 :M -.065(which )A 313 590 :M (has )S 334 590 :M -.163(vertex )A 368 590 :M -.109(set )A 386 590 :M .985(\(X)A f0_7 sf 0 3 rm .545(1)A 0 -3 rm f0_12 sf .614(,...,X)A f0_7 sf 0 3 rm .545(n)A 0 -3 rm f0_12 sf .849 .085(, )J 445 590 :M .964(T)A f0_7 sf 0 3 rm .46(1)A 0 -3 rm f0_12 sf .489(,...,T)A f0_7 sf 0 3 rm .46(n)A 0 -3 rm f0_12 sf <29>S 59 608 :M .564(G)A f0_7 sf 0 3 rm .191(Constsruct\(0\))A 0 -3 rm f0_12 sf (.)S 77 626 :M -.058(3. Let G)A f0_7 sf 0 3 rm -.033(Construct\(i\))A 0 -3 rm f0_12 sf -.054( be the the graph constructed after the i)A f0_7 sf 0 -5 rm -.032(th)A 0 5 rm f0_12 sf ( )S 344 626 :M -.22(iteration )A 386 626 :M (of )S 400 626 :M -.22(the )A 418 626 :M -.073(following )A 468 626 :M .168(step,)A 59 644 :M -.055(starting with i = 1: If r > i, and there is no trek )A 281 644 :M -.139(between )A 324 644 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 339 644 :M -.109(and )A 360 644 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 375 644 :M -.167(in )A 388 644 :M .285(G)A f0_7 sf 0 3 rm .093(Construct\(i-1\))A 0 -3 rm f0_12 sf .099 .01( )J 442 644 :M -.184(containing)A 59 662 :M -.056(a variable T)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.052(, where j < i, and )A f1_12 sf -.062(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.055( and )A f1_12 sf -.062(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.054( are uncorrelated in )A f1_12 sf -.084(S)A f0_12 sf -.058(, then remove the T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.14A f0_12 sf -.069( X)A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.068( edge.)A endp %%Page: 20 20 %%BeginPageSetup initializepage (peter; page: 20 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (20)S gR gS 0 0 552 730 rC 77 56 :M f0_12 sf .258 .026(For )J 101 56 :M -.056(inputs )A 137 56 :M 1.114 .111(G, )J 157 56 :M 1.114 .111(X, )J 177 56 :M .281 .028(Y, )J 196 56 :M -.109(and )A 220 56 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 239 56 :M (we )S 260 56 :M -.166(will )A 285 56 :M -.126(refer )A 314 56 :M -.167(to )A 330 56 :M -.22(the )A 351 56 :M -.111(output )A 388 56 :M (of )S 405 56 :M -.084(this )A 429 56 :M -.184(algorithm )A 481 56 :M (as)S 59 74 :M .573(G)A f0_7 sf 0 3 rm .2(Construct)A 0 -3 rm f0_12 sf .368(\(G,X,Y,)A f2_12 sf .529(Z)A f0_12 sf .551 .055(\). )J 159 74 :M -.081(Note )A 186 74 :M -.249(that )A 207 74 :M -.334(it )A 217 74 :M (follows )S 257 74 :M -.08(from )A 284 74 :M -.082(step )A 307 74 :M (2 )S 317 74 :M (of )S 331 74 :M -.22(the )A 349 74 :M -.109(construction )A 412 74 :M -.184(algorithm )A 462 74 :M -.249(that )A 484 74 :M -.327(if)A 59 92 :M .045 .004(there is a trek X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf ( T)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf ( X)S f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf .049 .005(, then j )J cF f1_12 sf .005A sf .049 .005( min\(i,k\).)J 77 110 :M -.038(Suppose for the graph in Figure 11 )A 247 110 :M (we )S 265 110 :M -.215(are )A 283 110 :M -.164(interested )A 332 110 :M -.167(in )A 345 110 :M -.092(whether )A 387 110 :M f1_12 sf .191(r)A f0_12 sf .607 .061(\(X,Y\) )J 427 110 :M .211 .021(= )J 438 110 :M (0 )S 448 110 :M .898 .09(\(i.e. )J 472 110 :M f2_12 sf (Z)S f0_12 sf ( )S 484 110 :M (=)S 59 128 :M f1_12 sf .662A f0_12 sf .469(\).)A 136 155 297 27 rC 136 164 :M (X)S 145 164 :M ( )S 148 164 :M ( )S 151 164 :M ( )S 154 164 :M ( )S 157 164 :M ( )S 160 164 :M ( )S 163 164 :M ( )S 166 164 :M ( )S 169 164 :M ( )S 172 164 :M ( )S 175 164 :M ( )S 178 164 :M ( )S 181 164 :M ( )S 184 164 :M ( )S 187 164 :M (A)S 195 164 :M ( )S 198 164 :M ( )S 201 164 :M ( )S 204 164 :M ( )S 207 164 :M ( )S 210 164 :M ( )S 213 164 :M ( )S 216 164 :M ( )S 219 164 :M ( )S 222 164 :M ( )S 225 164 :M ( )S 228 164 :M ( )S 231 164 :M ( )S 234 164 :M ( )S 237 164 :M ( )S 240 164 :M (B)S 248 164 :M ( )S 251 164 :M ( )S 254 164 :M ( )S 257 164 :M ( )S 260 164 :M ( )S 263 164 :M ( )S 266 164 :M ( )S 269 164 :M ( )S 272 164 :M ( )S 275 164 :M ( )S 278 164 :M ( )S 281 164 :M ( )S 284 164 :M ( )S 287 164 :M ( )S 290 164 :M (C)S 298 164 :M ( )S 301 164 :M ( )S 304 164 :M ( )S 307 164 :M ( )S 310 164 :M ( )S 313 164 :M ( )S 316 164 :M ( )S 319 164 :M ( )S 322 164 :M ( )S 325 164 :M ( )S 328 164 :M ( )S 331 164 :M ( )S 334 164 :M ( )S 337 164 :M ( )S 340 164 :M ( )S 343 164 :M ( )S 346 164 :M (D)S 354 164 :M ( )S 357 164 :M ( )S 360 164 :M ( )S 363 164 :M ( )S 366 164 :M ( )S 369 164 :M ( )S 372 164 :M ( )S 375 164 :M ( )S 378 164 :M ( )S 381 164 :M ( )S 384 164 :M ( )S 387 164 :M ( )S 390 164 :M ( )S 393 164 :M ( )S 396 164 :M ( )S 399 164 :M ( )S 402 164 :M (Y)S gR gS 133 149 301 57 rC 156 154 -1 1 179 153 1 156 153 @a np 176 149 :M 176 157 :L 184 153 :L 176 149 :L eofill -1 -1 177 158 1 1 176 149 @b -1 -1 177 158 1 1 184 153 @b 176 150 -1 1 185 153 1 176 149 @a np 158 157 :M 158 149 :L 150 153 :L 158 157 :L eofill -1 -1 159 158 1 1 158 149 @b -1 -1 151 154 1 1 158 149 @b 150 154 -1 1 159 157 1 150 153 @a 205 162 -1 1 228 161 1 205 161 @a np 225 157 :M 225 165 :L 233 161 :L 225 157 :L eofill -1 -1 226 166 1 1 225 157 @b -1 -1 226 166 1 1 233 161 @b 225 158 -1 1 234 161 1 225 157 @a np 207 165 :M 207 157 :L 199 161 :L 207 165 :L eofill -1 -1 208 166 1 1 207 157 @b -1 -1 200 162 1 1 207 157 @b 199 162 -1 1 208 165 1 199 161 @a 254 162 -1 1 277 161 1 254 161 @a np 274 157 :M 274 165 :L 282 161 :L 274 157 :L eofill -1 -1 275 166 1 1 274 157 @b -1 -1 275 166 1 1 282 161 @b 274 158 -1 1 283 161 1 274 157 @a np 256 165 :M 256 157 :L 248 161 :L 256 165 :L eofill -1 -1 257 166 1 1 256 157 @b -1 -1 249 162 1 1 256 157 @b 248 162 -1 1 257 165 1 248 161 @a 308 161 -1 1 331 160 1 308 160 @a np 328 156 :M 328 164 :L 336 160 :L 328 156 :L eofill -1 -1 329 165 1 1 328 156 @b -1 -1 329 165 1 1 336 160 @b 328 157 -1 1 337 160 1 328 156 @a np 310 164 :M 310 156 :L 302 160 :L 310 164 :L eofill -1 -1 311 165 1 1 310 156 @b -1 -1 303 161 1 1 310 156 @b 302 161 -1 1 311 164 1 302 160 @a 156 162 -1 1 180 161 1 156 161 @a np 158 165 :M 158 157 :L 150 161 :L 158 165 :L eofill -1 -1 159 166 1 1 158 157 @b -1 -1 151 162 1 1 158 157 @b 150 162 -1 1 159 165 1 150 161 @a 364 169 -1 1 393 168 1 364 168 @a np 390 164 :M 390 171 :L 398 168 :L 390 164 :L eofill -1 -1 391 172 1 1 390 164 @b -1 -1 391 172 1 1 398 168 @b 390 165 -1 1 399 168 1 390 164 @a -1 -1 138 179 1 1 137 176 @b np 133 178 :M 141 178 :L 137 170 :L 133 178 :L eofill 133 179 -1 1 142 178 1 133 178 @a 137 171 -1 1 142 178 1 137 170 @a -1 -1 134 179 1 1 137 170 @b -1 -1 150 179 1 1 149 176 @b np 145 178 :M 153 178 :L 149 170 :L 145 178 :L eofill 145 179 -1 1 154 178 1 145 178 @a 149 171 -1 1 154 178 1 149 170 @a -1 -1 146 179 1 1 149 170 @b 0 90 100 48 188.5 171.5 @n 90 180 94 36 196.5 177.5 @n 90 180 160 56 218.5 177.5 @n 0 90 148 70 217.5 170.5 @n 370 161 -1 1 393 160 1 370 160 @a np 390 157 :M 390 164 :L 398 160 :L 390 157 :L eofill -1 -1 391 165 1 1 390 157 @b -1 -1 391 165 1 1 398 160 @b 390 158 -1 1 399 160 1 390 157 @a np 372 164 :M 372 157 :L 364 160 :L 372 164 :L eofill -1 -1 373 165 1 1 372 157 @b -1 -1 365 161 1 1 372 157 @b 364 161 -1 1 373 164 1 364 160 @a gR gS 0 0 552 730 rC 279 221 :M f0_12 sf (G)S 247 239 :M f2_12 sf 3.34 .334(Figure 11)J 77 263 :M f0_12 sf -.135(Applying the first step of Algorithm Construct Latent Directed Graph to )A 418 263 :M .306 .031(G )J 431 263 :M -.167(in )A 444 263 :M -.054(Figure )A 479 263 :M (11)S 59 281 :M -.025(with vertex inputs X, Y, )A f1_12 sf -.051A f0_12 sf -.025(, results in the naming of the vertices shown in Figure 12.)A 163 315 297 34 rC 163 324 :M (X)S 172 327 :M f0_7 sf (1)S 176 324 :M f0_12 sf ( )S 179 324 :M ( )S 182 324 :M ( )S 185 324 :M ( )S 188 324 :M ( )S 191 324 :M ( )S 194 324 :M ( )S 197 324 :M ( )S 200 324 :M ( )S 203 324 :M ( )S 206 324 :M ( )S 209 324 :M ( )S 212 324 :M ( )S 215 324 :M ( )S 218 324 :M (X)S 227 327 :M f0_7 sf (3)S 231 324 :M f0_12 sf ( )S 234 324 :M ( )S 237 324 :M ( )S 240 324 :M ( )S 243 324 :M ( )S 246 324 :M ( )S 249 324 :M ( )S 252 324 :M ( )S 255 324 :M ( )S 258 324 :M ( )S 261 324 :M ( )S 264 324 :M ( )S 267 324 :M (X)S 276 327 :M f0_7 sf (5)S 280 324 :M f0_12 sf ( )S 283 324 :M ( )S 286 324 :M ( )S 289 324 :M ( )S 292 324 :M ( )S 295 324 :M ( )S 298 324 :M ( )S 301 324 :M ( )S 304 324 :M ( )S 307 324 :M ( )S 310 324 :M ( )S 313 324 :M ( )S 316 324 :M ( )S 319 324 :M (X)S 328 327 :M f0_7 sf (4)S 332 324 :M f0_12 sf ( )S 335 324 :M ( )S 338 324 :M ( )S 341 324 :M ( )S 344 324 :M ( )S 347 324 :M ( )S 350 324 :M ( )S 353 324 :M ( )S 356 324 :M ( )S 359 324 :M ( )S 362 324 :M ( )S 365 324 :M ( )S 368 324 :M ( )S 371 324 :M ( )S 374 324 :M ( )S 377 324 :M (X)S 386 327 :M f0_7 sf (6)S 390 324 :M f0_12 sf ( )S 393 324 :M ( )S 396 324 :M ( )S 399 324 :M ( )S 402 324 :M ( )S 405 324 :M ( )S 408 324 :M ( )S 411 324 :M ( )S 414 324 :M ( )S 417 324 :M ( )S 420 324 :M ( )S 423 324 :M ( )S 426 324 :M ( )S 429 324 :M ( )S 432 324 :M ( )S 435 324 :M ( )S 438 324 :M (X)S 447 327 :M f0_7 sf (2)S gR gS 96 302 375 68 rC 184 322 -1 1 207 321 1 184 321 @a np 204 317 :M 204 325 :L 212 321 :L 204 317 :L eofill -1 -1 205 326 1 1 204 317 @b -1 -1 205 326 1 1 212 321 @b 204 318 -1 1 213 321 1 204 317 @a np 186 325 :M 186 317 :L 178 321 :L 186 325 :L eofill -1 -1 187 326 1 1 186 317 @b -1 -1 179 322 1 1 186 317 @b 178 322 -1 1 187 325 1 178 321 @a 238 322 -1 1 261 321 1 238 321 @a np 258 317 :M 258 325 :L 266 321 :L 258 317 :L eofill -1 -1 259 326 1 1 258 317 @b -1 -1 259 326 1 1 266 321 @b 258 318 -1 1 267 321 1 258 317 @a np 240 325 :M 240 317 :L 232 321 :L 240 325 :L eofill -1 -1 241 326 1 1 240 317 @b -1 -1 233 322 1 1 240 317 @b 232 322 -1 1 241 325 1 232 321 @a 289 322 -1 1 312 321 1 289 321 @a np 309 317 :M 309 325 :L 317 321 :L 309 317 :L eofill -1 -1 310 326 1 1 309 317 @b -1 -1 310 326 1 1 317 321 @b 309 318 -1 1 318 321 1 309 317 @a np 291 325 :M 291 317 :L 283 321 :L 291 325 :L eofill -1 -1 292 326 1 1 291 317 @b -1 -1 284 322 1 1 291 317 @b 283 322 -1 1 292 325 1 283 321 @a 343 322 -1 1 366 321 1 343 321 @a np 363 317 :M 363 325 :L 371 321 :L 363 317 :L eofill -1 -1 364 326 1 1 363 317 @b -1 -1 364 326 1 1 371 321 @b 363 318 -1 1 372 321 1 363 317 @a np 345 325 :M 345 317 :L 337 321 :L 345 325 :L eofill -1 -1 346 326 1 1 345 317 @b -1 -1 338 322 1 1 345 317 @b 337 322 -1 1 346 325 1 337 321 @a 399 322 -1 1 422 321 1 399 321 @a np 419 317 :M 419 325 :L 427 321 :L 419 317 :L eofill -1 -1 420 326 1 1 419 317 @b -1 -1 420 326 1 1 427 321 @b 419 318 -1 1 428 321 1 419 317 @a np 401 325 :M 401 317 :L 393 321 :L 401 325 :L eofill -1 -1 402 326 1 1 401 317 @b -1 -1 394 322 1 1 401 317 @b 393 322 -1 1 402 325 1 393 321 @a 186 332 -1 1 210 331 1 186 331 @a np 188 335 :M 188 327 :L 180 331 :L 188 335 :L eofill -1 -1 189 336 1 1 188 327 @b -1 -1 181 332 1 1 188 327 @b 180 332 -1 1 189 335 1 180 331 @a 395 332 -1 1 424 331 1 395 331 @a np 421 327 :M 421 335 :L 429 331 :L 421 327 :L eofill -1 -1 422 336 1 1 421 327 @b -1 -1 422 336 1 1 429 331 @b 421 328 -1 1 430 331 1 421 327 @a -1 -1 169 342 1 1 168 339 @b np 164 341 :M 172 341 :L 168 333 :L 164 341 :L eofill 164 342 -1 1 173 341 1 164 341 @a 168 334 -1 1 173 341 1 168 333 @a -1 -1 165 342 1 1 168 333 @b -1 -1 181 342 1 1 180 339 @b np 176 341 :M 184 341 :L 180 333 :L 176 341 :L eofill 176 342 -1 1 185 341 1 176 341 @a 180 334 -1 1 185 341 1 180 333 @a -1 -1 177 342 1 1 180 333 @b 0 90 100 48 219.5 334.5 @n 90 180 94 36 227.5 340.5 @n 90 180 160 56 249.5 340.5 @n 0 90 148 70 248.5 333.5 @n 97 303 374 34 rC 97 312 :M f0_12 sf ( )S 100 312 :M ( )S 103 312 :M (O)S 111 312 :M (l)S 114 312 :M (d)S 120 312 :M ( )S 123 312 :M (N)S 132 312 :M (a)S 137 312 :M (m)S 146 312 :M (e)S 151 312 :M (:)S 154 312 :M ( )S 157 312 :M ( )S 160 312 :M ( )S 163 312 :M (X)S 172 312 :M ( )S 175 312 :M ( )S 178 312 :M ( )S 181 312 :M ( )S 184 312 :M ( )S 187 312 :M ( )S 190 312 :M ( )S 193 312 :M ( )S 196 312 :M ( )S 199 312 :M ( )S 202 312 :M ( )S 205 312 :M ( )S 208 312 :M ( )S 211 312 :M ( )S 214 312 :M ( )S 217 312 :M (A)S 225 312 :M ( )S 228 312 :M ( )S 231 312 :M ( )S 234 312 :M ( )S 237 312 :M ( )S 240 312 :M ( )S 243 312 :M ( )S 246 312 :M ( )S 249 312 :M ( )S 252 312 :M ( )S 255 312 :M ( )S 258 312 :M ( )S 261 312 :M ( )S 264 312 :M ( )S 267 312 :M (B)S 275 312 :M ( )S 278 312 :M ( )S 281 312 :M ( )S 284 312 :M ( )S 287 312 :M ( )S 290 312 :M ( )S 293 312 :M ( )S 296 312 :M ( )S 299 312 :M ( )S 302 312 :M ( )S 305 312 :M ( )S 308 312 :M ( )S 311 312 :M ( )S 314 312 :M ( )S 317 312 :M ( )S 320 312 :M (C)S 328 315 :M f0_7 sf .25 .025( )J 330 312 :M f0_12 sf ( )S 333 312 :M ( )S 336 312 :M ( )S 339 312 :M ( )S 342 312 :M ( )S 345 312 :M ( )S 348 312 :M ( )S 351 312 :M ( )S 354 312 :M ( )S 357 312 :M ( )S 360 312 :M ( )S 363 312 :M ( )S 366 312 :M ( )S 369 312 :M ( )S 372 312 :M ( )S 375 312 :M ( )S 378 312 :M (D)S 386 315 :M f0_7 sf .25 .025( )J 388 312 :M f0_12 sf ( )S 391 312 :M ( )S 394 312 :M ( )S 397 312 :M ( )S 400 312 :M ( )S 403 312 :M ( )S 406 312 :M ( )S 409 312 :M ( )S 412 312 :M ( )S 415 312 :M ( )S 418 312 :M ( )S 421 312 :M ( )S 424 312 :M ( )S 427 312 :M ( )S 430 312 :M ( )S 433 312 :M ( )S 436 312 :M ( )S 439 312 :M (Y)S 97 324 :M (N)S 106 324 :M (e)S 111 324 :M (w)S 120 324 :M ( )S 123 324 :M (N)S 132 324 :M (a)S 137 324 :M (m)S 146 324 :M (e)S 151 324 :M (:)S gR gS 0 0 552 730 rC 174 397 :M f2_12 sf 2.339 .234(Figure 12: G with vertices renamed)J 77 421 :M f0_12 sf -.166(Applying )A 125 421 :M (steps )S 153 421 :M (2 )S 163 421 :M -.109(and )A 184 421 :M (3 )S 195 421 :M (of )S 210 421 :M -.221(Algorithm )A 263 421 :M -.073(Construct )A 314 421 :M -.275(Latent )A 348 421 :M -.288(Directed )A 392 421 :M (Graph )S 427 421 :M -.046(results )A 463 421 :M -.167(in )A 477 421 :M -.33(the)A 59 439 :M -.029(directed graph shown in Figure 13.)A 101 461 366 112 rC 101 474 :M f1_12 sf (T)S 109 477 :M f1_7 sf (1)S 113 474 :M f1_12 sf ( )S 116 474 :M ( )S 119 474 :M ( )S 122 474 :M ( )S 125 474 :M ( )S 128 474 :M ( )S 131 474 :M ( )S 134 474 :M ( )S 137 474 :M ( )S 140 474 :M ( )S 143 474 :M ( )S 146 474 :M ( )S 149 474 :M ( )S 152 474 :M ( )S 155 474 :M ( )S 158 474 :M ( )S 161 474 :M ( )S 164 474 :M (T)S 172 477 :M f1_7 sf (3)S 176 474 :M f1_12 sf ( )S 179 474 :M ( )S 182 474 :M ( )S 185 474 :M ( )S 188 474 :M ( )S 191 474 :M ( )S 194 474 :M ( )S 197 474 :M ( )S 200 474 :M ( )S 203 474 :M ( )S 206 474 :M ( )S 209 474 :M ( )S 212 474 :M ( )S 215 474 :M ( )S 218 474 :M ( )S 221 474 :M ( )S 224 474 :M (T)S 232 477 :M f1_7 sf (5)S 236 474 :M f1_12 sf ( )S 239 474 :M ( )S 242 474 :M ( )S 245 474 :M ( )S 248 474 :M ( )S 251 474 :M ( )S 254 474 :M ( )S 257 474 :M ( )S 260 474 :M ( )S 263 474 :M ( )S 266 474 :M ( )S 269 474 :M ( )S 272 474 :M ( )S 275 474 :M ( )S 278 474 :M ( )S 281 474 :M (T)S 289 477 :M f1_7 sf (4)S 293 474 :M f1_12 sf ( )S 296 474 :M ( )S 299 474 :M ( )S 302 474 :M ( )S 305 474 :M ( )S 308 474 :M ( )S 311 474 :M ( )S 314 474 :M ( )S 317 474 :M ( )S 320 474 :M ( )S 323 474 :M ( )S 326 474 :M ( )S 329 474 :M ( )S 332 474 :M ( )S 335 474 :M ( )S 338 474 :M ( )S 341 474 :M ( )S 344 474 :M ( )S 347 474 :M (T)S 355 477 :M f1_7 sf (6)S 359 474 :M f1_12 sf ( )S 362 474 :M ( )S 365 474 :M ( )S 368 474 :M ( )S 371 474 :M ( )S 374 474 :M ( )S 377 474 :M ( )S 380 474 :M ( )S 383 474 :M ( )S 386 474 :M ( )S 389 474 :M ( )S 392 474 :M ( )S 395 474 :M ( )S 398 474 :M ( )S 401 474 :M ( )S 404 474 :M ( )S 407 474 :M ( )S 410 474 :M (T)S 418 477 :M f1_7 sf (2)S 101 550 :M f0_12 sf (X)S 110 553 :M f0_7 sf (1)S 114 550 :M f0_12 sf ( )S 117 550 :M ( )S 120 550 :M ( )S 123 550 :M ( )S 126 550 :M ( )S 129 550 :M ( )S 132 550 :M ( )S 135 550 :M ( )S 138 550 :M ( )S 141 550 :M ( )S 144 550 :M ( )S 147 550 :M ( )S 150 550 :M ( )S 153 550 :M ( )S 156 550 :M ( )S 159 550 :M ( )S 162 550 :M (X)S 171 553 :M f0_7 sf (3)S 175 550 :M f0_12 sf ( )S 178 550 :M ( )S 181 550 :M ( )S 184 550 :M ( )S 187 550 :M ( )S 190 550 :M ( )S 193 550 :M ( )S 196 550 :M ( )S 199 550 :M ( )S 202 550 :M ( )S 205 550 :M ( )S 208 550 :M ( )S 211 550 :M ( )S 214 550 :M ( )S 217 550 :M ( )S 220 550 :M ( )S 223 550 :M ( )S 226 550 :M (X)S 235 553 :M f0_7 sf (5)S 239 550 :M f0_12 sf ( )S 242 550 :M ( )S 245 550 :M ( )S 248 550 :M ( )S 251 550 :M ( )S 254 550 :M ( )S 257 550 :M ( )S 260 550 :M ( )S 263 550 :M ( )S 266 550 :M ( )S 269 550 :M ( )S 272 550 :M ( )S 275 550 :M ( )S 278 550 :M ( )S 281 550 :M ( )S 284 550 :M (X)S 293 553 :M f0_7 sf (4)S 297 550 :M f0_12 sf ( )S 300 550 :M ( )S 303 550 :M ( )S 306 550 :M ( )S 309 550 :M ( )S 312 550 :M ( )S 315 550 :M ( )S 318 550 :M ( )S 321 550 :M ( )S 324 550 :M ( )S 327 550 :M ( )S 330 550 :M ( )S 333 550 :M ( )S 336 550 :M ( )S 339 550 :M ( )S 342 550 :M ( )S 345 550 :M ( )S 348 550 :M (X)S 357 553 :M f0_7 sf (6)S 361 550 :M f0_12 sf ( )S 364 550 :M ( )S 367 550 :M ( )S 370 550 :M ( )S 373 550 :M ( )S 376 550 :M ( )S 379 550 :M ( )S 382 550 :M ( )S 385 550 :M ( )S 388 550 :M ( )S 391 550 :M ( )S 394 550 :M ( )S 397 550 :M ( )S 400 550 :M ( )S 403 550 :M ( )S 406 550 :M ( )S 409 550 :M (X)S 418 553 :M f0_7 sf (2)S gR gS 100 460 368 136 rC -.75 -.75 107.75 525.75 .75 .75 107 481 @b np 111 523 :M 103 523 :L 107 531 :L 111 523 :L .75 lw eofill 103 523.75 -.75 .75 111.75 523 .75 103 523 @a 103 523.75 -.75 .75 107.75 531 .75 103 523 @a -.75 -.75 107.75 531.75 .75 .75 111 523 @b -.75 -.75 170.75 524.75 .75 .75 170 480 @b np 174 522 :M 166 522 :L 170 530 :L 174 522 :L eofill 166 522.75 -.75 .75 174.75 522 .75 166 522 @a 166 522.75 -.75 .75 170.75 530 .75 166 522 @a -.75 -.75 170.75 530.75 .75 .75 174 522 @b -.75 -.75 229.75 525.75 .75 .75 229 481 @b np 233 523 :M 225 523 :L 229 531 :L 233 523 :L eofill 225 523.75 -.75 .75 233.75 523 .75 225 523 @a 225 523.75 -.75 .75 229.75 531 .75 225 523 @a -.75 -.75 229.75 531.75 .75 .75 233 523 @b -.75 -.75 289.75 525.75 .75 .75 289 481 @b np 293 523 :M 285 523 :L 289 531 :L 293 523 :L eofill 285 523.75 -.75 .75 293.75 523 .75 285 523 @a 285 523.75 -.75 .75 289.75 531 .75 285 523 @a -.75 -.75 289.75 531.75 .75 .75 293 523 @b -.75 -.75 352.75 524.75 .75 .75 352 480 @b np 356 522 :M 348 522 :L 352 530 :L 356 522 :L eofill 348 522.75 -.75 .75 356.75 522 .75 348 522 @a 348 522.75 -.75 .75 352.75 530 .75 348 522 @a -.75 -.75 352.75 530.75 .75 .75 356 522 @b -.75 -.75 414.75 526.75 .75 .75 414 482 @b np 418 524 :M 410 524 :L 414 532 :L 418 524 :L eofill 410 524.75 -.75 .75 418.75 524 .75 410 524 @a 410 524.75 -.75 .75 414.75 532 .75 410 524 @a -.75 -.75 414.75 532.75 .75 .75 418 524 @b 107 481.75 -.75 .75 159.75 524 .75 107 481 @a np 160 520 :M 156 526 :L 164 528 :L 160 520 :L eofill -.75 -.75 156.75 526.75 .75 .75 160 520 @b 156 526.75 -.75 .75 164.75 528 .75 156 526 @a 160 520.75 -.75 .75 164.75 528 .75 160 520 @a -.75 -.75 364.75 527.75 .75 .75 415 481 @b np 368 528 :M 364 523 :L 360 531 :L 368 528 :L eofill 364 523.75 -.75 .75 368.75 528 .75 364 523 @a -.75 -.75 360.75 531.75 .75 .75 364 523 @b -.75 -.75 360.75 531.75 .75 .75 368 528 @b 170 479.75 -.75 .75 221.75 524 .75 170 479 @a np 221 520 :M 217 525 :L 225 528 :L 221 520 :L eofill -.75 -.75 217.75 525.75 .75 .75 221 520 @b 217 525.75 -.75 .75 225.75 528 .75 217 525 @a 221 520.75 -.75 .75 225.75 528 .75 221 520 @a -.75 -.75 241.75 524.75 .75 .75 290 479 @b np 245 525 :M 240 520 :L 237 528 :L 245 525 :L eofill 240 520.75 -.75 .75 245.75 525 .75 240 520 @a -.75 -.75 237.75 528.75 .75 .75 240 520 @b -.75 -.75 237.75 528.75 .75 .75 245 525 @b 291 478.75 -.75 .75 341.75 524 .75 291 478 @a np 342 520 :M 337 525 :L 345 528 :L 342 520 :L eofill -.75 -.75 337.75 525.75 .75 .75 342 520 @b 337 525.75 -.75 .75 345.75 528 .75 337 525 @a 342 520.75 -.75 .75 345.75 528 .75 342 520 @a 230 480.75 -.75 .75 336.75 529 .75 230 480 @a np 335 524 :M 332 531 :L 341 531 :L 335 524 :L eofill -.75 -.75 332.75 531.75 .75 .75 335 524 @b 332 531.75 -.75 .75 341.75 531 .75 332 531 @a 335 524.75 -.75 .75 341.75 531 .75 335 524 @a 123 546.75 -.75 .75 155.75 546 .75 123 546 @a np 125 550 :M 125 542 :L 117 546 :L 125 550 :L eofill -.75 -.75 125.75 550.75 .75 .75 125 542 @b -.75 -.75 117.75 546.75 .75 .75 125 542 @b 117 546.75 -.75 .75 125.75 550 .75 117 546 @a 366 546.75 -.75 .75 398.75 546 .75 366 546 @a np 396 542 :M 396 550 :L 404 546 :L 396 542 :L eofill -.75 -.75 396.75 550.75 .75 .75 396 542 @b -.75 -.75 396.75 550.75 .75 .75 404 546 @b 396 542.75 -.75 .75 404.75 546 .75 396 542 @a -.75 -.75 104.75 561.75 .75 .75 104 561 @b np 100 563 :M 108 563 :L 104 555 :L 100 563 :L eofill 100 563.75 -.75 .75 108.75 563 .75 100 563 @a 104 555.75 -.75 .75 108.75 563 .75 104 555 @a -.75 -.75 100.75 563.75 .75 .75 104 555 @b -.75 -.75 114.75 561.75 .75 .75 114 561 @b np 110 563 :M 118 563 :L 114 555 :L 110 563 :L eofill 110 563.75 -.75 .75 118.75 563 .75 110 563 @a 114 555.75 -.75 .75 118.75 563 .75 114 555 @a -.75 -.75 110.75 563.75 .75 .75 114 555 @b 90 180 160 62 184.5 563.5 @n 0 90 208 84 183.5 553.5 @n 90 180 90 30 159.5 563.5 @n 0 90 146 46 160.5 555.5 @n gR gS 0 0 552 730 rC 201 615 :M f2_12 sf 1.453 .145(Figure 13: )J f0_12 sf .697(G)A f0_7 sf 0 3 rm .243(Construct)A 0 -3 rm f0_12 sf 1.525 .153(\(X,Y, )J f1_12 sf .795A f0_12 sf <29>S 77 646 :M -.145(As an example of an application of )A 244 646 :M -.082(step )A 267 646 :M .833 .083(3, )J 281 646 :M -.22(the )A 299 646 :M -.163(edge )A 325 646 :M -.08(from )A 352 646 :M .09(T)A f0_7 sf 0 3 rm (3)S 0 -3 rm f0_12 sf ( )S 367 646 :M -.167(to )A 380 646 :M .478(X)A f0_7 sf 0 3 rm .193(4)A 0 -3 rm f0_12 sf .166 .017( )J 397 646 :M (is )S 409 646 :M -.14(removed )A 454 646 :M -.162(because)A 59 664 :M -.167(in )A 72 664 :M .361(G)A f0_7 sf 0 3 rm .123(Construct\(2\))A 0 -3 rm f0_12 sf .125 .012( )J 122 664 :M -.196(there )A 149 664 :M (is )S 161 664 :M (no )S 177 664 :M -.163(trek )A 199 664 :M -.139(between )A 242 664 :M .478(X)A f0_7 sf 0 3 rm .193(3)A 0 -3 rm f0_12 sf .166 .017( )J 259 664 :M -.109(and )A 281 664 :M .478(X)A f0_7 sf 0 3 rm .193(4)A 0 -3 rm f0_12 sf .166 .017( )J 299 664 :M -.249(that )A 321 664 :M -.123(contains )A 365 664 :M .09(T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S 381 664 :M (or )S 396 664 :M .51(T)A f0_7 sf 0 3 rm .243(2)A 0 -3 rm f0_12 sf .379 .038(, )J 416 664 :M -.109(and )A 438 664 :M -.196(there )A 466 664 :M (is )S 479 664 :M (no)S 59 682 :M -.006(double-headed arrow between X)A f0_7 sf 0 3 rm (3)S 0 -3 rm f0_12 sf ( and X)S f0_7 sf 0 3 rm (4)S 0 -3 rm f0_12 sf ( in G.)S endp %%Page: 21 21 %%BeginPageSetup initializepage (peter; page: 21 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (21)S gR gS 0 0 552 730 rC 77 56 :M f0_12 sf -.219(The )A 101 56 :M -.165(next )A 127 56 :M -.052(series )A 160 56 :M (of )S 176 56 :M -.22(lemmas )A 219 56 :M .669 .067(shows )J 257 56 :M .259 .026(how )J 285 56 :M -.167(to )A 301 56 :M -.109(construct )A 351 56 :M -.326(a )A 363 56 :M -.223(SEM )A 394 56 :M .119(M\325\(M,X,Y,)A f2_12 sf .154(Z)A f0_12 sf .122 .012(\) )J 470 56 :M -.11(with)A 59 74 :M -.122(measured )A 112 74 :M -.145(variables )A 162 74 :M f2_12 sf .25(V)A f0_12 sf .087 .009( )J 179 74 :M -.109(and )A 204 74 :M -.276(latent )A 237 74 :M -.145(variables )A 287 74 :M f2_12 sf (T)S f0_12 sf ( )S 303 74 :M .909 .091(, )J 315 74 :M .277 .028(so )J 334 74 :M -.249(that )A 359 74 :M -.22(the )A 382 74 :M -.206(marginal )A 432 74 :M -.08(over )A 462 74 :M f2_12 sf 1.215 .122(V )J 481 74 :M f0_12 sf (of)S 59 92 :M f1_12 sf .218(S)A f0_12 sf .183(\(M\325\(M,X,Y,)A f2_12 sf .246(Z)A f0_12 sf .255 .025(\)\) = )J f1_12 sf .218(S)A f0_12 sf .87 .087(\(M\), and G\(M\325\(M,X,Y,)J f2_12 sf .246(Z)A f0_12 sf .347 .035(\)\) = G)J f0_7 sf 0 3 rm .093(Construct)A 0 -3 rm f0_12 sf .177(\(G\(M\),X,Y,)A f2_12 sf .246(Z)A f0_12 sf .215(\).)A 77 128 :M f2_12 sf -.033(Lemma 4:)A f0_12 sf -.018( If )A f1_12 sf (S)S f0_12 sf -.023( is a positive definite matrix, then there )A 341 128 :M -.055(exists )A 372 128 :M -.326(a )A 381 128 :M -.124(positive )A 422 128 :M -.206(definite )A 461 128 :M -.264(matrix)A 59 146 :M f1_12 sf (S)S f0_12 sf .03 .003(\325 = )J f1_12 sf (S)S f0_12 sf ( - )S f1_12 sf (d)S f0_12 sf .062 .006(I, where )J f1_12 sf (d)S f0_12 sf .069 .007( is a real positive number.)J 77 164 :M f2_12 sf -.027(Proof)A f0_12 sf -.022(. Suppose that )A f1_12 sf (S)S f0_12 sf -.021( is a positive definite matrix. It follows then that for all solutions)A 59 182 :M -.009(of det\()A f1_12 sf (S)S f0_12 sf ( - )S f1_12 sf (l)S f0_12 sf -.008(I\) = 0, )A f1_12 sf (l)S f0_12 sf -.009( is positive. Let the smallest solution of det\()A f1_12 sf (S)S f0_12 sf ( - )S f1_12 sf (l)S f0_12 sf -.009(I\) = 0 be )A 432 182 :M f1_12 sf .784(l)A f0_7 sf 0 3 rm .416(1)A 0 -3 rm f0_12 sf .649 .065(. )J 451 182 :M -.33(Let )A 470 182 :M f1_12 sf (d)S f0_12 sf ( )S 480 182 :M -.326(be)A 59 200 :M .037 .004(less than )J f1_12 sf (l)S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .034 .003( and greater than 0. Let )J f1_12 sf (S)S f0_12 sf (\325 = )S f1_12 sf (S)S f0_12 sf ( - )S f1_12 sf (d)S f0_12 sf .037 .004(I. We will now show )J 380 200 :M -.249(that )A 401 200 :M -.331(all )A 416 200 :M (of )S 430 200 :M -.22(the )A 448 200 :M -.042(solutions)A 59 218 :M .245 .024(of det\()J f1_12 sf .093(S)A f0_12 sf .077 .008(\325 - )J f1_12 sf .087(l)A f0_12 sf .174 .017(\325I\) = 0 are positive. )J f1_12 sf .093(S)A f0_12 sf .077 .008(\325 - )J f1_12 sf .087(l)A f0_12 sf .109 .011(\325I = )J f1_12 sf .093(S)A f0_12 sf .057 .006( - )J f1_12 sf .078(d)A f0_12 sf .077 .008(I - )J f1_12 sf .087(l)A f0_12 sf .114 .011(\325I = )J 331 218 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 343 218 :M (- )S 351 218 :M .097<28>A f1_12 sf .16(l)A f0_12 sf .155 .015J 370 218 :M .211 .021(+ )J 381 218 :M f1_12 sf .323(d)A f0_12 sf .587 .059(\)I. )J 403 218 :M (If )S 415 218 :M (we )S 433 218 :M -.109(set )A 450 218 :M f1_12 sf .204(l)A f0_12 sf .197 .02J 465 218 :M .211 .021(= )J 476 218 :M f1_12 sf .285(l)A f0_12 sf .13 .013( )J 487 218 :M (-)S 59 236 :M f1_12 sf (d)S f0_12 sf .095 .009(, then for )J 113 236 :M -.245(each )A 138 236 :M -.084(solution )A 180 236 :M (of )S 194 236 :M .04(det\()A f1_12 sf .061(S)A f0_12 sf ( )S 224 236 :M (- )S 232 236 :M f1_12 sf .16(l)A f0_12 sf .223 .022(I\) )J 251 236 :M .211 .021(= )J 262 236 :M .833 .083(0, )J 276 236 :M -.196(there )A 303 236 :M (is )S 315 236 :M -.326(a )A 324 236 :M -.084(solution )A 366 236 :M (of )S 380 236 :M .04(det\()A f1_12 sf .061(S)A f0_12 sf ( )S 410 236 :M (- )S 418 236 :M .097<28>A f1_12 sf .16(l)A f0_12 sf .155 .015J 437 236 :M .211 .021(+ )J 448 236 :M f1_12 sf (d)S f0_12 sf .052 .005(\)I\) )J 470 236 :M .211 .021(= )J 481 236 :M 1(0.)A 59 254 :M .253 .025(Since )J f1_12 sf .09(l)A f0_12 sf .095 .01(\325 = )J f1_12 sf .09(l)A f0_12 sf .059 .006( - )J f1_12 sf .081(d)A f0_12 sf .138 .014(, and )J f1_12 sf .081(d)A f0_12 sf .148 .015( is less than )J f1_12 sf .09(l)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .074 .007(, )J 245 254 :M -.22(the )A 263 254 :M -.165(smallest )A 305 254 :M -.084(solution )A 347 254 :M (of )S 361 254 :M .036(det\()A f1_12 sf .055(S)A f0_12 sf .049 .005J 395 254 :M (- )S 403 254 :M f1_12 sf .133(l)A f0_12 sf .232 .023J 426 254 :M .211 .021(= )J 437 254 :M (0 )S 447 254 :M (is )S 459 254 :M -.216(greater)A 59 272 :M (than 0. )S f1_12 sf <5C>S 77 290 :M f0_12 sf -.002(A linear transformation of a set of random variables is )A f2_12 sf (lower triangular)S f0_12 sf ( if and )S 459 290 :M -.083(only )A 484 290 :M -.327(if)A 59 308 :M -.085(there is an ordering of )A 167 308 :M -.22(the )A 185 308 :M -.145(variables )A 231 308 :M (such )S 257 308 :M -.249(that )A 278 308 :M -.22(the )A 296 308 :M -.22(matrix )A 330 308 :M -.108(representing )A 392 308 :M -.22(the )A 410 308 :M -.117(transformation )A 483 308 :M (is)S 59 326 :M -.035(zero for all entries a)A f0_7 sf 0 3 rm (ij)S 0 -3 rm f0_12 sf -.038(, when j > i.)A 77 362 :M f2_12 sf .724 .072(Lemma 5:)J f0_12 sf .226 .023( If X)J f0_7 sf 0 3 rm .087(1)A 0 -3 rm f0_12 sf .125 .012(, )J f1_12 sf .299A f0_12 sf .281 .028(, X)J f0_7 sf 0 3 rm .087(n)A 0 -3 rm f0_12 sf .291 .029( have a joint )J 259 362 :M -.165(normal )A 296 362 :M -.111(distribution )A 354 362 :M .454(N\(0,)A f1_12 sf .595(S)A f0_12 sf .698 .07(\), )J 397 362 :M -.062(where )A 430 362 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 442 362 :M (is )S 454 362 :M -.142(positive)A 59 380 :M -.111(definite, then there is a set of n mutually independent standard normal )A 391 380 :M -.145(variables )A 437 380 :M .51(T)A f0_7 sf 0 3 rm .243(1)A 0 -3 rm f0_12 sf .379 .038(, )J 456 380 :M f1_12 sf .667A f0_12 sf .303 .03(, )J 476 380 :M .62(T)A f0_7 sf 0 3 rm .296(n)A 0 -3 rm f0_12 sf (,)S 59 398 :M -.049(such that X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.03(, )A f1_12 sf -.119A f0_12 sf -.049(, X)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf -.047( are a lower triangular linear transformation of )A 377 398 :M .51(T)A f0_7 sf 0 3 rm .243(1)A 0 -3 rm f0_12 sf .379 .038(, )J 396 398 :M f1_12 sf .667A f0_12 sf .303 .03(, )J 416 398 :M .09(T)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf ( )S 431 398 :M -.109(and )A 452 398 :M (for )S 470 398 :M -.326(each)A 59 416 :M -.083(i, the coefficient of T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.088( in the equation for X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.087( is not equal to zero.)A 77 434 :M f2_12 sf .719(Proof)A f0_12 sf .685 .068(. )J 119 434 :M .258 .026(For )J 142 434 :M -.129(every )A 174 434 :M -.124(positive )A 217 434 :M -.206(definite )A 258 434 :M -.179(correlation )A 314 434 :M -.22(matrix )A 350 434 :M f1_12 sf 1.029(S)A f0_12 sf .79 .079(, )J 368 434 :M -.196(there )A 397 434 :M (is )S 411 434 :M -.326(a )A 422 434 :M -.223(SEM )A 453 434 :M -.667(M )A 470 434 :M -.11(with)A 59 452 :M -.089(correlation matrix )A f1_12 sf -.135(S)A f0_12 sf -.1(\(M\) = )A f1_12 sf -.135(S)A f0_12 sf -.093(, and directed acyclic graph G\(M\) that has )A 394 452 :M -.245(each )A 419 452 :M -.163(pair )A 441 452 :M (of )S 455 452 :M -.187(vertices)A 59 470 :M -.167(in )A 72 470 :M -.079(G\(M\) )A 103 470 :M -.247(adjacent )A 145 470 :M -.04(\(Spirtes )A 186 470 :M -.33(et )A 198 470 :M .261 .026(al. )J 215 470 :M .629 .063(1993\). )J 252 470 :M -.219(The )A 275 470 :M -.139(reduced )A 317 470 :M -.08(form )A 345 470 :M (of )S 360 470 :M -.326(a )A 370 470 :M -.248(complete )A 417 470 :M -.205(directed )A 459 470 :M -.329(acyclic)A 59 488 :M -.098(graph is a lower triangular transformation of independent error )A 358 488 :M -.145(variables )A 404 488 :M -.109(\(in )A 421 488 :M -.084(this )A 442 488 :M -.161(case )A 466 488 :M -.22(the )A 484 488 :M (T)S 59 506 :M -.077(variables\) that is non-zero on the diagonal, because )A f1_12 sf -.117(S)A f0_12 sf -.07( is positive definite. )A f1_12 sf <5C>S 77 542 :M f0_12 sf -.195(There )A 108 542 :M (is )S 120 542 :M -.326(a )A 129 542 :M -.166(simple )A 164 542 :M -.163(rule )A 186 542 :M (for )S 204 542 :M -.24(calculating )A 258 542 :M .38 .038(Cov\(X,Y\) )J 311 542 :M -.08(from )A 338 542 :M -.326(a )A 347 542 :M -.165(path )A 371 542 :M -.188(diagram )A 413 542 :M -.083(with )A 438 542 :M (no )S 454 542 :M -.234(directed)A 59 560 :M -.09(cycles that is used in the following lemmas. )A 269 560 :M -.195(There )A 300 560 :M (is )S 312 560 :M -.163(an )A 327 560 :M -.163(edge )A 353 560 :M -.205(directed )A 394 560 :M -.167(into )A 416 560 :M -.326(a )A 425 560 :M -.163(vertex )A 458 560 :M -.663(A )A 470 560 :M (on )S 486 560 :M (a)S 59 578 :M -.165(path )A 83 578 :M .299 .03(P )J 94 578 :M -.164(if )A 105 578 :M -.109(and )A 126 578 :M -.083(only )A 151 578 :M -.164(if )A 162 578 :M .299 .03(P )J 173 578 :M -.123(contains )A 216 578 :M -.163(an )A 231 578 :M -.163(edge )A 257 578 :M -.663(A )A 269 578 :M f1_12 sf .126A f0_12 sf ( )S 285 578 :M (B )S 297 578 :M (or )S 311 578 :M -.663(A )A 323 578 :M f1_12 sf .4A f0_12 sf .096 .01( )J 340 578 :M .83 .083(B. )J 356 578 :M -.219(The )A 378 578 :M f2_12 sf .563(source)A f0_12 sf .304 .03( )J 419 578 :M (of )S 433 578 :M -.326(a )A 442 578 :M -.163(trek )A 464 578 :M (is )S 477 578 :M -.33(the)A 59 596 :M -.114(vertex on the trek with )A 169 596 :M (no )S 185 596 :M -.064(edges )A 216 596 :M -.205(directed )A 257 596 :M -.167(into )A 279 596 :M .255 .026(it, )J 293 596 :M -.164(if )A 304 596 :M -.196(there )A 331 596 :M (such )S 357 596 :M -.326(a )A 366 596 :M (vertex. )S 403 596 :M .245 .024(\(For )J 428 596 :M -.235(example )A 471 596 :M (B )S 483 596 :M (is)S 59 614 :M -.054(the source of the trek A )A f1_12 sf -.137A f0_12 sf -.054( B )A f1_12 sf -.137A f0_12 sf -.054( C, A is the source of A )A f1_12 sf -.137A f0_12 sf -.054( B )A f1_12 sf -.137A f0_12 sf -.054( C, and the trek A )A f1_12 sf -.137A f0_12 sf ( )S 466 614 :M (B )S 478 614 :M f1_12 sf S 59 632 :M f0_12 sf .054(C)A f1_12 sf .08A f0_12 sf ( )S 84 632 :M -.663(D )A 97 632 :M (has )S 118 632 :M (no )S 135 632 :M .386 .039(source.\) )J 179 632 :M -.164(Associated )A 235 632 :M -.083(with )A 261 632 :M -.245(each )A 288 632 :M -.163(edge )A 316 632 :M -.663(A )A 330 632 :M f1_12 sf .126A f0_12 sf ( )S 348 632 :M (B )S 362 632 :M -.167(in )A 377 632 :M -.326(a )A 388 632 :M -.064(graph )A 421 632 :M (is )S 435 632 :M -.326(a )A 446 632 :M -.264(label )A 474 632 :M -.331(that)A 59 650 :M -.112(corresponds to the coefficient of A in )A 238 650 :M -.22(the )A 256 650 :M -.165(equation )A 300 650 :M (for )S 318 650 :M .83 .083(B, )J 334 650 :M -.109(and )A 355 650 :M -.131(associated )A 407 650 :M -.083(with )A 432 650 :M -.245(each )A 457 650 :M -.163(edge )A 483 650 :M (A)S 59 668 :M f1_12 sf .4A f0_12 sf .096 .01( )J 78 668 :M (B )S 92 668 :M (is )S 106 668 :M -.326(a )A 117 668 :M -.264(label )A 145 668 :M -.249(that )A 168 668 :M (corresponds )S 232 668 :M -.167(to )A 247 668 :M -.22(the )A 267 668 :M -.179(correlation )A 323 668 :M (of )S 339 668 :M -.22(the )A 359 668 :M -.061(error )A 388 668 :M -.131(terms )A 420 668 :M (for )S 440 668 :M -.663(A )A 455 668 :M -.109(and )A 479 668 :M .996(B.)A 59 686 :M -.067(Cov\(X,Y\)is equal to the sum over all of )A 251 686 :M -.22(the )A 269 686 :M .425 .042(treks, )J 300 686 :M (of )S 314 686 :M -.22(the )A 332 686 :M -.093(product )A 372 686 :M (of )S 386 686 :M -.22(the )A 404 686 :M -.163(edge )A 430 686 :M -.165(labels )A 461 686 :M (on )S 477 686 :M -.33(the)A endp %%Page: 22 22 %%BeginPageSetup initializepage (peter; page: 22 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (22)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.075(trek, times the variance of the source of the trek \(if there is one\). For )A 386 56 :M -.081(example, )A 433 56 :M -.167(in )A 446 56 :M -.054(Figure )A 481 56 :M 1(4,)A 59 64 130 14 rC 189 78 :M psb currentpoint pse 59 64 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4160 div 448 3 -1 roll exch div scale currentpoint translate 64 36 translate -12 284 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (Cov) show 629 284 moveto 384 /Times-Roman f1 (\() show 769 284 moveto 384 /Times-Roman f1 (Y) show 1046 284 moveto 384 /Times-Roman f1 (,) show 1185 284 moveto 384 /Times-Roman f1 (Z) show 1435 284 moveto 384 /Times-Roman f1 (\)) show 1652 284 moveto 384 /Symbol f1 (=) show 1952 284 moveto 384 /Times-Roman f1 (\() show 2085 284 moveto 384 /Symbol f1 (ab) show 2617 284 moveto 384 /Symbol f1 (+) show 2918 284 moveto 384 /Symbol f1 (g) show 3124 284 moveto 384 /Times-Roman f1 (\)) show 3258 284 moveto 384 /Times-Roman f1 (V) show 3541 284 moveto 384 /Times-Roman f1 (\() show 3687 284 moveto 384 /Times-Roman f1 (Z) show 3937 284 moveto 384 /Times-Roman f1 (\)) show end pse gR gS 0 0 552 730 rC 189 74 :M f0_12 sf -.039(. For a proof of the case without correlated errors, see )A 448 74 :M -.054(Glymour)A 59 92 :M -.103(et al. 1987; the case with correlated errors is a simple modification of the latter proof.)A 77 128 :M f2_12 sf 1.122 .112(Lemma )J 123 128 :M .387(6:)A f0_12 sf .232 .023( )J 139 128 :M -.195(There )A 171 128 :M (is )S 185 128 :M -.326(a )A 196 128 :M -.223(SEM )A 226 128 :M .119(M\325\(M,X,Y,)A f2_12 sf .154(Z)A f0_12 sf .122 .012(\) )J 301 128 :M -.083(with )A 328 128 :M -.122(measured )A 379 128 :M -.145(variables )A 427 128 :M f2_12 sf 1.215 .122(V )J 443 128 :M f0_12 sf -.109(and )A 466 128 :M -.331(latent)A 59 146 :M -.024(variables )A f2_12 sf (T)S f0_12 sf (, )S 120 146 :M (such )S 146 146 :M -.249(that )A 167 146 :M .121(G\(M\325\(M,X,Y,)A f2_12 sf .157(Z)A f0_12 sf .179 .018(\)\) )J 257 146 :M .211 .021(= )J 268 146 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\), )J 386 146 :M -.109(and )A 407 146 :M -.22(the )A 425 146 :M -.206(marginal )A 470 146 :M -.106(over)A 59 164 :M f2_12 sf .115(V)A f0_12 sf .088 .009( of )J f1_12 sf .094(S)A f0_12 sf .079(\(M\325\(M,X,Y,)A f2_12 sf .106(Z)A f0_12 sf .152 .015(\)\) is equal to )J f1_12 sf .094(S)A f0_12 sf .096(\(M\).)A 77 182 :M f2_12 sf -.058(Proof)A f0_12 sf -.049(. Let the correlation matrix among the error terms of )A 358 182 :M -.667(M )A 372 182 :M -.163(be )A 387 182 :M f1_12 sf 1.029(S)A f0_12 sf .79 .079(. )J 403 182 :M (If )S 415 182 :M -.22(the )A 433 182 :M -.11(equations )A 482 182 :M -.334(in)A 59 200 :M -.329(M are:)A 239 203 91 27 rC 330 230 :M psb currentpoint pse 239 203 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2912 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (X) -4 346 sh (b) 1337 346 sh (X) 1720 346 sh 224 ns (i) 313 442 sh (ij) 1565 442 sh (j) 2053 442 sh (j) 941 719 sh (i) 1151 719 sh (i) 2754 442 sh 384 /Symbol f1 (=) 542 346 sh (+) 2247 346 sh 224 ns (\271) 1010 719 sh 576 ns (\345) 865 433 sh 384 /Symbol f1 (e) 2545 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 330 215 :M f0_12 sf ( )S 455 215 :M (\(1\))S 59 245 :M -.05(\(where )A 97 245 :M -.082(some )A 127 245 :M (of )S 142 245 :M -.22(the )A 161 245 :M .051(b)A f0_7 sf 0 3 rm (ij)S 0 -3 rm f0_12 sf ( )S 176 245 :M -.22(may )A 201 245 :M -.197(equal )A 231 245 :M .236 .024(zero, )J 260 245 :M -.109(and )A 282 245 :M -.082(some )A 312 245 :M (of )S 327 245 :M -.22(the )A 346 245 :M f1_12 sf -.11(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 358 245 :M -.22(may )A 383 245 :M -.163(be )A 399 245 :M -.177(correlated\) )A 454 245 :M (we )S 473 245 :M -.222(will)A 59 263 :M -.069(construct equations in M\325\(M,X,Y,)A f2_12 sf -.107(Z)A f0_12 sf -.062(\) that are:)A 239 285 147 27 rC 386 312 :M psb currentpoint pse 239 285 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4704 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (X) -4 346 sh (b) 1337 346 sh (X) 1720 346 sh (a) 3006 346 sh (T) 3374 346 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 4261 346 sh (\251) 4389 346 sh 224 ns (i) 313 442 sh (ij) 1565 442 sh (j) 2053 442 sh (j) 941 719 sh (i) 1151 719 sh (ij) 3221 442 sh (j) 3616 442 sh (j) 2624 719 sh (i) 2828 719 sh (i) 4527 442 sh 384 /Symbol f1 (=) 542 346 sh (+) 2247 346 sh (+) 3810 346 sh 224 ns (\271) 1010 719 sh (\243) 2690 719 sh 576 ns (\345) 865 433 sh (\345) 2545 433 sh 384 /Symbol f1 (e) 4108 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 455 297 :M f1_12 sf (\(2\))S 59 327 :M f0_12 sf -.096(by showing that there is a latent variable model of )A f1_12 sf -.146(S)A f0_12 sf -.107( of the form)A 239 348 91 27 rC 330 375 :M psb currentpoint pse 239 348 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2912 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (e) -8 346 sh (e) 2316 346 sh 224 /Times-Roman f1 (i) 201 442 sh (ij) 1429 442 sh (j) 1824 442 sh (j) 832 719 sh (i) 1036 719 sh (i) 2735 442 sh 384 ns (a) 1214 346 sh (T) 1582 346 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 2469 346 sh (\251) 2597 346 sh 384 /Symbol f1 (=) 430 346 sh (+) 2018 346 sh 224 ns (\243) 898 719 sh 576 ns (\345) 753 433 sh end MTsave restore pse gR gS 0 0 552 730 rC 455 360 :M f0_12 sf (\(3\))S 59 390 :M -.099(where each of the T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.092( and )A f1_12 sf -.104(e)A f0_12 sf -.079A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.097( are uncorrelated.)A 77 408 :M .394 .039(By hypothesis, )J f1_12 sf .132(S)A f0_12 sf .056 .006( )J 164 408 :M (is )S 176 408 :M -.326(a )A 185 408 :M -.124(positive )A 226 408 :M -.206(definite )A 265 408 :M -.046(matrix. )A 303 408 :M (By )S 321 408 :M -.33(Lemma )A 360 408 :M (4 )S 370 408 :M -.196(there )A 397 408 :M (is )S 409 408 :M -.326(a )A 418 408 :M -.109(set )A 435 408 :M (of )S 449 408 :M -.163(variables)A 59 426 :M f1_12 sf .06(e)A f0_12 sf S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .034(,...,)A f1_12 sf .06(e)A f0_12 sf S f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf .201 .02( with positive definite matrix )J f1_12 sf .081(S)A f0_12 sf .08 .008(\325 = )J f1_12 sf .081(S)A f0_12 sf .05 .005( - )J f1_12 sf .068(d)A f0_12 sf .167 .017(I, where )J f1_12 sf .068(d)A f0_12 sf .145 .015( > 0. So we can write)J 239 444 :M f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( = )S f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (\325 + )S f1_12 sf (e)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf S 455 444 :M (\(4\))S 59 462 :M -.062(where )A 92 462 :M -.22(the )A 110 462 :M f1_12 sf -.058(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.06A 129 462 :M -.215(are )A 147 462 :M -.163(uncorrelated )A 209 462 :M -.083(with )A 234 462 :M -.245(each )A 259 462 :M -.131(other )A 287 462 :M -.109(and )A 308 462 :M -.22(the )A 326 462 :M f1_12 sf -.076(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.101A 341 462 :M -.031(variables, )A 391 462 :M -.245(each )A 416 462 :M f1_12 sf -.058(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.06A 436 462 :M (is )S 449 462 :M -.189(normally)A 59 480 :M -.12(distributed with mean zero and )A 208 480 :M -.204(variance )A 251 480 :M f1_12 sf .533(d)A f0_12 sf .491 .049(. )J 265 480 :M -.219(The )A 287 480 :M f1_12 sf -.082(e)A f0_12 sf -.086A 304 480 :M -.145(variables )A 350 480 :M -.166(will )A 372 480 :M -.062(serve )A 401 480 :M (as )S 415 480 :M -.22(the )A 433 480 :M -.178(uncorrelated)A 59 498 :M -.061(error )A 88 498 :M -.131(terms )A 120 498 :M -.167(in )A 136 498 :M -.22(the )A 157 498 :M (new )S 184 498 :M -.199(model )A 220 498 :M -.249(that )A 244 498 :M (we )S 265 498 :M -.131(construct; )A 318 498 :M -.22(the )A 339 498 :M f1_12 sf -.112(e)A f0_12 sf -.149A 355 498 :M -.145(variables )A 404 498 :M -.215(are )A 425 498 :M (used )S 454 498 :M -.083(only )A 482 498 :M -.334(in)A 59 516 :M -.109(intermediate stages of constuction, and have the )A 288 516 :M -.163(same )A 316 516 :M -.196(covariance )A 370 516 :M -.22(matrix )A 404 516 :M (as )S 418 516 :M -.22(the )A 436 516 :M f1_12 sf -.17(e)A f0_12 sf ( )S 445 516 :M -.034(variables,)A 59 534 :M -.145(except that the variances of the variables have been )A 302 534 :M -.143(decreased )A 352 534 :M (by )S 368 534 :M -.326(a )A 377 534 :M -.199(small )A 406 534 :M -.166(amount )A 445 534 :M f1_12 sf .402(d,)A f0_12 sf .27 .027( )J 459 534 :M .957 .096(i.e. )J 479 534 :M f1_12 sf .579(S)A f0_12 sf S 59 552 :M .211 .021(= )J 70 552 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 82 552 :M (- )S 90 552 :M f1_12 sf .402(d)A f0_12 sf .565 .056(I. )J 108 552 :M -.165(As )A 125 552 :M -.326(a )A 134 552 :M -.064(first )A 157 552 :M -.082(step )A 180 552 :M -.167(to )A 193 552 :M -.109(constructing )A 255 552 :M -.326(a )A 264 552 :M -.276(latent )A 293 552 :M -.205(variable )A 334 552 :M -.199(model )A 367 552 :M (of )S 381 552 :M f2_12 sf .79(V)A f0_12 sf .497 .05(, )J 398 552 :M (we )S 416 552 :M -.166(will )A 438 552 :M -.109(construct )A 486 552 :M (a)S 59 570 :M -.128(latent variable model of )A f1_12 sf -.144(e)A f0_12 sf -.191A 77 588 :M (By )S 95 588 :M -.33(Lemma )A 134 588 :M .833 .083(5, )J 148 588 :M -.196(there )A 175 588 :M (is )S 187 588 :M -.326(a )A 196 588 :M -.109(set )A 214 588 :M (of )S 229 588 :M -.145(variables )A 276 588 :M f2_12 sf .351(T)A f0_12 sf .29(={T)A f0_7 sf 0 3 rm .153(1)A 0 -3 rm f0_12 sf .239 .024(, )J 317 588 :M 2.857 .286(..., )J 338 588 :M .153(T)A f0_7 sf 0 3 rm .073(n)A 0 -3 rm f0_12 sf .167 .017(} )J 360 588 :M (such )S 387 588 :M -.249(that )A 409 588 :M f1_12 sf .348(e)A f0_12 sf .264A f0_7 sf 0 3 rm .231(1)A 0 -3 rm f0_12 sf .36 .036(, )J 431 588 :M 2.857 .286(..., )J 452 588 :M f1_12 sf .08(e)A f0_12 sf .061A f0_7 sf 0 3 rm .053(n)A 0 -3 rm f0_12 sf ( )S 470 588 :M -.11(with)A 59 606 :M -.119(correlation matrix )A f1_12 sf -.18(S)A f0_12 sf -.119(\325 are a lower triangular )A 264 606 :M -.219(linear )A 294 606 :M -.117(transformation )A 367 606 :M (of )S 381 606 :M 1.314(T)A f0_7 sf 0 3 rm .628(1)A 0 -3 rm f0_12 sf 2.152 .215(,..., )J 416 606 :M .09(T)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf ( )S 431 606 :M -.109(and )A 452 606 :M (for )S 470 606 :M -.326(each)A 59 624 :M -.095(i, the coefficient of T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.096( in the equation for )A f1_12 sf -.113(e)A f0_12 sf -.086A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.099( is not equal to zero. That is)A 239 645 65 27 rC 304 672 :M psb currentpoint pse 239 645 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2080 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (e) -8 346 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 145 346 sh (a) 1246 346 sh (T) 1614 346 sh 224 ns (i) 283 442 sh (ij) 1461 442 sh (j) 1856 442 sh (j) 864 719 sh (i) 1069 719 sh 384 /Symbol f1 (=) 462 346 sh 224 ns (<) 931 719 sh 576 ns (\345) 785 433 sh end MTsave restore pse gR gS 0 0 552 730 rC 455 657 :M f0_12 sf (\(5\))S endp %%Page: 23 23 %%BeginPageSetup initializepage (peter; page: 23 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (23)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf .312 .031(where a)J f0_7 sf 0 3 rm .027(ii)A 0 -3 rm f0_12 sf ( )S f1_12 sf .093A f0_12 sf .141 .014( 0.)J 77 74 :M -.128(There is a directed graph H that represents this latent variable model of the )A 430 74 :M f1_12 sf -.076(e)A f0_12 sf -.058A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 445 74 :M -.034(variables,)A 59 92 :M -.05(in which there is an edge from T)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.04( to )A f1_12 sf -.054(e)A f0_12 sf (')S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 244 92 :M -.083(only )A 269 92 :M -.164(if )A 280 92 :M -.334(j )A 287 92 :M cF f1_12 sf .038A sf .376 .038( )J 298 92 :M .555 .055(i. )J 309 92 :M (From )S 339 92 :M -.22(the )A 357 92 :M -.109(construction )A 419 92 :M (of )S 433 92 :M 1.114 .111(H, )J 450 92 :M -.196(there )A 477 92 :M -.323(are)A 59 110 :M -.047(no edges from T)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.035( to )A f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S 171 110 :M (unless )S 205 110 :M -.334(j )A 212 110 :M .211 .021(= )J 223 110 :M .833 .083(1. )J 237 110 :M .224 .022(Hence, )J 275 110 :M (for )S 293 110 :M -.129(every )A 323 110 :M -.334(j )A 330 110 :M f1_12 sf 1.347A f0_12 sf 2.359 .236(\3121, )J 357 110 :M -.167(in )A 370 110 :M .306 .031(H )J 383 110 :M -.129(every )A 413 110 :M -.163(trek )A 435 110 :M -.139(between )A 478 110 :M f1_12 sf .099(e)A f0_12 sf .075A f0_7 sf 0 3 rm (1)S 0 -3 rm 59 128 :M f0_12 sf -.019(and )A f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.018( contains T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf (. )S 154 128 :M -.164(It )A 165 128 :M (follows )S 205 128 :M -.249(that )A 226 128 :M -.196(there )A 253 128 :M (is )S 265 128 :M -.33(at )A 277 128 :M -.084(most )A 304 128 :M -.109(one )A 325 128 :M -.163(trek )A 347 128 :M -.139(between )A 390 128 :M f1_12 sf .08(e)A f0_12 sf .061A f0_7 sf 0 3 rm .053(1)A 0 -3 rm f0_12 sf ( )S 407 128 :M -.109(and )A 428 128 :M f1_12 sf .243(e)A f0_12 sf .184A f0_7 sf 0 3 rm .09(j)A 0 -3 rm f0_12 sf .252 .025(. )J 447 128 :M -.219(The )A 469 128 :M -.217(edge)A 59 146 :M -.08(from )A 86 146 :M .09(T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S 101 146 :M -.167(to )A 114 146 :M f1_12 sf .08(e)A f0_12 sf .061A f0_7 sf 0 3 rm .053(1)A 0 -3 rm f0_12 sf ( )S 131 146 :M (is )S 143 146 :M -.111(not )A 162 146 :M .236 .024(zero. )J 190 146 :M -.128(Hence )A 224 146 :M -.334(it )A 234 146 :M (follows )S 274 146 :M -.08(from )A 301 146 :M -.22(the )A 319 146 :M -.163(trek )A 341 146 :M -.163(rule )A 363 146 :M (for )S 381 146 :M -.24(calculating )A 436 146 :M -.163(covariances)A 59 164 :M -.109(from a path diagram, that if )A f1_12 sf -.123(e)A f0_7 sf 0 3 rm -.082(1)A 0 -3 rm f0_12 sf -.119( and)A f1_12 sf -.097( e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.111( are not correlated )A 318 164 :M -.167(in )A 331 164 :M f1_12 sf .632(S)A f0_12 sf .267 .027( )J 343 164 :M .898 .09(\(i.e. )J 367 164 :M -.196(there )A 394 164 :M (is )S 406 164 :M (no )S 422 164 :M -.136(double-headed)A 59 182 :M (arrow )S 91 182 :M -.139(between )A 134 182 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 151 182 :M -.093(and)A f1_12 sf ( )S 172 182 :M f0_12 sf .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 187 182 :M -.167(in )A 200 182 :M -.062(G\(M\)\) )A 235 182 :M -.165(then )A 259 182 :M -.22(the )A 277 182 :M -.163(edge )A 303 182 :M -.08(from )A 330 182 :M .09(T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S 346 182 :M -.167(to )A 360 182 :M f1_12 sf -.11(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S 372 182 :M (is )S 385 182 :M .236 .024(zero. )J 414 182 :M (\(In )S 433 182 :M -.22(the )A 452 182 :M -.274(example)A 59 200 :M .464 .046(from Figure 12, a)J f0_7 sf 0 3 rm .091(12)A 0 -3 rm f0_12 sf .196 .02( = a)J f0_7 sf 0 3 rm .091(14)A 0 -3 rm f0_12 sf .196 .02( = a)J f0_7 sf 0 3 rm .091(15)A 0 -3 rm f0_12 sf .196 .02( = a)J f0_7 sf 0 3 rm .091(16)A 0 -3 rm f0_12 sf .267 .027( = 0.\))J 77 218 :M -.166(Applying )A 125 218 :M -.084(this )A 146 218 :M -.123(strategy )A 187 218 :M -.167(to )A 200 218 :M -.245(each )A 225 218 :M (of )S 239 218 :M -.22(the )A 257 218 :M -.164(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 270 218 :M -.145(variables )A 316 218 :M -.167(in )A 330 218 :M .448 .045(turn, )J 358 218 :M (we )S 377 218 :M -.217(can )A 398 218 :M .259 .026(now )J 424 218 :M .479 .048(show )J 455 218 :M -.249(that )A 477 218 :M (for)S 59 236 :M -.082(each i and r > i, if there is no trek between )A 262 236 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 277 236 :M -.109(and )A 298 236 :M f1_12 sf -.076(e)A f0_12 sf -.058A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 313 236 :M -.165(containing )A 366 236 :M -.326(a )A 375 236 :M -.205(variable )A 416 236 :M .348(T)A f0_7 sf 0 3 rm .092(j)A 0 -3 rm f0_12 sf .259 .026(, )J 433 236 :M -.062(where )A 466 236 :M -.334(j )A 473 236 :M .211 .021(< )J 484 236 :M .666(i,)A 59 254 :M -.1(and )A f1_12 sf -.104(e)A f0_12 sf -.079A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.092( and )A f1_12 sf -.104(e)A f0_12 sf -.079A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.102( are )A 145 254 :M -.163(uncorrelated )A 207 254 :M -.167(in )A 220 254 :M f1_12 sf 1.029(S)A f0_12 sf .79 .079(, )J 236 254 :M -.165(then )A 260 254 :M -.196(there )A 287 254 :M (is )S 299 254 :M (no )S 315 254 :M f1_12 sf .433(T)A f0_7 sf 0 3 rm .115(i)A 0 -3 rm f0_12 sf .177 .018( )J 329 254 :M f1_12 sf .126A f0_12 sf ( )S 345 254 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 360 254 :M -.163(edge )A 386 254 :M -.167(in )A 399 254 :M 1.114 .111(H. )J 416 254 :M .197 .02(Suppose )J 461 254 :M (on )S 477 254 :M -.33(the)A 59 272 :M -.103(contrary that in H there is no trek )A 219 272 :M -.139(between )A 262 272 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 277 272 :M -.109(and )A 298 272 :M f1_12 sf -.076(e)A f0_12 sf -.058A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 313 272 :M -.165(containing )A 366 272 :M -.326(a )A 375 272 :M -.205(variable )A 416 272 :M .348(T)A f0_7 sf 0 3 rm .092(j)A 0 -3 rm f0_12 sf .259 .026(, )J 433 272 :M -.062(where )A 466 272 :M -.334(j )A 473 272 :M .211 .021(< )J 484 272 :M .666(i,)A 59 290 :M -.078(and )A f1_12 sf -.081(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.071( and )A f1_12 sf -.081(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.072( are uncorrelated in M, but the T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.181A f0_12 sf ( )S f1_12 sf -.081(e)A f0_12 sf -.061A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.075( edge is in H. By )A 384 290 :M -.22(the )A 402 290 :M -.109(construction )A 464 290 :M (of )S 478 290 :M 1.337(H,)A 59 308 :M -.045(if k > i, then there is no edge from T)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.037( to )A f1_12 sf -.051(e)A f0_12 sf S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.044(. It follows that )A 338 308 :M -.164(if )A 349 308 :M -.167(in )A 362 308 :M .306 .031(H )J 375 308 :M -.196(there )A 402 308 :M (is )S 414 308 :M (no )S 430 308 :M -.163(trek )A 452 308 :M -.163(between)A 59 326 :M f1_12 sf -.138(e)A f0_12 sf -.105A f0_7 sf 0 3 rm -.061(r)A 0 -3 rm f0_12 sf -.153( and )A 94 326 :M f1_12 sf -.076(e)A f0_12 sf -.058A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 109 326 :M -.165(containing )A 162 326 :M -.326(a )A 171 326 :M -.205(variable )A 212 326 :M .348(T)A f0_7 sf 0 3 rm .092(j)A 0 -3 rm f0_12 sf .259 .026(, )J 229 326 :M -.062(where )A 262 326 :M -.334(j )A 269 326 :M .211 .021(< )J 280 326 :M .555 .055(i, )J 291 326 :M -.165(then )A 315 326 :M -.129(every )A 345 326 :M -.163(trek )A 367 326 :M -.139(between )A 410 326 :M f1_12 sf -.076(e)A f0_12 sf -.058A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 425 326 :M -.109(and )A 446 326 :M -.109(any )A 467 326 :M -.163(other)A 59 344 :M -.205(variable )A 100 344 :M -.123(contains )A 143 344 :M -.22(the )A 161 344 :M -.163(edge )A 187 344 :M -.08(from )A 214 344 :M -.164(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 227 344 :M -.167(to )A 240 344 :M f1_12 sf .243(e)A f0_12 sf .184A f0_7 sf 0 3 rm .09(i)A 0 -3 rm f0_12 sf .252 .025(, )J 259 344 :M -.065(which )A 292 344 :M (is )S 304 344 :M -.167(in )A 317 344 :M .306 .031(H )J 330 344 :M -.131(since )A 358 344 :M -.094(a)A f0_7 sf 0 3 rm -.034(ii)A 0 -3 rm f0_12 sf ( )S 372 344 :M cF f1_12 sf .038A sf .376 .038( )J 384 344 :M .833 .083(0. )J 399 344 :M -.219(The )A 422 344 :M -.164(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 436 344 :M f1_12 sf .126A f0_12 sf ( )S 453 344 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 469 344 :M -.217(edge)A 59 362 :M -.067(exists by hypothesis, so there is exactly one trek between )A 333 362 :M f1_12 sf -.076(e)A f0_12 sf -.058A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 348 362 :M -.109(and )A 369 362 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 384 362 :M -.167(in )A 397 362 :M 1.114 .111(H. )J 414 362 :M .224 .022(Hence, )J 452 362 :M -.167(in )A 465 362 :M -.161(every)A 59 380 :M .036 .004(SEM L with vertices {)J f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf (,...,)S f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf .035 .003(} and directed graph G\(L\) = H, )J f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .022 .002( and )J f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( are )S 432 380 :M -.196(correlated )A 482 380 :M -.334(in)A 59 398 :M f1_12 sf -.136(S)A f0_12 sf -.092(\(L\). \(Note that this could not be claimed if there were more )A 349 398 :M -.165(than )A 373 398 :M -.109(one )A 394 398 :M -.163(trek )A 416 398 :M -.139(between )A 459 398 :M f1_12 sf -.076(e)A f0_12 sf -.058A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S 474 398 :M -.163(and)A 59 416 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 75 416 :M -.131(since )A 104 416 :M -.167(in )A 118 416 :M -.249(that )A 140 416 :M -.161(case )A 165 416 :M -.22(the )A 184 416 :M -.064(treks )A 212 416 :M -.2(might )A 244 416 :M -.273(cancel )A 278 416 :M -.245(each )A 304 416 :M .208 .021(other.\) )J 341 416 :M -.131(Since )A 372 416 :M -.22(the )A 391 416 :M -.148(covariances )A 452 416 :M -.163(between)A 59 434 :M -.166(distinct )A 97 434 :M f1_12 sf -.112(e)A f0_12 sf -.149A 110 434 :M -.145(variables )A 156 434 :M -.215(are )A 174 434 :M -.197(equal )A 203 434 :M -.167(to )A 216 434 :M -.22(the )A 234 434 :M -.136(correlations )A 293 434 :M -.139(between )A 336 434 :M -.22(the )A 354 434 :M -.049(corresponding )A 426 434 :M f1_12 sf -.17(e)A f0_12 sf ( )S 435 434 :M -.031(variables, )A 485 434 :M -.668(it)A 59 452 :M -.091(follows that )A f1_12 sf -.104(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.092( and )A f1_12 sf -.104(e)A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.087( are correlated in )A f1_12 sf -.14(S)A f0_12 sf -.093(, and hence there is a )A 347 452 :M -.125(double-headed )A 420 452 :M (arrow )S 452 452 :M -.163(between)A 59 470 :M f1_12 sf -.061(e)A f0_12 sf S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.054( and )A f1_12 sf -.061(e)A f0_12 sf S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.055( in G\(M\). This is a contradiction.)A 77 488 :M -.043(The graph H for the path diagram in Figure 12 is shown in Figure 14.)A 101 492 366 93 rC 101 505 :M f1_12 sf (T)S 109 508 :M f1_7 sf (1)S 113 505 :M f1_12 sf ( )S 116 505 :M ( )S 119 505 :M ( )S 122 505 :M ( )S 125 505 :M ( )S 128 505 :M ( )S 131 505 :M ( )S 134 505 :M ( )S 137 505 :M ( )S 140 505 :M ( )S 143 505 :M ( )S 146 505 :M ( )S 149 505 :M ( )S 152 505 :M ( )S 155 505 :M ( )S 158 505 :M ( )S 161 505 :M ( )S 164 505 :M (T)S 172 508 :M f1_7 sf (3)S 176 505 :M f1_12 sf ( )S 179 505 :M ( )S 182 505 :M ( )S 185 505 :M ( )S 188 505 :M ( )S 191 505 :M ( )S 194 505 :M ( )S 197 505 :M ( )S 200 505 :M ( )S 203 505 :M ( )S 206 505 :M ( )S 209 505 :M ( )S 212 505 :M ( )S 215 505 :M ( )S 218 505 :M ( )S 221 505 :M ( )S 224 505 :M (T)S 232 508 :M f1_7 sf (5)S 236 505 :M f1_12 sf ( )S 239 505 :M ( )S 242 505 :M ( )S 245 505 :M ( )S 248 505 :M ( )S 251 505 :M ( )S 254 505 :M ( )S 257 505 :M ( )S 260 505 :M ( )S 263 505 :M ( )S 266 505 :M ( )S 269 505 :M ( )S 272 505 :M ( )S 275 505 :M ( )S 278 505 :M ( )S 281 505 :M (T)S 289 508 :M f1_7 sf (4)S 293 505 :M f1_12 sf ( )S 296 505 :M ( )S 299 505 :M ( )S 302 505 :M ( )S 305 505 :M ( )S 308 505 :M ( )S 311 505 :M ( )S 314 505 :M ( )S 317 505 :M ( )S 320 505 :M ( )S 323 505 :M ( )S 326 505 :M ( )S 329 505 :M ( )S 332 505 :M ( )S 335 505 :M ( )S 338 505 :M ( )S 341 505 :M ( )S 344 505 :M ( )S 347 505 :M (T)S 355 508 :M f1_7 sf (6)S 359 505 :M f1_12 sf ( )S 362 505 :M ( )S 365 505 :M ( )S 368 505 :M ( )S 371 505 :M ( )S 374 505 :M ( )S 377 505 :M ( )S 380 505 :M ( )S 383 505 :M ( )S 386 505 :M ( )S 389 505 :M ( )S 392 505 :M ( )S 395 505 :M ( )S 398 505 :M ( )S 401 505 :M ( )S 404 505 :M ( )S 407 505 :M ( )S 410 505 :M (T)S 418 508 :M f1_7 sf (2)S 101 573 :M f1_12 sf (e)S 106 573 :M f0_12 sf (')S 110 576 :M f0_7 sf (X)S 115 576 :M (1)S 119 573 :M f0_12 sf ( )S 122 573 :M ( )S 125 573 :M ( )S 128 573 :M ( )S 131 573 :M ( )S 134 573 :M ( )S 137 573 :M ( )S 140 573 :M ( )S 143 573 :M ( )S 146 573 :M ( )S 149 573 :M ( )S 152 573 :M ( )S 155 573 :M ( )S 158 573 :M ( )S 161 573 :M ( )S 164 573 :M f1_12 sf (e)S 169 573 :M f0_12 sf (')S 173 576 :M f0_7 sf (X)S 178 576 :M (3)S 182 573 :M f0_12 sf ( )S 185 573 :M ( )S 188 573 :M ( )S 191 573 :M ( )S 194 573 :M ( )S 197 573 :M ( )S 200 573 :M ( )S 203 573 :M ( )S 206 573 :M ( )S 209 573 :M ( )S 212 573 :M ( )S 215 573 :M ( )S 218 573 :M ( )S 221 573 :M f1_12 sf (e)S 226 573 :M f0_12 sf (')S 230 576 :M f0_7 sf (X)S 235 576 :M (5)S 239 573 :M f0_12 sf ( )S 242 573 :M ( )S 245 573 :M ( )S 248 573 :M ( )S 251 573 :M ( )S 254 573 :M ( )S 257 573 :M ( )S 260 573 :M ( )S 263 573 :M ( )S 266 573 :M ( )S 269 573 :M ( )S 272 573 :M ( )S 275 573 :M ( )S 278 573 :M ( )S 281 573 :M ( )S 284 573 :M f1_12 sf (e)S 289 573 :M f0_12 sf (')S 293 576 :M f0_7 sf (X)S 298 576 :M (4)S 302 573 :M f0_12 sf ( )S 305 573 :M ( )S 308 573 :M ( )S 311 573 :M ( )S 314 573 :M ( )S 317 573 :M ( )S 320 573 :M ( )S 323 573 :M ( )S 326 573 :M ( )S 329 573 :M ( )S 332 573 :M ( )S 335 573 :M ( )S 338 573 :M ( )S 341 573 :M ( )S 344 573 :M ( )S 347 573 :M f1_12 sf (e)S 352 573 :M f0_12 sf (')S 356 576 :M f0_7 sf (X)S 361 576 :M (6)S 365 573 :M f0_12 sf ( )S 368 573 :M ( )S 371 573 :M ( )S 374 573 :M ( )S 377 573 :M ( )S 380 573 :M ( )S 383 573 :M ( )S 386 573 :M ( )S 389 573 :M ( )S 392 573 :M ( )S 395 573 :M ( )S 398 573 :M ( )S 401 573 :M ( )S 404 573 :M ( )S 407 573 :M ( )S 410 573 :M f1_12 sf (e)S 415 573 :M f0_12 sf (')S 419 576 :M f0_7 sf (X)S 424 576 :M (2)S gR gS 100 491 368 94 rC -.75 -.75 107.75 556.75 .75 .75 107 512 @b np 111 554 :M 103 554 :L 107 562 :L 111 554 :L .75 lw eofill 103 554.75 -.75 .75 111.75 554 .75 103 554 @a 103 554.75 -.75 .75 107.75 562 .75 103 554 @a -.75 -.75 107.75 562.75 .75 .75 111 554 @b -.75 -.75 170.75 555.75 .75 .75 170 511 @b np 174 553 :M 166 553 :L 170 561 :L 174 553 :L eofill 166 553.75 -.75 .75 174.75 553 .75 166 553 @a 166 553.75 -.75 .75 170.75 561 .75 166 553 @a -.75 -.75 170.75 561.75 .75 .75 174 553 @b -.75 -.75 229.75 556.75 .75 .75 229 512 @b np 233 554 :M 225 554 :L 229 562 :L 233 554 :L eofill 225 554.75 -.75 .75 233.75 554 .75 225 554 @a 225 554.75 -.75 .75 229.75 562 .75 225 554 @a -.75 -.75 229.75 562.75 .75 .75 233 554 @b -.75 -.75 289.75 556.75 .75 .75 289 512 @b np 293 554 :M 285 554 :L 289 562 :L 293 554 :L eofill 285 554.75 -.75 .75 293.75 554 .75 285 554 @a 285 554.75 -.75 .75 289.75 562 .75 285 554 @a -.75 -.75 289.75 562.75 .75 .75 293 554 @b -.75 -.75 352.75 555.75 .75 .75 352 511 @b np 356 553 :M 348 553 :L 352 561 :L 356 553 :L eofill 348 553.75 -.75 .75 356.75 553 .75 348 553 @a 348 553.75 -.75 .75 352.75 561 .75 348 553 @a -.75 -.75 352.75 561.75 .75 .75 356 553 @b -.75 -.75 414.75 557.75 .75 .75 414 513 @b np 418 555 :M 410 555 :L 414 563 :L 418 555 :L eofill 410 555.75 -.75 .75 418.75 555 .75 410 555 @a 410 555.75 -.75 .75 414.75 563 .75 410 555 @a -.75 -.75 414.75 563.75 .75 .75 418 555 @b 107 512.75 -.75 .75 159.75 555 .75 107 512 @a np 160 551 :M 156 557 :L 164 559 :L 160 551 :L eofill -.75 -.75 156.75 557.75 .75 .75 160 551 @b 156 557.75 -.75 .75 164.75 559 .75 156 557 @a 160 551.75 -.75 .75 164.75 559 .75 160 551 @a -.75 -.75 364.75 558.75 .75 .75 415 512 @b np 368 559 :M 364 554 :L 360 562 :L 368 559 :L eofill 364 554.75 -.75 .75 368.75 559 .75 364 554 @a -.75 -.75 360.75 562.75 .75 .75 364 554 @b -.75 -.75 360.75 562.75 .75 .75 368 559 @b 170 510.75 -.75 .75 221.75 555 .75 170 510 @a np 221 551 :M 217 556 :L 225 559 :L 221 551 :L eofill -.75 -.75 217.75 556.75 .75 .75 221 551 @b 217 556.75 -.75 .75 225.75 559 .75 217 556 @a 221 551.75 -.75 .75 225.75 559 .75 221 551 @a -.75 -.75 241.75 555.75 .75 .75 290 510 @b np 245 556 :M 240 551 :L 237 559 :L 245 556 :L eofill 240 551.75 -.75 .75 245.75 556 .75 240 551 @a -.75 -.75 237.75 559.75 .75 .75 240 551 @b -.75 -.75 237.75 559.75 .75 .75 245 556 @b 291 509.75 -.75 .75 341.75 555 .75 291 509 @a np 342 551 :M 337 556 :L 345 559 :L 342 551 :L eofill -.75 -.75 337.75 556.75 .75 .75 342 551 @b 337 556.75 -.75 .75 345.75 559 .75 337 556 @a 342 551.75 -.75 .75 345.75 559 .75 342 551 @a 230 511.75 -.75 .75 336.75 560 .75 230 511 @a np 335 555 :M 332 562 :L 341 562 :L 335 555 :L eofill -.75 -.75 332.75 562.75 .75 .75 335 555 @b 332 562.75 -.75 .75 341.75 562 .75 332 562 @a 335 555.75 -.75 .75 341.75 562 .75 335 555 @a gR gS 0 0 552 730 rC 238 600 :M f2_12 sf 2.479 .248(Figure 14: H)J 59 624 :M f0_12 sf -.054(From the latent variable model of the )A f1_12 sf -.059(e)A f0_12 sf -.058(' variables, we can now form a model M\325\(M,X,Y,)A f2_12 sf -.09(Z)A f0_12 sf <29>S 59 642 :M -.014(with measured variables )A f2_12 sf (V)S f0_12 sf -.013( and latent variables T)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.01(,...,T)A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf -.013(, but without correlated errors.)A endp %%Page: 24 24 %%BeginPageSetup initializepage (peter; page: 24 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (24)S gR gS 210 41 147 27 rC 357 68 :M psb currentpoint pse 210 41 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4704 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (X) -4 346 sh (b) 1337 346 sh (X) 1720 346 sh (a) 3006 346 sh (T) 3374 346 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 4261 346 sh (\251) 4389 346 sh 224 ns (i) 313 442 sh (ij) 1565 442 sh (j) 2053 442 sh (j) 941 719 sh (i) 1151 719 sh (ij) 3221 442 sh (j) 3616 442 sh (j) 2624 719 sh (i) 2829 719 sh (i) 4527 442 sh 384 /Symbol f1 (=) 542 346 sh (+) 2247 346 sh (+) 3810 346 sh 224 ns (\271) 1010 719 sh (<) 2691 719 sh 576 ns (\345) 865 433 sh (\345) 2545 433 sh 384 /Symbol f1 (e) 4108 346 sh end MTsave restore pse gR gS 0 0 552 730 rC 77 83 :M f0_12 sf -.164(It )A 92 83 :M (follows )S 136 83 :M -.08(from )A 167 83 :M -.11(equations )A 220 83 :M .724 .072(\(1\), )J 246 83 :M .724 .072(\(4\), )J 272 83 :M -.109(and )A 297 83 :M (\(5\) )S 319 83 :M -.249(that )A 345 83 :M -.22(the )A 368 83 :M -.206(marginal )A 418 83 :M -.111(distribution )A 481 83 :M (of)S 59 101 :M f2_12 sf (V)S f0_12 sf .036(={X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .041<2CC958>A f0_7 sf 0 3 rm (n)S 0 -3 rm f0_12 sf .116 .012(} in M\325\(M,X,Y,)J f2_12 sf (Z)S f0_12 sf .073 .007(\) is the same as the distribution of )J f2_12 sf (V)S f0_12 sf .06 .006( in M.)J 77 119 :M -.326(We )A 97 119 :M -.166(will )A 119 119 :M .259 .026(now )J 144 119 :M .479 .048(show )J 174 119 :M -.249(that )A 195 119 :M .121(G\(M\325\(M,X,Y,)A f2_12 sf .157(Z)A f0_12 sf .179 .018(\)\) )J 285 119 :M .211 .021(= )J 296 119 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\). )J 414 119 :M .258 .026(For )J 436 119 :M -.145(variables )A 483 119 :M (A)S 59 137 :M -.109(and )A 83 137 :M (B )S 98 137 :M -.167(in )A 114 137 :M f2_12 sf .79(V)A f0_12 sf .497 .05(, )J 134 137 :M (by )S 153 137 :M -.22(the )A 174 137 :M -.109(construction )A 239 137 :M (of )S 256 137 :M .261 .026(M\325, )J 282 137 :M -.196(there )A 313 137 :M (is )S 329 137 :M -.163(an )A 348 137 :M -.163(edge )A 378 137 :M -.139(between )A 425 137 :M -.663(A )A 441 137 :M -.109(and )A 466 137 :M (B )S 482 137 :M -.334(in)A 59 155 :M -.073(G\(M\325\(M,X,Y,)A f2_12 sf -.094(Z)A f0_12 sf -.056(\)\) if and only if there is an edge between A and B in )A 385 155 :M 1.114 .111(G, )J 402 155 :M -.109(and )A 423 155 :M -.196(hence )A 454 155 :M -.163(an )A 469 155 :M -.217(edge)A 59 173 :M -.139(between )A 103 173 :M -.663(A )A 116 173 :M -.109(and )A 139 173 :M (B )S 153 173 :M -.167(in )A 168 173 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\). )J 288 173 :M -.106(\(Hence )A 328 173 :M -.22(the )A 348 173 :M -.122(ancestor )A 393 173 :M -.146(relations )A 439 173 :M -.132(among )A 477 173 :M -.33(the)A 59 191 :M -.053(substantive variables in G\(M\325\(M,X,Y,)A f2_12 sf -.082(Z)A f0_12 sf -.05(\)\) are the same as )A 336 191 :M -.22(the )A 354 191 :M -.122(ancestor )A 397 191 :M -.146(relations )A 441 191 :M -.132(among )A 477 191 :M -.33(the)A 59 209 :M -.049(corresponding )A 132 209 :M -.145(variables )A 179 209 :M -.167(in )A 193 209 :M .432 .043(G\(M\).\) )J 233 209 :M -.195(There )A 265 209 :M (is )S 278 209 :M -.163(an )A 294 209 :M -.163(edge )A 322 209 :M -.139(between )A 367 209 :M -.326(a )A 378 209 :M -.205(variable )A 421 209 :M -.33(T )A 434 209 :M -.167(in )A 449 209 :M f2_12 sf (T)S f0_12 sf ( )S 463 209 :M -.109(and )A 486 209 :M (a)S 59 227 :M -.07(variable A in )A f2_12 sf -.13(V)A f0_12 sf -.083( in G\(M\325\(M,X,Y,)A f2_12 sf -.12(Z)A f0_12 sf -.07(\)\) if and only if there is an edge between T and )A f1_12 sf -.079(e)A f0_12 sf -.06A f0_7 sf 0 3 rm -.076(A)A 0 -3 rm f0_12 sf ( )S 465 227 :M -.167(in )A 478 227 :M 1.337(H.)A 59 245 :M -.326(We )A 79 245 :M -.163(have )A 105 245 :M -.187(already )A 143 245 :M .447 .045(shown )J 179 245 :M -.249(that )A 200 245 :M (for )S 218 245 :M -.245(each )A 243 245 :M -.334(i )A 250 245 :M -.109(and )A 272 245 :M (r )S 281 245 :M .211 .021(> )J 293 245 :M .555 .055(i, )J 305 245 :M -.164(if )A 317 245 :M -.196(there )A 345 245 :M (is )S 358 245 :M (no )S 375 245 :M -.163(trek )A 398 245 :M -.139(between )A 442 245 :M f1_12 sf -.213(e)A f0_12 sf -.162A f0_7 sf 0 3 rm -.094(r)A 0 -3 rm f0_12 sf ( )S 458 245 :M -.109(and )A 480 245 :M f1_12 sf -.097(e)A f0_12 sf -.073A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 263 :M f0_12 sf -.062(containing a variable T)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.057(, where j < i, and )A f1_12 sf -.068(e)A f0_12 sf -.052A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.06( and )A f1_12 sf -.068(e)A f0_12 sf -.052A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.059( are uncorrelated in )A f1_12 sf -.092(S)A f0_12 sf -.056(, then there is no )A f1_12 sf -.095(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 281 :M f1_12 sf S f0_12 sf ( )S f1_12 sf (e)S f0_12 sf S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.014( edge in H. It follows )A 191 281 :M -.249(that )A 212 281 :M (for )S 230 281 :M -.245(each )A 255 281 :M -.334(i )A 262 281 :M -.109(and )A 283 281 :M (r )S 291 281 :M .211 .021(> )J 302 281 :M .555 .055(i, )J 313 281 :M -.164(if )A 324 281 :M -.196(there )A 351 281 :M (is )S 363 281 :M (no )S 379 281 :M -.163(trek )A 401 281 :M -.139(between )A 444 281 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 459 281 :M -.109(and )A 480 281 :M .32(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 299 :M f0_12 sf -.075(containing a variable T)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.069(, where j < i, and )A f1_12 sf -.082(e)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.073( and )A f1_12 sf -.082(e)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( )S 294 299 :M -.215(are )A 312 299 :M -.163(uncorrelated )A 374 299 :M -.167(in )A 387 299 :M f1_12 sf 1.029(S)A f0_12 sf .79 .079(, )J 403 299 :M -.165(then )A 427 299 :M -.196(there )A 454 299 :M (is )S 466 299 :M (no )S 482 299 :M -.217(T)A f0_7 sf 0 3 rm (i)S 0 -3 rm 59 317 :M f1_12 sf .126A f0_12 sf ( )S 78 317 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 96 317 :M -.163(edge )A 125 317 :M -.167(in )A 141 317 :M .186(G\(M\325\(M,X,Y,)A f2_12 sf .24(Z)A f0_12 sf .323 .032(\)\). )J 238 317 :M -.065(\(This )A 270 317 :M -.275(latter )A 300 317 :M -.081(property )A 347 317 :M (is )S 362 317 :M -.22(the )A 383 317 :M -.081(property )A 430 317 :M -.148(obtaining )A 482 317 :M -.334(in)A 59 335 :M .358(G)A f0_7 sf 0 3 rm .125(Construct)A 0 -3 rm f0_12 sf .238(\(G\(M\),X,Y,)A f2_12 sf .331(Z)A f0_12 sf .263 .026(\) )J 173 335 :M (by )S 189 335 :M -.21(application )A 244 335 :M (of )S 258 335 :M (steps )S 286 335 :M (2 )S 296 335 :M -.109(and )A 317 335 :M .775 .077(3.\) )J 335 335 :M -.128(Hence )A 369 335 :M .358(G)A f0_7 sf 0 3 rm .125(Construct)A 0 -3 rm f0_12 sf .238(\(G\(M\),X,Y,)A f2_12 sf .331(Z)A f0_12 sf .263 .026(\) )J 484 335 :M (=)S 59 353 :M .086(G\(M\325\(M,X,Y,)A f2_12 sf .111(Z)A f0_12 sf .117 .012(\)\) )J f1_12 sf <5C>S 77 371 :M f0_12 sf -.219(The )A 99 371 :M -.165(next )A 123 371 :M -.052(series )A 154 371 :M (of )S 168 371 :M -.22(lemmas )A 208 371 :M .479 .048(show )J 238 371 :M -.249(that )A 259 371 :M -.164(if )A 270 371 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 287 371 :M -.109(and )A 308 371 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 325 371 :M -.215(are )A 343 371 :M -.117(d-separated )A 402 371 :M -.132(given )A 433 371 :M f2_12 sf (Z)S f0_12 sf ( )S 446 371 :M -.167(in )A 460 371 :M .171(G\(M\),)A 59 389 :M .032 .003(then X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .019 .002( and X)J f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf .027 .003( are d-separated given )J f2_12 sf (Z)S f0_12 sf .038 .004( in G\(M\325\(M,X,Y,)J f2_12 sf (Z)S f0_12 sf (\)\).)S 77 407 :M -.326(We )A 99 407 :M -.166(will )A 123 407 :M -.33(call )A 145 407 :M -.326(a )A 156 407 :M -.163(trek )A 180 407 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 197 407 :M f1_12 sf .126A f0_12 sf ( )S 215 407 :M -.36(T)A f0_7 sf 0 3 rm -.267(m)A 0 -3 rm f0_12 sf ( )S 233 407 :M f1_12 sf .126A f0_12 sf ( )S 251 407 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 268 407 :M -.249(that )A 291 407 :M -.123(contains )A 336 407 :M -.326(a )A 347 407 :M -.33(T )A 360 407 :M -.205(variable )A 404 407 :M -.326(a )A 416 407 :M f2_12 sf .304(latent)A f2_7 sf 0 3 rm .109 .011( )J 0 -3 rm 453 407 :M f2_12 sf .15(trek)A f0_12 sf .084 .008( )J 482 407 :M -.334(in)A 59 425 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\). )J 177 425 :M (In )S 191 425 :M .456 .046(G\(M\), )J 226 425 :M -.326(a )A 235 425 :M f2_12 sf 2.186 .219(correlated )J 296 425 :M 1.13 .113(error )J 329 425 :M 1.203 .12(trek )J 356 425 :M .589(sequence)A f0_12 sf .307 .031( )J 411 425 :M (is )S 423 425 :M -.326(a )A 433 425 :M -.121(sequence )A 481 425 :M (of)S 59 443 :M -.04(vertices such that no pair of vertices adjacent in the sequence are )A 427 443 :M -.131(identical, )A 474 443 :M -.163(and)A 59 461 :M -.09(for each consecutive pair of vertices X)A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf -.101( and X)A f0_7 sf 0 3 rm -.052(s)A 0 -3 rm f0_12 sf ( )S 281 461 :M -.167(in )A 294 461 :M -.22(the )A 312 461 :M (sequence, )S 363 461 :M -.196(there )A 390 461 :M (is )S 402 461 :M -.163(an )A 417 461 :M -.163(edge )A 443 461 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 458 461 :M f1_12 sf .4A f0_12 sf .096 .01( )J 475 461 :M .972(X)A f0_7 sf 0 3 rm .306(s)A 0 -3 rm f0_12 sf (.)S 59 479 :M .258 .026(For )J 80 479 :M -.235(example )A 123 479 :M -.167(in )A 136 479 :M -.054(Figure )A 171 479 :M .769 .077(11, )J 191 479 :M -.22(the )A 209 479 :M -.121(sequence )A 256 479 :M (of )S 270 479 :M -.163(vertices )A 310 479 :M 1.654 .165( )J 397 479 :M (is )S 409 479 :M -.326(a )A 418 479 :M -.196(correlated )A 468 479 :M -.076(error)A 59 497 :M -.078(trek sequence between X and Y.)A 77 515 :M f2_12 sf 1.122 .112(Lemma )J 122 515 :M .387(7:)A f0_12 sf .232 .023( )J 137 515 :M (If )S 149 515 :M -.196(there )A 176 515 :M (is )S 188 515 :M -.326(a )A 197 515 :M -.276(latent )A 226 515 :M -.163(trek )A 248 515 :M -.139(between )A 291 515 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 307 515 :M -.109(and )A 329 515 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 345 515 :M -.167(in )A 359 515 :M .358(G)A f0_7 sf 0 3 rm .125(Construct)A 0 -3 rm f0_12 sf .238(\(G\(M\),X,Y,)A f2_12 sf .331(Z)A f0_12 sf .263 .026(\) )J 474 515 :M -.331(that)A 59 533 :M -.123(contains )A 102 533 :M -.326(a )A 111 533 :M -.205(variable )A 152 533 :M .159(T)A f0_7 sf 0 3 rm .051(r)A 0 -3 rm f0_12 sf .119 .012(, )J 169 533 :M .957 .096(i.e. )J 189 533 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 204 533 :M f1_12 sf .126A f0_12 sf ( )S 221 533 :M -.382(T)A f0_7 sf 0 3 rm -.121(r)A 0 -3 rm f0_12 sf ( )S 235 533 :M f1_12 sf .126A f0_12 sf ( )S 252 533 :M .726(X)A f0_7 sf 0 3 rm .163(j)A 0 -3 rm f0_12 sf .457 .046(, )J 272 533 :M -.165(then )A 297 533 :M -.167(in )A 311 533 :M -.079(G\(M\) )A 343 533 :M -.196(there )A 371 533 :M (is )S 384 533 :M -.326(a )A 394 533 :M -.196(correlated )A 445 533 :M -.061(error )A 473 533 :M -.218(trek)A 59 551 :M -.08(sequence between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.078( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.069(, such that every variable )A 314 551 :M -.167(in )A 327 551 :M -.22(the )A 345 551 :M -.196(correlated )A 395 551 :M -.061(error )A 422 551 :M -.163(trek )A 444 551 :M (sequence,)S 59 569 :M -.012(with the possible exception of the endpoints, X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.014( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.011(,, has index \(i.e. subscript\) less than)A 59 587 :M (or )S 75 587 :M -.197(equal )A 106 587 :M -.167(to )A 121 587 :M (r )S 131 587 :M -.117(\(henceforth )A 192 587 :M -.119(referred )A 236 587 :M -.167(to )A 252 587 :M (as )S 269 587 :M -.22(the )A 290 587 :M -.196(correlated )A 343 587 :M -.061(error )A 373 587 :M -.163(trek )A 398 587 :M -.121(sequence )A 448 587 :M -.167(in )A 464 587 :M -.105(G\(M\))A 59 605 :M .038 .004(corresponding to the latent trek between X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .023 .002( and X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( in G)S f0_7 sf 0 3 rm .006(Construct)A 0 -3 rm f0_12 sf .011(\(G\(M\),X,Y,)A f2_12 sf (Z)S f0_12 sf (\).\))S 77 623 :M f2_12 sf .065(Proof.)A f0_12 sf .175 .017( The proof is by induction on r. Suppose first that r )J 360 623 :M .211 .021(= )J 371 623 :M .833 .083(1. )J 385 623 :M (From )S 415 623 :M -.22(the )A 433 623 :M -.119(construction)A 59 641 :M -.184(algorithm )A 110 641 :M (for )S 130 641 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\), )J 250 641 :M -.164(if )A 263 641 :M -.196(there )A 292 641 :M (is )S 307 641 :M -.326(a )A 319 641 :M -.276(latent )A 351 641 :M -.163(trek )A 376 641 :M -.139(between )A 422 641 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 440 641 :M -.109(and )A 464 641 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 482 641 :M -.334(in)A 59 659 :M (G)S f0_7 sf 0 3 rm .017(Construct)A 0 -3 rm f0_12 sf .031(\(G\(M\),X,Y,)A f2_12 sf (Z)S f0_12 sf .091 .009(\) that contains T)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .082 .008( then there are edges X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf .068A f0_12 sf .053 .005( X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S 391 659 :M -.109(and )A 412 659 :M -.109(and )A 433 659 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 448 659 :M f1_12 sf .4A f0_12 sf .096 .01( )J 465 659 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 482 659 :M -.334(in)A 59 677 :M .456 .046(G\(M\). )J 95 677 :M -.219(The )A 118 677 :M -.203(concatenation )A 187 677 :M (of )S 202 677 :M -.131(these )A 231 677 :M (two )S 254 677 :M -.064(edges )A 286 677 :M (forms )S 319 677 :M -.326(a )A 329 677 :M -.196(correlated )A 380 677 :M -.061(error )A 409 677 :M -.163(trek )A 433 677 :M -.121(sequence )A 482 677 :M -.334(in)A endp %%Page: 25 25 %%BeginPageSetup initializepage (peter; page: 25 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (25)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.135(which \(trivially\) )A 139 56 :M -.129(every )A 169 56 :M -.205(variable )A 210 56 :M -.167(in )A 223 56 :M -.22(the )A 241 56 :M -.121(sequence )A 288 56 :M -.219(except )A 322 56 :M (for )S 340 56 :M -.22(the )A 358 56 :M -.073(endpoints )A 408 56 :M (has )S 428 56 :M -.163(an )A 443 56 :M -.132(index )A 473 56 :M (less)S 59 74 :M -.165(than )A 83 74 :M (or )S 97 74 :M -.197(equal )A 126 74 :M -.167(to )A 139 74 :M .833 .083(1. )J 153 74 :M -.219(The )A 175 74 :M -.148(induction )A 223 74 :M -.033(hypothesis )A 278 74 :M (is )S 290 74 :M -.249(that )A 311 74 :M (for )S 329 74 :M -.331(all )A 344 74 :M (r )S 353 74 :M cF f1_12 sf .038A sf .376 .038( )J 365 74 :M .833 .083(n, )J 380 74 :M -.164(if )A 392 74 :M -.196(there )A 420 74 :M (is )S 433 74 :M -.326(a )A 443 74 :M -.276(latent )A 473 74 :M -.218(trek)A 59 92 :M -.139(between )A 103 92 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 119 92 :M -.109(and )A 141 92 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 157 92 :M -.167(in )A 171 92 :M .358(G)A f0_7 sf 0 3 rm .125(Construct)A 0 -3 rm f0_12 sf .238(\(G\(M\),X,Y,)A f2_12 sf .331(Z)A f0_12 sf .263 .026(\) )J 286 92 :M -.249(that )A 308 92 :M -.123(contains )A 352 92 :M .159(T)A f0_7 sf 0 3 rm .051(r)A 0 -3 rm f0_12 sf .119 .012(, )J 370 92 :M -.165(then )A 395 92 :M -.167(in )A 410 92 :M -.079(G\(M\) )A 443 92 :M -.196(there )A 472 92 :M (is )S 486 92 :M (a)S 59 110 :M -.064(correlated error trek sequence between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.069( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.066(, such )A 321 110 :M -.249(that )A 342 110 :M -.129(every )A 372 110 :M -.205(variable )A 413 110 :M -.167(in )A 426 110 :M -.22(the )A 444 110 :M (sequence,)S 59 128 :M -.09(with the possible exception of the endpoints has an )A 303 128 :M -.132(index )A 333 128 :M (less )S 355 128 :M -.165(than )A 379 128 :M .839 .084(r. )J 391 128 :M .197 .02(Suppose )J 436 128 :M .259 .026(now )J 461 128 :M -.249(that )A 482 128 :M -.334(in)A 59 146 :M .224(G)A f0_7 sf 0 3 rm .078(Construct)A 0 -3 rm f0_12 sf .149(\(G\(M\),X,Y,)A f2_12 sf .207(Z)A f0_12 sf .326 .033(\) there )J 199 146 :M (is )S 211 146 :M -.326(a )A 220 146 :M -.276(latent )A 249 146 :M -.163(trek )A 271 146 :M -.139(between )A 314 146 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 329 146 :M -.109(and )A 350 146 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 365 146 :M (such )S 391 146 :M -.249(that )A 412 146 :M -.22(the )A 430 146 :M -.163(trek )A 452 146 :M -.141(contains)A 59 164 :M .52(T)A f0_7 sf 0 3 rm .259(n+1)A 0 -3 rm f0_12 sf .387 .039(, )J 93 164 :M -.062(where )A 133 164 :M .555 .055(i, )J 151 164 :M -.334(j )A 165 164 :M cF f1_12 sf .038A sf .376 .038( )J 183 164 :M .88 .088(n+1. )J 217 164 :M -.131(Since )A 254 164 :M -.22(the )A 279 164 :M -.163(edge )A 312 164 :M -.139(between )A 362 164 :M .249(T)A f0_7 sf 0 3 rm .124(n+1)A 0 -3 rm f0_12 sf .102 .01( )J 392 164 :M -.109(and )A 420 164 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 443 164 :M -.055(exists )A 482 164 :M -.334(in)A 59 182 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\), )J 190 182 :M -.334(it )A 213 182 :M (follows )S 266 182 :M -.08(from )A 306 182 :M -.22(the )A 338 182 :M -.109(construction )A 414 182 :M -.184(algorithm )A 477 182 :M (for)S 59 200 :M .358(G)A f0_7 sf 0 3 rm .125(Construct)A 0 -3 rm f0_12 sf .238(\(G\(M\),X,Y,)A f2_12 sf .331(Z)A f0_12 sf .263 .026(\) )J 177 200 :M -.249(that )A 202 200 :M -.219(either )A 236 200 :M -.196(there )A 267 200 :M (is )S 283 200 :M -.326(a )A 297 200 :M -.276(latent )A 331 200 :M -.163(trek )A 358 200 :M -.139(between )A 406 200 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 426 200 :M -.109(and )A 452 200 :M .532(X)A f0_7 sf 0 3 rm .224(n+1)A 0 -3 rm f0_12 sf .184 .018( )J 482 200 :M -.334(in)A 59 218 :M .234(G)A f0_7 sf 0 3 rm .082(Construct)A 0 -3 rm f0_12 sf .155(\(G\(M\),X,Y,)A f2_12 sf .216(Z)A f0_12 sf .29 .029(\) that )J 193 218 :M -.123(contains )A 236 218 :M -.082(some )A 265 218 :M .159(T)A f0_7 sf 0 3 rm .051(r)A 0 -3 rm f0_12 sf .119 .012(, )J 282 218 :M (r )S 290 218 :M .211 .021(< )J 301 218 :M .88 .088(n+1, )J 328 218 :M (or )S 342 218 :M -.196(there )A 369 218 :M (is )S 381 218 :M -.326(a )A 390 218 :M -.125(double-headed )A 463 218 :M (arrow)S 59 236 :M -.139(between )A 102 236 :M .532(X)A f0_7 sf 0 3 rm .224(n+1)A 0 -3 rm f0_12 sf .184 .018( )J 127 236 :M -.109(and )A 148 236 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 163 236 :M -.167(in )A 176 236 :M .456 .046(G\(M\). )J 211 236 :M (In )S 225 236 :M -.22(the )A 243 236 :M -.107(former )A 279 236 :M .236 .024(case, )J 307 236 :M (by )S 323 236 :M -.22(the )A 341 236 :M -.148(induction )A 389 236 :M -.033(hypothesis )A 445 236 :M -.196(there )A 473 236 :M (is )S 486 236 :M (a)S 59 254 :M -.106(correlated error trek sequence between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.126( and )A 278 254 :M .532(X)A f0_7 sf 0 3 rm .224(n+1)A 0 -3 rm f0_12 sf .184 .018( )J 303 254 :M (that, )S 328 254 :M -.219(except )A 362 254 :M (for )S 380 254 :M -.22(the )A 398 254 :M .17 .017(endpoints, )J 452 254 :M -.141(contains)A 59 272 :M -.077(only vertices whose indices are less than or equal to r, and hence less than )A 412 272 :M (or )S 426 272 :M -.197(equal )A 455 272 :M -.167(to )A 468 272 :M .411(n+1.)A 59 290 :M (In )S 74 290 :M -.22(the )A 93 290 :M -.275(latter )A 121 290 :M .236 .024(case, )J 150 290 :M .503( )J 205 290 :M (is )S 218 290 :M -.326(a )A 228 290 :M -.196(correlated )A 279 290 :M -.061(error )A 308 290 :M -.163(trek )A 332 290 :M -.121(sequence )A 381 290 :M -.139(between )A 426 290 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 443 290 :M -.109(and )A 466 290 :M .915(X)A f0_7 sf 0 3 rm .386(n+1)A 0 -3 rm f0_12 sf (.)S 59 308 :M -.114(Similarly, there is a correlated error trek sequence between )A 339 308 :M .532(X)A f0_7 sf 0 3 rm .224(n+1)A 0 -3 rm f0_12 sf .184 .018( )J 364 308 :M -.109(and )A 385 308 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 400 308 :M (that, )S 425 308 :M -.219(except )A 459 308 :M (for )S 477 308 :M -.33(the)A 59 326 :M .17 .017(endpoints, )J 113 326 :M -.123(contains )A 156 326 :M -.083(only )A 181 326 :M -.163(vertices )A 221 326 :M .229 .023(whose )J 256 326 :M -.141(indices )A 293 326 :M -.215(are )A 311 326 :M (less )S 333 326 :M -.165(than )A 357 326 :M (or )S 371 326 :M -.197(equal )A 400 326 :M -.167(to )A 413 326 :M .88 .088(n+1. )J 440 326 :M -.13(These )A 473 326 :M (two)S 59 344 :M -.133(correlated error trek sequences can be concatenated to form a )A 349 344 :M -.196(correlated )A 399 344 :M -.061(error )A 426 344 :M -.163(trek )A 448 344 :M -.139(sequence)A 59 362 :M -.07(between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.065( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.058( that, except )A 207 362 :M (for )S 225 362 :M -.22(the )A 243 362 :M .17 .017(endpoints, )J 297 362 :M -.123(contains )A 340 362 :M -.083(only )A 365 362 :M -.163(vertices )A 405 362 :M .229 .023(whose )J 440 362 :M -.141(indices )A 477 362 :M -.323(are)A 59 380 :M -.037(less than or equal to n+1. )A f1_12 sf <5C>S 77 398 :M f0_12 sf .064 .006(For G)J f0_7 sf 0 3 rm .01(Construct)A 0 -3 rm f0_12 sf .02(\(G\(M\),X,Y,)A f2_12 sf (Z)S f0_12 sf .051 .005(\) shown in Figure 13, there is a latent trek between )J 447 398 :M .478(X)A f0_7 sf 0 3 rm .193(5)A 0 -3 rm f0_12 sf .166 .017( )J 464 398 :M f1_12 sf .126A f0_12 sf ( )S 480 398 :M .115(T)A f0_7 sf 0 3 rm (5)S 0 -3 rm 59 416 :M f1_12 sf .371A f0_12 sf .304 .03( X)J f0_7 sf 0 3 rm .11(6)A 0 -3 rm f0_12 sf .329 .033(, and )J 115 416 :M -.326(a )A 124 416 :M -.049(corresponding )A 196 416 :M -.196(correlated )A 246 416 :M -.061(error )A 273 416 :M -.163(trek )A 295 416 :M -.121(sequence )A 342 416 :M .65( )J 407 416 :M -.167(in )A 420 416 :M -.22(the )A 438 416 :M -.064(graph )A 469 416 :M .306 .031(G )J 482 416 :M -.334(in)A 59 434 :M .356 .036(Figure 12.)J 77 452 :M -.326(We )A 99 452 :M -.166(will )A 123 452 :M -.247(make )A 154 452 :M (use )S 176 452 :M (of )S 192 452 :M -.22(the )A 212 452 :M -.073(following )A 264 452 :M -.33(Lemma )A 305 452 :M -.065(which )A 340 452 :M (is )S 354 452 :M -.326(a )A 365 452 :M -.166(simple )A 403 452 :M -.11(extension )A 455 452 :M -.167(to )A 471 452 :M -.22(path)A 59 470 :M -.085(diagrams with directed cycles of Lemma 3.3.1 in Spirtes )A f5_12 sf -.069(et al.)A f0_12 sf ( )S 356 470 :M .596 .06(\(1993\). )J 396 470 :M -.083(This )A 421 470 :M -.33(Lemma )A 460 470 :M -.065(allows)A 59 488 :M -.144(us to concatenate \324small\325 )A 179 488 :M -.137(d-connecting )A 244 488 :M -.065(paths )A 273 488 :M -.167(to )A 286 488 :M -.08(form )A 313 488 :M -.326(a )A 322 488 :M -.162(larger )A 353 488 :M -.137(d-connecting )A 418 488 :M .226 .023(path. )J 446 488 :M -.326(We )A 466 488 :M (say )S 486 488 :M (a)S 59 506 :M -.024(path is )A f2_12 sf -.027(into)A f0_12 sf -.027( endpoint X if the path contains some edge X )A f1_12 sf -.068A f0_12 sf -.027( Y or X )A f1_12 sf -.065A f0_12 sf -.04( Y.)A 59 530 :M f2_12 sf (Lemma 8: )S f0_12 sf .007 .001(In a path diagram G over a set of vertices )J f2_12 sf (V)S f0_12 sf (, if:)S 86 552 :M ( \(a\) )S f2_12 sf (Q)S f0_12 sf -.003( is a sequence of vertices in )A f2_12 sf (V)S f0_12 sf ( from A to B, )S f2_12 sf (Q)S f0_12 sf ( )S f1_12 sf S f0_12 sf (S f0_12 sf (X)S f0_9 sf 0 2 rm (0)S 0 -2 rm f0_12 sf <2CC958>S f0_10 sf 0 2 rm (n)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm f1_12 sf S f0_12 sf (B>, such that)S 68 568 :M f1_12 sf .149(")A f0_12 sf .204 .02<692CCA30CA>J cF f1_12 sf .02A sf .204 .02( i )J cF f1_12 sf .02A sf .204 .02( n, X)J f0_10 sf 0 2 rm (i)S 0 -2 rm f0_12 sf .047 .005( )J f1_12 sf .115A f0_12 sf .169 .017( X)J f0_10 sf 0 2 rm (i)S 0 -2 rm f0_9 sf 0 2 rm .083(+1)A 0 -2 rm f0_12 sf .215 .021( \(the X)J f0_10 sf 0 2 rm (i)S 0 -2 rm f0_12 sf .196 .02( are only )J f5_12 sf .514 .051(pairwise distinct)J f0_12 sf .302 .03( , i.e. not necessarily distinct\),)J 86 584 :M .311<286229CA>A f2_12 sf .671 .067(Z )J f1_12 sf .626A f0_12 sf .199 .02( )J f2_12 sf .634(V)A f0_12 sf .457(\\{A,B},)A 86 600 :M -.046(\(c\) )A f2_12 sf -.058(P )A f0_12 sf -.053(is a set of undirected paths such that)A 100 616 :M 1.584 .158(\(i\)\312for )J 136 616 :M -.245(each )A 162 616 :M -.163(pair )A 185 616 :M (of )S 200 616 :M -.149(consecutive )A 260 616 :M -.163(vertices )A 301 616 :M -.167(in )A 315 616 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(, )J 333 616 :M .335(X)A f0_10 sf 0 2 rm .107(i)A 0 -2 rm f0_12 sf .116 .012( )J 350 616 :M -.109(and )A 373 616 :M .636(X)A f0_10 sf 0 2 rm .204(i)A 0 -2 rm f0_9 sf 0 2 rm .351(+1)A 0 -2 rm f0_12 sf .4 .04(, )J 405 616 :M -.196(there )A 434 616 :M (is )S 448 616 :M -.326(a )A 459 616 :M -.132(unique)A 82 632 :M .017 .002(undirected path in )J f2_12 sf (P)S f0_12 sf .017 .002( that d-connects X)J f0_10 sf 0 2 rm (i)S 0 -2 rm f0_12 sf ( and X)S f0_10 sf 0 2 rm (i)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm f0_12 sf ( given )S f2_12 sf (Z)S f0_12 sf (\\{X)S f0_10 sf 0 2 rm (i)S 0 -2 rm f0_12 sf ( , X)S f0_10 sf 0 2 rm (i)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm f0_12 sf (},)S 100 648 :M 1.187 .119(\(ii\)\312if )J 131 648 :M -.082(some )A 160 648 :M -.163(vertex )A 193 648 :M .175(X)A f0_10 sf 0 2 rm .101(k)A 0 -2 rm f0_12 sf .061 .006( )J 211 648 :M -.167(in )A 224 648 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(, )J 241 648 :M (is )S 253 648 :M -.167(in )A 267 648 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 284 648 :M -.165(then )A 309 648 :M -.22(the )A 328 648 :M -.065(paths )A 358 648 :M -.167(in )A 372 648 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 385 648 :M -.249(that )A 407 648 :M -.189(contain )A 446 648 :M .175(X)A f0_10 sf 0 2 rm .101(k)A 0 -2 rm f0_12 sf .061 .006( )J 465 648 :M (as )S 480 648 :M -.326(an)A 82 664 :M -.073(endpoint collide at X)A f0_10 sf 0 2 rm -.076(k)A 0 -2 rm f0_12 sf -.067(, \(i.e. all such paths are directed into X)A f0_10 sf 0 2 rm -.076(k)A 0 -2 rm f0_12 sf <29>S endp %%Page: 26 26 %%BeginPageSetup initializepage (peter; page: 26 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (26)S gR gS 0 0 552 730 rC 100 54 :M f0_12 sf .364 .036(\(iii\)\312if for three vertices X)J f0_10 sf 0 2 rm .101(k)A 0 -2 rm f0_9 sf 0 2 rm .091A 0 -2 rm f0_12 sf .228 .023(, X)J f0_10 sf 0 2 rm .101(k)A 0 -2 rm f0_12 sf .228 .023(, X)J f0_10 sf 0 2 rm .101(k)A 0 -2 rm f0_9 sf 0 2 rm .097(+1)A 0 -2 rm f0_12 sf .061 .006( )J 295 54 :M -.108(occurring )A 344 54 :M -.167(in )A 357 54 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 370 54 :M -.22(the )A 388 54 :M -.137(d-connecting )A 453 54 :M -.065(paths )A 482 54 :M -.334(in)A 82 70 :M f2_12 sf (P)S f0_12 sf .084 .008( between X)J f0_10 sf 0 2 rm (k)S 0 -2 rm f0_9 sf 0 2 rm S 0 -2 rm f0_12 sf .057 .006( and X)J f0_10 sf 0 2 rm (k)S 0 -2 rm f0_12 sf .06 .006(, and X)J f0_10 sf 0 2 rm (k)S 0 -2 rm f0_12 sf .057 .006( and X)J f0_10 sf 0 2 rm (k)S 0 -2 rm f0_9 sf 0 2 rm (+1)S 0 -2 rm f0_12 sf .059 .006(, collide at )J 337 70 :M .175(X)A f0_10 sf 0 2 rm .101(k)A 0 -2 rm f0_12 sf .061 .006( )J 355 70 :M -.165(then )A 379 70 :M .175(X)A f0_10 sf 0 2 rm .101(k)A 0 -2 rm f0_12 sf .061 .006( )J 397 70 :M (has )S 417 70 :M -.326(a )A 426 70 :M -.131(descendant )A 482 70 :M -.334(in)A 82 86 :M f2_12 sf .724(Z)A f0_12 sf (,)S 59 102 :M -.034(then there is a path U in G)A f5_12 sf ( )S f0_12 sf -.037(that d-connects A)A f1_12 sf S f0_12 sf -.064(X)A f0_9 sf 0 2 rm (0)S 0 -2 rm f0_12 sf -.038( and B)A f1_12 sf S f0_12 sf -.064(X)A f0_10 sf 0 2 rm (n)S 0 -2 rm f0_9 sf 0 2 rm -.035(+1)A 0 -2 rm f0_12 sf -.034( given )A f2_12 sf -.059(Z)A f0_12 sf (.)S 77 120 :M -.086(Note that we do not )A 174 120 :M -.139(require )A 211 120 :M -.249(that )A 232 120 :M -.326(a )A 241 120 :M -.163(vertex )A 274 120 :M -.129(occur )A 304 120 :M -.083(only )A 329 120 :M -.163(once )A 355 120 :M -.167(in )A 368 120 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(. )J 385 120 :M -.128(Hence )A 419 120 :M -.109(one )A 440 120 :M -.18(occurrence)A 59 138 :M -.11(of a vertex in )A f2_12 sf -.231(Q)A f5_12 sf -.074( )A f0_12 sf -.119(may be a collider, and another occurrence of the same vertex in )A f2_12 sf -.231(Q)A f0_12 sf -.145( may be )A 486 138 :M (a)S 59 156 :M -.049(non-collider. )A 124 156 :M -.215(\(We )A 148 156 :M (say )S 168 156 :M -.249(that )A 189 156 :M -.093(Y)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf ( )S 206 156 :M (is )S 219 156 :M -.326(a )A 229 156 :M -.206(collider )A 269 156 :M -.116(\(non-collider\) )A 339 156 :M -.167(in )A 353 156 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 367 156 :M -.164(if )A 379 156 :M -.22(the )A 398 156 :M -.163(pair )A 421 156 :M (of )S 436 156 :M -.164(consecutive)A 59 174 :M -.096(paths in )A f2_12 sf -.155(P)A f0_12 sf -.1( that contain Y)A f0_7 sf 0 3 rm -.074(k)A 0 -3 rm f0_12 sf -.098( as an endpoint collide \(do not collide\) at Y)A f0_7 sf 0 3 rm -.074(k)A 0 -3 rm f0_12 sf -.148(.\))A 77 192 :M f2_12 sf 1.122 .112(Lemma )J 123 192 :M .387(9:)A f0_12 sf .232 .023( )J 139 192 :M (If )S 152 192 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 170 192 :M f1_12 sf .285A f0_12 sf .13 .013( )J 182 192 :M .306 .031(X )J 196 192 :M -.109(and )A 218 192 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 236 192 :M f1_12 sf .285A f0_12 sf .13 .013( )J 248 192 :M -.663(Y )A 261 192 :M -.215(are )A 280 192 :M -.148(d-connected )A 342 192 :M -.132(given )A 373 192 :M f2_12 sf (Z)S f0_12 sf ( )S 386 192 :M -.167(in )A 401 192 :M -.22(the )A 421 192 :M -.205(directed )A 464 192 :M -.08(graph)A 59 210 :M .156(G)A f0_7 sf 0 3 rm .054(Construct)A 0 -3 rm f0_12 sf .104(\(G\(M\),X,Y,)A f2_12 sf .144(Z)A f0_12 sf .216 .022(\), then X and Y are )J 260 210 :M -.148(d-connected )A 321 210 :M -.132(given )A 351 210 :M f2_12 sf (Z)S f0_12 sf ( )S 363 210 :M -.167(in )A 376 210 :M -.22(the )A 394 210 :M -.165(path )A 418 210 :M -.188(diagram )A 460 210 :M .171(G\(M\).)A 59 228 :M -.062(\(G\(M\) )A 94 228 :M (has )S 114 228 :M -.163(vertex )A 147 228 :M -.109(set )A 164 228 :M f2_12 sf .79(V)A f0_12 sf .497 .05(, )J 182 228 :M .358(G)A f0_7 sf 0 3 rm .125(Construct)A 0 -3 rm f0_12 sf .238(\(G\(M\),X,Y,)A f2_12 sf .331(Z)A f0_12 sf .263 .026(\) )J 297 228 :M (has )S 318 228 :M -.163(vertex )A 352 228 :M -.109(set )A 370 228 :M f2_12 sf .498(V)A f1_12 sf .529A f3_12 sf .421(T)A f0_12 sf .313 .031(, )J 405 228 :M -.109(and )A 427 228 :M .2({X,Y})A f1_12 sf .289A f3_12 sf .23(Z)A f1_12 sf .094 .009( )J 482 228 :M S 59 246 :M f2_12 sf .108(V.)A f0_12 sf .074<29>A f2_12 sf 0 18 rm .098(Proof.)A 0 -18 rm f0_12 sf 0 18 rm .303 .03( Suppose that there is an undirected )J 0 -18 rm 285 264 :M -.165(path )A 309 264 :M .306 .031(U )J 322 264 :M -.249(that )A 343 264 :M -.097(d-connects )A 398 264 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 415 264 :M -.109(and )A 436 264 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 453 264 :M -.132(given )A 483 264 :M f2_12 sf (Z)S 59 282 :M f0_12 sf -.167(in )A 72 282 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\). )J 190 282 :M -.326(We )A 210 282 :M -.166(will )A 232 282 :M -.064(prove )A 263 282 :M -.249(that )A 284 282 :M .306 .031(X )J 297 282 :M -.109(and )A 318 282 :M -.663(Y )A 330 282 :M -.215(are )A 348 282 :M -.148(d-connected )A 409 282 :M -.132(given )A 439 282 :M f2_12 sf (Z)S f0_12 sf ( )S 451 282 :M -.167(in )A 464 282 :M -.105(G\(M\))A 59 300 :M (by )S 76 300 :M -.109(constructing )A 139 300 :M -.326(a )A 149 300 :M -.121(sequence )A 197 300 :M (of )S 212 300 :M -.163(vertices )A 253 300 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 267 300 :M -.109(and )A 289 300 :M -.326(a )A 299 300 :M -.109(set )A 317 300 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 330 300 :M (of )S 346 300 :M -.065(paths )A 377 300 :M -.167(in )A 392 300 :M .306 .031(G )J 407 300 :M -.139(between )A 452 300 :M -.064(pairs )A 481 300 :M (of)S 59 318 :M -.085(consecutive vertices in )A f2_12 sf -.112(Q )A f0_12 sf -.092(satisfying the conditions of Lemma 8.)A 77 336 :M -.219(Our )A 99 336 :M -.064(first )A 122 336 :M -.082(step )A 145 336 :M -.166(will )A 167 336 :M -.163(be )A 182 336 :M -.167(to )A 195 336 :M (use )S 216 336 :M .306 .031(U )J 230 336 :M -.167(to )A 244 336 :M -.109(construct )A 292 336 :M -.326(a )A 302 336 :M -.121(sequence )A 350 336 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 369 336 :M -.109(and )A 391 336 :M -.326(a )A 401 336 :M -.109(set )A 419 336 :M (of )S 434 336 :M -.065(paths )A 464 336 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 482 336 :M -.334(in)A 59 354 :M .358(G)A f0_7 sf 0 3 rm .125(Construct)A 0 -3 rm f0_12 sf .238(\(G\(M\),X,Y,)A f2_12 sf .331(Z)A f0_12 sf .263 .026(\) )J 173 354 :M -.08(from )A 200 354 :M -.065(which )A 233 354 :M (we )S 251 354 :M -.166(will )A 273 354 :M -.165(then )A 297 354 :M -.109(construct )A 344 354 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 356 354 :M -.109(and )A 377 354 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(. )J 394 354 :M -.082(Intuitively, )A 450 354 :M (we )S 468 354 :M -.107(form)A 59 372 :M f2_12 sf <51D5>S f0_12 sf .034 .003( and )J f2_12 sf <50D5>S f0_12 sf .055 .006( by breaking U into pieces, such )J 266 372 :M -.249(that )A 287 372 :M -.245(each )A 312 372 :M -.276(latent )A 341 372 :M -.163(trek )A 363 372 :M -.052(occurs )A 398 372 :M (as )S 412 372 :M -.326(a )A 421 372 :M -.162(separate )A 463 372 :M -.062(piece.)A 59 390 :M -.097(More formally, form a sequence )A 215 390 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 233 390 :M (of )S 247 390 :M -.163(vertices )A 287 390 :M -.109(and )A 308 390 :M -.163(an )A 323 390 :M -.131(associated )A 375 390 :M -.121(sequence )A 422 390 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 439 390 :M (of )S 453 390 :M -.065(paths )A 482 390 :M -.334(in)A 59 408 :M .358(G)A f0_7 sf 0 3 rm .125(Construct)A 0 -3 rm f0_12 sf .238(\(G\(M\),X,Y,)A f2_12 sf .331(Z)A f0_12 sf .263 .026(\) )J 173 408 :M -.083(with )A 198 408 :M -.22(the )A 216 408 :M -.073(following )A 266 408 :M -.119(properties: )A 320 408 :M -.107(\(i\) )A 335 408 :M -.129(every )A 366 408 :M -.163(vertex )A 400 408 :M -.167(in )A 414 408 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 433 408 :M (is )S 446 408 :M -.167(in )A 460 408 :M f2_12 sf .25(V)A f0_12 sf .087 .009( )J 474 408 :M -.163(and)A 59 426 :M -.072(occurs on U; \(ii\) no vertex occurs in )A f2_12 sf -.103<51D5>A f0_12 sf -.08( more than once; )A 329 426 :M -.198(\(iii\) )A 350 426 :M -.164(if )A 361 426 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 376 426 :M -.052(occurs )A 411 426 :M -.106(before )A 445 426 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 460 426 :M -.167(in )A 473 426 :M f2_12 sf .683<51D5>A f0_12 sf (,)S 59 444 :M -.036(then X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.034( occurs before X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.033( on U; \(iv\) )A 228 444 :M -.164(if )A 239 444 :M -.22(the )A 257 444 :M -.047(subpath )A 298 444 :M (of )S 312 444 :M .306 .031(U )J 325 444 :M -.139(between )A 368 444 :M .25(X)A f0_7 sf 0 3 rm .056(i)A 0 -3 rm f0_12 sf .086 .009( )J 383 444 :M -.109(and )A 404 444 :M .25(X)A f0_7 sf 0 3 rm .056(j)A 0 -3 rm f0_12 sf .086 .009( )J 419 444 :M (is )S 431 444 :M -.326(a )A 440 444 :M -.276(latent )A 469 444 :M .087(trek,)A 59 462 :M (X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( )S f1_12 sf S f0_12 sf (T)S f0_7 sf 0 3 rm (r)S 0 -3 rm f1_12 sf S f0_12 sf (X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (, then X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( and X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( both occur in that order in )S f2_12 sf <51D5>S f0_12 sf (. )S 342 462 :M -.219(The )A 364 462 :M -.165(path )A 388 462 :M -.167(in )A 401 462 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 418 462 :M -.131(associated )A 470 462 :M -.11(with)A 59 480 :M -.039(a pair X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.043( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.037( of consecutive vertices in )A f2_12 sf -.054<51D5>A f0_12 sf -.04( is the subpath of U between X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.043( and X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (. )S 463 480 :M (In )S 477 480 :M -.33(the)A 59 498 :M .03 .003(example in Figure 13, in G)J f0_7 sf 0 3 rm (Construct)S 0 -3 rm f0_12 sf .01(\(G\(M\),X,Y,)A f2_12 sf (Z)S f0_12 sf .033 .003(\) the d-connecting )J 373 498 :M -.165(path )A 397 498 :M -.139(between )A 440 498 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 457 498 :M -.109(and )A 478 498 :M .596(X)A f0_7 sf 0 3 rm (2)S 0 -3 rm 59 516 :M f0_12 sf .782 .078(given )J f2_12 sf .338(Z)A f0_12 sf .234 .023( = )J f1_12 sf .417A f0_12 sf .382 .038( is X)J f0_7 sf 0 3 rm .148(1)A 0 -3 rm f0_12 sf .115 .012( )J f1_12 sf .5A f0_12 sf .41 .041( X)J f0_7 sf 0 3 rm .148(5)A 0 -3 rm f0_12 sf .115 .012( )J f1_12 sf .5A f0_12 sf .363 .036( T)J f0_7 sf 0 3 rm .148(4)A 0 -3 rm f0_12 sf .115 .012( )J f1_12 sf .5A f0_12 sf .41 .041( X)J f0_7 sf 0 3 rm .148(6)A 0 -3 rm f0_12 sf .115 .012( )J f1_12 sf .5A f0_12 sf .41 .041( X)J f0_7 sf 0 3 rm .148(2)A 0 -3 rm f0_12 sf .211 .021(, )J f2_12 sf .281<51D5>A f0_12 sf .476 .048( = , and )J f2_12 sf .239<50D5>A f0_12 sf .245 .024( = )J 434 516 :M .376(A f0_12 sf ( )S 474 516 :M 1.049(X)A f0_7 sf 0 3 rm .424(5)A 0 -3 rm f0_12 sf (,)S 59 534 :M .092(X)A f0_7 sf 0 3 rm (5)S 0 -3 rm f0_12 sf ( )S f1_12 sf .126A f0_12 sf .091 .009( T)J f0_7 sf 0 3 rm (4)S 0 -3 rm f0_12 sf ( )S f1_12 sf .126A f0_12 sf .103 .01( X)J f0_7 sf 0 3 rm (6)S 0 -3 rm f0_12 sf .119 .012(, X)J f0_7 sf 0 3 rm (6)S 0 -3 rm f0_12 sf ( )S f1_12 sf .126A f0_12 sf .103 .01( X)J f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf .163 .016(>. In this example, there are no colliders in )J f2_12 sf .071<51D5>A f0_12 sf (.)S 77 552 :M -.049(Because U is a path that d-connects X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.053( and X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf ( )S 300 552 :M -.132(given )A 330 552 :M f2_12 sf (Z)S f0_12 sf ( )S 342 552 :M -.167(in )A 355 552 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\), )J 473 552 :M -.334(it )A 483 552 :M (is)S 59 570 :M -.071(clear that the paths in )A f2_12 sf -.092<50D5>A f0_12 sf ( )S 178 570 :M -.163(have )A 204 570 :M -.22(the )A 222 570 :M -.073(following )A 272 570 :M -.097(properties )A 323 570 :M -.167(in )A 336 570 :M .322(G)A f0_7 sf 0 3 rm .112(Construct)A 0 -3 rm f0_12 sf .214(\(G\(M\),X,Y,)A f2_12 sf .297(Z)A f0_12 sf .319 .032(\): )J 453 570 :M -.107(\(i\) )A 468 570 :M -.328(Each)A 59 588 :M (path in )S f2_12 sf <50D5>S f0_12 sf ( d-connects its endpoints X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf ( and X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf ( given )S f2_12 sf (Z)S f0_12 sf (\\{X)S f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (,X)S f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (}; )S 360 588 :M -.164(\(ii\) )A 378 588 :M -.164(if )A 389 588 :M -.065(paths )A 418 588 :M -.167(in )A 431 588 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 448 588 :M -.236(collide )A 483 588 :M -.66(at)A 59 606 :M -.083(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.048( then X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.045( has a descendant in )A f2_12 sf -.076(Z)A f0_12 sf -.04(; and \(iii\) if X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.036( is in )A f2_12 sf -.076(Z)A f0_12 sf -.043( then the paths in )A f2_12 sf -.046<50D520>A f0_12 sf -.044(collide at X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf (.)S 77 624 :M -.089(We will now )A 142 624 :M .479 .048(show )J 172 624 :M .259 .026(how )J 197 624 :M -.167(to )A 210 624 :M -.109(construct )A 257 624 :M -.326(a )A 266 624 :M -.121(sequence )A 313 624 :M (of )S 327 624 :M -.163(vertices )A 367 624 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 380 624 :M -.109(and )A 401 624 :M -.326(a )A 410 624 :M -.109(set )A 427 624 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 439 624 :M (of )S 453 624 :M -.065(paths )A 482 624 :M -.334(in)A 59 642 :M -.108(G\(M\) between )A 132 642 :M -.064(pairs )A 159 642 :M (of )S 173 642 :M -.149(consecutive )A 232 642 :M -.163(vertices )A 272 642 :M -.167(in )A 285 642 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 298 642 :M -.066(satisfying )A 348 642 :M -.22(the )A 366 642 :M -.1(conditions )A 419 642 :M (of )S 433 642 :M -.33(Lemma )A 472 642 :M -.167(8; )A 485 642 :M -.668(it)A 59 660 :M -.077(follows then that X and Y are d-connected given )A f2_12 sf -.125(Z)A f0_12 sf -.084( in G.)A endp %%Page: 27 27 %%BeginPageSetup initializepage (peter; page: 27 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (27)S gR gS 0 0 552 730 rC 77 56 :M f0_12 sf -.326(We )A 97 56 :M -.166(will )A 119 56 :M -.272(create )A 150 56 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 163 56 :M (by )S 179 56 :M -.139(several )A 216 56 :M -.152(modifications )A 284 56 :M (of )S 298 56 :M f2_12 sf .577<51D5>A f0_12 sf .472 .047(. )J 320 56 :M -.083(Step )A 345 56 :M (\(1\) )S 363 56 :M -.167(in )A 376 56 :M -.205(creating )A 417 56 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 431 56 :M (is )S 444 56 :M -.167(to )A 458 56 :M -.272(replace)A 59 74 :M .021 .002(each subsequence of )S f2_12 sf <51D5>S f0_12 sf ( such that )S 266 74 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 281 74 :M -.109(and )A 302 74 :M .37(X)A f0_7 sf 0 3 rm .116(s)A 0 -3 rm f0_12 sf .128 .013( )J 318 74 :M -.215(are )A 336 74 :M -.22(the )A 354 74 :M -.073(endpoints )A 404 74 :M (of )S 418 74 :M ( )S 422 74 :M -.326(a )A 431 74 :M -.276(latent )A 460 74 :M -.163(trek )A 482 74 :M -.334(in)A 59 92 :M f2_12 sf .875<50D5>A f0_12 sf .843 .084(, )J 80 92 :M -.083(with )A 105 92 :M -.22(the )A 123 92 :M -.049(corresponding )A 195 92 :M -.196(correlated )A 246 92 :M -.061(error )A 274 92 :M -.163(trek )A 297 92 :M -.121(sequence )A 345 92 :M .403(A f0_12 sf .303 .03(, )J 393 92 :M .347(X)A f0_7 sf 0 3 rm .109(s)A 0 -3 rm f0_12 sf .356 .036(> )J 417 92 :M -.167(in )A 431 92 :M .456 .046(G\(M\). )J 467 92 :M -.219(Then)A 59 110 :M -.233(replace )A 96 110 :M -.22(the )A 114 110 :M -.276(latent )A 143 110 :M -.163(trek )A 165 110 :M -.167(in )A 178 110 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 195 110 :M -.083(with )A 220 110 :M -.22(the )A 238 110 :M -.049(corresponding )A 310 110 :M -.196(correlated )A 361 110 :M -.061(error )A 389 110 :M -.163(trek )A 412 110 :M -.121(sequence )A 460 110 :M -.167(in )A 474 110 :M f2_12 sf 1.058<50D5>A f0_12 sf (.)S 59 128 :M -.081(Note )A 86 128 :M -.249(that )A 107 128 :M -.245(each )A 132 128 :M -.162(occurrence )A 187 128 :M (of )S 201 128 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 218 128 :M -.139(between )A 261 128 :M .403(A f0_12 sf .298(,X)A f0_7 sf 0 3 rm .139(s)A 0 -3 rm f0_12 sf .453 .045(> )J 326 128 :M (is )S 338 128 :M -.326(a )A 347 128 :M -.206(collider )A 386 128 :M -.167(in )A 399 128 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(. )J 416 128 :M (In )S 430 128 :M -.22(the )A 448 128 :M -.092(example,)A 59 146 :M .412 .041(after the first step )J f2_12 sf .269(Q)A f0_12 sf .325 .033( = and )J f2_12 sf .211(P)A f0_12 sf .086 .009( )J 296 146 :M .211 .021(= )J 307 146 :M .376(A f0_12 sf ( )S 347 146 :M .876(X)A f0_7 sf 0 3 rm .354(5)A 0 -3 rm f0_12 sf .552 .055(, )J 368 146 :M .478(X)A f0_7 sf 0 3 rm .193(5)A 0 -3 rm f0_12 sf .166 .017( )J 385 146 :M f1_12 sf .4A f0_12 sf .096 .01( )J 402 146 :M .876(X)A f0_7 sf 0 3 rm .354(4)A 0 -3 rm f0_12 sf .552 .055(, )J 423 146 :M .478(X)A f0_7 sf 0 3 rm .193(4)A 0 -3 rm f0_12 sf .166 .017( )J 440 146 :M f1_12 sf .4A f0_12 sf .096 .01( )J 457 146 :M .876(X)A f0_7 sf 0 3 rm .354(6)A 0 -3 rm f0_12 sf .552 .055(, )J 478 146 :M .596(X)A f0_7 sf 0 3 rm (6)S 0 -3 rm 59 164 :M f1_12 sf .569A f0_12 sf .467 .047( X)J f0_7 sf 0 3 rm .168(2)A 0 -3 rm f0_12 sf .57 .057(>, i.e. we )J 138 164 :M -.204(replaced )A 181 164 :M -.22(the )A 199 164 :M -.058(subsequence )A 263 164 :M .552( )J 311 164 :M -.167(in )A 324 164 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 342 164 :M (by )S 358 164 :M .743(, )J 427 164 :M -.109(and )A 448 164 :M -.22(the )A 466 164 :M -.331(latent)A 59 182 :M .593 .059(trek X)J f0_7 sf 0 3 rm .11(5)A 0 -3 rm f0_12 sf .085 .009( )J f1_12 sf .371A f0_12 sf .269 .027( T)J f0_7 sf 0 3 rm .11(4)A 0 -3 rm f0_12 sf .085 .009( )J f1_12 sf .371A f0_12 sf .304 .03( X)J f0_7 sf 0 3 rm .11(6)A 0 -3 rm f0_12 sf .334 .033( by X)J f0_7 sf 0 3 rm .11(5)A 0 -3 rm f0_12 sf .085 .009( )J f1_12 sf .391A f0_12 sf .304 .03( X)J f0_7 sf 0 3 rm .11(4)A 0 -3 rm f0_12 sf .385 .038( and X)J f0_7 sf 0 3 rm .11(4)A 0 -3 rm f0_12 sf .085 .009( )J f1_12 sf .391A f0_12 sf .304 .03( X)J f0_7 sf 0 3 rm .11(6)A 0 -3 rm f0_12 sf .297 .03( in)J f2_12 sf .393 .039<2051D5>J f0_12 sf (.)S 77 200 :M -.123(Recall that the ancestor relations among the variables in )A f2_12 sf -.227(V)A f0_12 sf -.135( \(which includes )A 431 200 :M -.22(the )A 449 200 :M -.163(variables)A 59 218 :M .245 .024(in )J f2_12 sf .206(Z)A f0_12 sf .278 .028(\) in G)J f0_7 sf 0 3 rm .078(Construct)A 0 -3 rm f0_12 sf .149(\(G\(M\),X,Y,)A f2_12 sf .206(Z)A f0_12 sf .164 .016(\) )J 212 218 :M -.215(are )A 230 218 :M -.22(the )A 248 218 :M -.163(same )A 276 218 :M (as )S 290 218 :M -.22(the )A 308 218 :M -.122(ancestor )A 351 218 :M -.146(relations )A 395 218 :M -.132(among )A 431 218 :M -.22(the )A 449 218 :M -.163(variables)A 59 236 :M -.167(in )A 73 236 :M .456 .046(G\(M\). )J 109 236 :M -.262(After )A 138 236 :M -.131(stage )A 167 236 :M (\(1\) )S 186 236 :M -.167(in )A 200 236 :M -.205(creating )A 242 236 :M f2_12 sf .405(Q)A f0_12 sf .237 .024(, )J 260 236 :M -.164(if )A 272 236 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 290 236 :M (is )S 303 236 :M -.111(not )A 323 236 :M -.163(an )A 339 236 :M -.122(ancestor )A 384 236 :M (of )S 400 236 :M f2_12 sf (Z)S f0_12 sf ( )S 414 236 :M -.167(in )A 429 236 :M -.079(G\(M\) )A 462 236 :M (\(or )S 482 236 :M -.334(in)A 59 254 :M .412(G)A f0_7 sf 0 3 rm .144(Construct)A 0 -3 rm f0_12 sf .274(\(G\(M\),X,Y,)A f2_12 sf .381(Z)A f0_12 sf .512 .051(\)\), )J 181 254 :M -.111(but )A 200 254 :M (has )S 220 254 :M -.163(an )A 235 254 :M -.162(occurrence )A 290 254 :M -.167(in )A 303 254 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 316 254 :M -.249(that )A 337 254 :M (is )S 350 254 :M -.326(a )A 360 254 :M -.072(collider, )A 404 254 :M -.334(it )A 415 254 :M (follows )S 456 254 :M -.249(that )A 478 254 :M .596(X)A f0_7 sf 0 3 rm (k)S 0 -3 rm 59 272 :M f0_12 sf -.094(was added to )A f2_12 sf -.174(Q)A f0_12 sf -.094( by replacing )A 198 272 :M -.326(a )A 207 272 :M -.058(subsequence )A 271 272 :M .384( )J 316 272 :M (of )S 330 272 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 348 272 :M (by )S 364 272 :M -.326(a )A 373 272 :M -.049(corresponding )A 445 272 :M -.218(correlated)A 59 290 :M -.061(error )A 86 290 :M -.163(trek )A 108 290 :M -.121(sequence )A 155 290 :M .403(A f0_12 sf .298(,X)A f0_7 sf 0 3 rm .139(s)A 0 -3 rm f0_12 sf .453 .045(> )J 220 290 :M -.167(in )A 233 290 :M .456 .046(G\(M\). )J 268 290 :M -.128(Hence )A 302 290 :M -.109(any )A 323 290 :M (such )S 349 290 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 366 290 :M -.165(lies )A 386 290 :M -.139(between )A 429 290 :M -.082(some )A 458 290 :M -.163(pair )A 481 290 :M (of)S 59 308 :M -.163(vertices )A 99 308 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 114 308 :M -.109(and )A 135 308 :M .37(X)A f0_7 sf 0 3 rm .116(s)A 0 -3 rm f0_12 sf .128 .013( )J 152 308 :M -.249(that )A 174 308 :M -.215(are )A 193 308 :M -.247(adjacent )A 236 308 :M -.167(in )A 250 308 :M f2_12 sf .577<51D5>A f0_12 sf .472 .047(. )J 273 308 :M -.139(Because )A 317 308 :M -.129(every )A 348 308 :M -.163(vertex )A 382 308 :M -.167(in )A 396 308 :M .403(A f0_12 sf .303 .03(, )J 444 308 :M .347(X)A f0_7 sf 0 3 rm .109(s)A 0 -3 rm f0_12 sf .356 .036(> )J 468 308 :M -.167(in )A 482 308 :M f2_12 sf (Q)S 59 326 :M f0_12 sf -.187(\(except )A 97 326 :M (for )S 115 326 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 130 326 :M -.109(and )A 151 326 :M .293(X)A f0_7 sf 0 3 rm .092(s)A 0 -3 rm f0_12 sf .215 .021(\) )J 171 326 :M (has )S 191 326 :M -.163(an )A 206 326 :M -.132(index )A 236 326 :M (less )S 258 326 :M -.165(than )A 282 326 :M (r )S 290 326 :M -.109(and )A 311 326 :M 1.111 .111(s, )J 324 326 :M -.109(and )A 345 326 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 362 326 :M (is )S 374 326 :M -.111(not )A 394 326 :M -.163(an )A 410 326 :M -.122(ancestor )A 454 326 :M (of )S 469 326 :M f2_12 sf (Z)S f0_12 sf ( )S 482 326 :M -.334(in)A 59 344 :M (G)S f0_7 sf 0 3 rm (Construct)S 0 -3 rm f0_12 sf (\(G\(M\),X,Y,)S f2_12 sf (Z)S f0_12 sf .011 .001(\), it follows from the ordering of the variables that )J 406 344 :M (we )S 424 344 :M .425 .043(chose, )J 459 344 :M -.249(that )A 480 344 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm 59 362 :M f0_12 sf .178 .018(and X)J f0_7 sf 0 3 rm (s)S 0 -3 rm f0_12 sf .128 .013( are not ancestors of )J f2_12 sf .074(Z)A f0_12 sf .089 .009( in G)J f0_7 sf 0 3 rm .028(Construct)A 0 -3 rm f0_12 sf .053(\(G\(M\),X,Y,)A f2_12 sf .074(Z)A f0_12 sf .094 .009(\). If a path )J 372 362 :M .306 .031(U )J 385 362 :M -.097(d-connects )A 440 362 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 457 362 :M -.109(and )A 478 362 :M .596(X)A f0_7 sf 0 3 rm (2)S 0 -3 rm 59 380 :M f0_12 sf -.016(given )A f2_12 sf (Z)S f0_12 sf -.015(, then every vertex on U is an ancestor of X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.016( or X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.013( or )A f2_12 sf (Z)S f0_12 sf -.017(. Because X)A f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 424 380 :M -.109(and )A 445 380 :M .37(X)A f0_7 sf 0 3 rm .116(s)A 0 -3 rm f0_12 sf .128 .013( )J 461 380 :M -.215(are )A 479 380 :M (on)S 59 398 :M .179 .018(U, but not ancestors of )J f2_12 sf .095(Z)A f0_12 sf .114 .011( in G)J f0_7 sf 0 3 rm .036(Construct)A 0 -3 rm f0_12 sf .068(\(G\(M\),X,Y,)A f2_12 sf .095(Z)A f0_12 sf .205 .021(\), and U d-connects X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .146 .015( and X)J f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf .149 .015( given )J 479 398 :M f2_12 sf .724(Z)A f0_12 sf (,)S 59 416 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 74 416 :M -.109(and )A 95 416 :M .37(X)A f0_7 sf 0 3 rm .116(s)A 0 -3 rm f0_12 sf .128 .013( )J 111 416 :M -.215(are )A 129 416 :M -.083(both )A 154 416 :M -.071(ancestors )A 202 416 :M (of )S 216 416 :M .668({X)A f0_7 sf 0 3 rm .324(1)A 0 -3 rm f0_12 sf .54(,X)A f0_7 sf 0 3 rm .324(2)A 0 -3 rm f0_12 sf .908 .091(}. )J 266 416 :M -.139(Because )A 309 416 :M -.167(in )A 322 416 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\), )J 441 416 :M (X)S f0_7 sf 0 3 rm (r)S 0 -3 rm f0_12 sf ( )S 457 416 :M -.109(and )A 479 416 :M .468(X)A f0_7 sf 0 3 rm (s)S 0 -3 rm 59 434 :M f0_12 sf -.215(are )A 78 434 :M -.083(both )A 104 434 :M -.071(ancestors )A 153 434 :M (of )S 168 434 :M .668({X)A f0_7 sf 0 3 rm .324(1)A 0 -3 rm f0_12 sf .54(,X)A f0_7 sf 0 3 rm .324(2)A 0 -3 rm f0_12 sf .908 .091(}, )J 219 434 :M -.109(and )A 241 434 :M (k )S 252 434 :M .211 .021(< )J 264 434 :M (r )S 274 434 :M -.109(and )A 297 434 :M 1.111 .111(s, )J 312 434 :M -.334(it )A 324 434 :M (follows )S 366 434 :M -.08(from )A 395 434 :M -.22(the )A 415 434 :M -.081(ordering )A 461 434 :M (of )S 477 434 :M -.33(the)A 59 452 :M .313 .031(variables that X)J f0_7 sf 0 3 rm .052(k)A 0 -3 rm f0_12 sf .209 .021( is also an ancestor of {X)J f0_7 sf 0 3 rm .052(1)A 0 -3 rm f0_12 sf .087(,X)A f0_7 sf 0 3 rm .052(2)A 0 -3 rm f0_12 sf .17 .017(} in G)J f0_7 sf 0 3 rm .045(Construct)A 0 -3 rm f0_12 sf .086(\(G\(M\),X,Y,)A f2_12 sf .119(Z)A f0_12 sf .124 .012(\). )J 417 452 :M -.128(Hence )A 451 452 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 468 452 :M (is )S 480 452 :M -.326(an)A 59 470 :M .112 .011(ancestor of {X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .032(,X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf .086 .009(} in G\(M\). In the example, in G\(M\), X)J f0_7 sf 0 3 rm (4)S 0 -3 rm f0_12 sf .051 .005( is not an )J 387 470 :M -.122(ancestor )A 430 470 :M (of )S 444 470 :M -.22(the )A 462 470 :M -.249(empty)A 59 488 :M -.094(set but )A 94 488 :M (is )S 106 488 :M -.163(an )A 121 488 :M -.122(ancestor )A 164 488 :M (of )S 178 488 :M .876(X)A f0_7 sf 0 3 rm .354(1)A 0 -3 rm f0_12 sf .552 .055(, )J 199 488 :M -.109(and )A 220 488 :M -.334(it )A 230 488 :M (is )S 242 488 :M -.139(between )A 285 488 :M (two )S 307 488 :M -.163(vertices )A 347 488 :M .478(X)A f0_7 sf 0 3 rm .193(5)A 0 -3 rm f0_12 sf .166 .017( )J 364 488 :M -.109(and )A 385 488 :M .478(X)A f0_7 sf 0 3 rm .193(6)A 0 -3 rm f0_12 sf .166 .017( )J 402 488 :M -.065(which )A 435 488 :M -.082(also )A 458 488 :M -.215(are )A 476 488 :M -.167(not)A 59 506 :M -.027(ancestors of the empty set but are ancestors of X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.029( or X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf (.)S 77 524 :M -.068(Thus, if there is some vertex X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.054( in )A f2_12 sf -.132(Q)A f0_12 sf -.065( that is not an ancestor )A 361 524 :M (of )S 375 524 :M f2_12 sf 1.141(Z)A f0_12 sf .428(,)A f2_12 sf .428 .043( )J 392 524 :M f0_12 sf -.111(but )A 411 524 :M -.052(occurs )A 446 524 :M -.167(in )A 459 524 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 472 524 :M (as )S 486 524 :M (a)S 59 542 :M -.017(collider then X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.016( is an ancestor of X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.017( or X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.017(. Let X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 299 542 :M -.163(be )A 314 542 :M -.22(the )A 332 542 :M -.165(last )A 352 542 :M -.162(occurrence )A 407 542 :M (of )S 421 542 :M -.326(a )A 430 542 :M -.206(collider )A 469 542 :M -.167(in )A 482 542 :M f2_12 sf (Q)S 59 560 :M f0_12 sf -.249(that )A 80 560 :M (is )S 92 560 :M -.163(an )A 107 560 :M -.122(ancestor )A 150 560 :M (of )S 164 560 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 181 560 :M -.111(but )A 200 560 :M -.111(not )A 219 560 :M (of )S 233 560 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 249 560 :M -.164(if )A 260 560 :M -.196(there )A 287 560 :M (is )S 299 560 :M .481 .048(one, )J 324 560 :M -.072(otherwise )A 374 560 :M -.331(let )A 389 560 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 405 560 :M .211 .021(= )J 416 560 :M .876(X)A f0_7 sf 0 3 rm .354(1)A 0 -3 rm f0_12 sf .552 .055(. )J 437 560 :M -.083(Step )A 463 560 :M (\(2\) )S 482 560 :M -.334(in)A 59 578 :M -.035(forming )A f2_12 sf -.063(Q)A f0_12 sf -.031( and )A f2_12 sf (P)S f0_12 sf -.031( is )A 155 578 :M -.167(to )A 168 578 :M -.233(replace )A 205 578 :M -.22(the )A 223 578 :M -.058(subsequence )A 287 578 :M .466(A f0_12 sf .352(,X)A f0_7 sf 0 3 rm .188(a)A 0 -3 rm f0_12 sf .536 .054(> )J 350 578 :M (by )S 366 578 :M .449( )J 413 578 :M -.164(if )A 424 578 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 440 578 :M cF f1_12 sf 1.151A sf 1.151A f0_7 sf 0 3 rm .662(1)A 0 -3 rm f0_12 sf 1.032 .103(, )J 474 578 :M -.163(and)A 59 596 :M -.085(replacing the corresponding paths in )A f2_12 sf -.13(P)A f0_12 sf -.087( by a directed path from X)A f0_7 sf 0 3 rm -.055(a)A 0 -3 rm f0_12 sf ( )S 372 596 :M -.167(to )A 385 596 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 402 596 :M -.164(if )A 413 596 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 429 596 :M cF f1_12 sf 1.151A sf 1.151A f0_7 sf 0 3 rm .662(1)A 0 -3 rm f0_12 sf 1.032 .103(. )J 463 596 :M (\(Such)S 59 614 :M -.326(a )A 69 614 :M -.205(directed )A 111 614 :M -.165(path )A 136 614 :M -.055(exists )A 168 614 :M -.164(if )A 180 614 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 197 614 :M cF f1_12 sf 1.045A sf 1.045A f0_7 sf 0 3 rm .601(1)A 0 -3 rm f0_12 sf .515 .052( )J 228 614 :M -.139(because )A 270 614 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 288 614 :M (is )S 302 614 :M -.163(an )A 319 614 :M -.122(ancestor )A 364 614 :M (of )S 380 614 :M .721(X)A f0_7 sf 0 3 rm .291(1)A 0 -3 rm f0_12 sf .693 .069(.\) )J 407 614 :M -.083(This )A 434 614 :M -.092(removes )A 480 614 :M -.497(all)A 59 632 :M -.079(occurrences of vertices between X)A f0_7 sf 0 3 rm -.056(1)A 0 -3 rm f0_12 sf -.093( and )A 249 632 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 265 632 :M -.249(that )A 286 632 :M -.215(are )A 304 632 :M -.111(not )A 323 632 :M -.071(ancestors )A 371 632 :M (of )S 385 632 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 401 632 :M -.111(but )A 420 632 :M -.215(are )A 438 632 :M -.146(colliders )A 482 632 :M -.334(in)A 59 650 :M f2_12 sf .261(Q)A f0_12 sf .437 .044(. In the example, X)J f0_7 sf 0 3 rm .087(a)A 0 -3 rm f0_12 sf .25 .025( = X)J f0_7 sf 0 3 rm .098(4)A 0 -3 rm f0_12 sf .343 .034(, and after step 2, )J f2_12 sf .261(Q)A f0_12 sf .316 .032( = and )J f2_12 sf .205(P)A f0_12 sf .316 .032( = A f0_12 sf .272 .027( X)J f0_7 sf 0 3 rm .098(4)A 0 -3 rm f0_12 sf .153 .015(, )J 478 650 :M .596(X)A f0_7 sf 0 3 rm (4)S 0 -3 rm 59 668 :M f1_12 sf .788A f0_12 sf .613 .061( X)J f0_7 sf 0 3 rm .221(6)A 0 -3 rm f0_12 sf .711 .071(, X)J f0_7 sf 0 3 rm .221(6)A 0 -3 rm f0_12 sf .172 .017( )J f1_12 sf .747A f0_12 sf .613 .061( X)J f0_7 sf 0 3 rm .221(2)A 0 -3 rm f0_12 sf .616(>.)A endp %%Page: 28 28 %%BeginPageSetup initializepage (peter; page: 28 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (28)S gR gS 0 0 552 730 rC 77 56 :M f0_12 sf (By )S 96 56 :M -.06(definition, )A 150 56 :M -.129(every )A 181 56 :M -.163(vertex )A 215 56 :M -.249(that )A 237 56 :M -.052(occurs )A 273 56 :M (as )S 288 56 :M -.326(a )A 298 56 :M -.206(collider )A 338 56 :M -.139(between )A 382 56 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 399 56 :M -.109(and )A 421 56 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 439 56 :M -.167(in )A 453 56 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 467 56 :M (is )S 480 56 :M -.326(an)A 59 74 :M -.047(ancestor of )A f2_12 sf -.08(Z)A f0_12 sf -.047( or of X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.048(. Let X)A f0_7 sf 0 3 rm (b)S 0 -3 rm f0_12 sf -.044( be the first vertex after X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf -.038( in )A f2_12 sf -.093(Q)A f0_12 sf ( )S 354 74 :M -.249(that )A 375 74 :M (is )S 387 74 :M -.163(an )A 402 74 :M -.122(ancestor )A 445 74 :M (of )S 459 74 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 476 74 :M -.167(but)A 59 92 :M -.039(not of )A f2_12 sf -.071(Z)A f0_12 sf -.039(, if there is one, otherwise let )A 240 92 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 257 92 :M .211 .021(= )J 268 92 :M .876(X)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(. )J 289 92 :M -.083(Step )A 314 92 :M (\(3\) )S 332 92 :M -.167(in )A 345 92 :M -.093(forming )A 387 92 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 400 92 :M -.109(and )A 421 92 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 433 92 :M (is )S 445 92 :M -.167(to )A 458 92 :M -.272(replace)A 59 110 :M -.22(the )A 77 110 :M -.058(subsequence )A 141 110 :M .64(A f0_12 sf .327(,X)A f0_7 sf 0 3 rm .196(2)A 0 -3 rm f0_12 sf .497 .05(> )J 209 110 :M (by )S 225 110 :M .552( )J 273 110 :M -.164(if )A 284 110 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 301 110 :M f1_12 sf .285A f0_12 sf .13 .013( )J 312 110 :M .876(X)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 334 110 :M -.109(and )A 356 110 :M -.182(replacing )A 404 110 :M -.22(the )A 423 110 :M -.053(corresponding)A 59 128 :M (paths in )S f2_12 sf (P)S f0_12 sf ( )S 111 128 :M (by )S 127 128 :M -.326(a )A 136 128 :M -.205(directed )A 177 128 :M -.165(path )A 201 128 :M -.08(from )A 228 128 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 245 128 :M -.167(to )A 258 128 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 275 128 :M -.164(if )A 286 128 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 303 128 :M cF f1_12 sf 1.151A sf 1.151A f0_7 sf 0 3 rm .662(2)A 0 -3 rm f0_12 sf 1.032 .103(. )J 337 128 :M -.083(This )A 362 128 :M -.092(removes )A 406 128 :M -.331(all )A 421 128 :M -.117(occurrences )A 481 128 :M (of)S 59 146 :M -.146(colliders )A 104 146 :M -.139(between )A 148 146 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 166 146 :M -.109(and )A 188 146 :M .478(X)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 206 146 :M -.249(that )A 228 146 :M -.215(are )A 247 146 :M -.111(not )A 267 146 :M -.071(ancestors )A 316 146 :M (of )S 332 146 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 350 146 :M -.081(Note )A 379 146 :M -.249(that )A 402 146 :M -.331(all )A 419 146 :M -.117(occurrences )A 481 146 :M (of)S 59 164 :M -.089(colliders that are left are between X)A f0_7 sf 0 3 rm -.059(a)A 0 -3 rm f0_12 sf -.102( and X)A f0_7 sf 0 3 rm -.067(b)A 0 -3 rm f0_12 sf -.095(, and every occurrence of )A 388 164 :M -.326(a )A 397 164 :M -.206(collider )A 436 164 :M -.139(between )A 479 164 :M .169(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm 59 182 :M f0_12 sf -.03(and X)A f0_7 sf 0 3 rm (b)S 0 -3 rm f0_12 sf -.023( is an ancestor of )A f2_12 sf (Z)S f0_12 sf -.025( by construction. In the example, X)A f0_7 sf 0 3 rm (b)S 0 -3 rm f0_12 sf ( )S 359 182 :M .211 .021(= )J 370 182 :M .876(X)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 391 182 :M -.109(and )A 412 182 :M -.195(after )A 437 182 :M -.082(step )A 460 182 :M .724 .072(\(3\), )J 482 182 :M f2_12 sf (Q)S 59 200 :M f0_12 sf -.014(and )A f2_12 sf (P)S f0_12 sf -.015( are unchanged.)A 77 218 :M -.326(We )A 98 218 :M -.166(will )A 121 218 :M .259 .026(now )J 147 218 :M .479 .048(show )J 178 218 :M -.249(that )A 200 218 :M -.129(every )A 231 218 :M -.165(path )A 256 218 :M -.139(between )A 300 218 :M -.326(a )A 310 218 :M -.163(pair )A 333 218 :M (of )S 348 218 :M -.145(variables )A 395 218 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 413 218 :M -.109(and )A 435 218 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 453 218 :M -.167(in )A 467 218 :M f2_12 sf .475(P)A f0_12 sf .195 .019( )J 481 218 :M (d-)S 59 236 :M .191 .019(connects X)J f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .088 .009( and X)J f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .086 .009( given )J f2_12 sf .057(Z)A f0_12 sf .042(\\{X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .042(,X)A f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .111 .011(}. If the path between X)J f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf ( )S 354 236 :M -.109(and )A 375 236 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 392 236 :M (is )S 404 236 :M -.082(also )A 427 236 :M -.167(in )A 440 236 :M f2_12 sf .875<50D5>A f0_12 sf .843 .084(, )J 461 236 :M -.165(then )A 485 236 :M -.668(it)A 59 254 :M .388 .039(d-connects X)J f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .165 .017( and X)J f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .162 .016( given )J f2_12 sf .107(Z)A f0_12 sf .079(\\{X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .078(,X)A f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .107 .011(} )J 251 254 :M -.139(because )A 292 254 :M -.129(every )A 322 254 :M -.165(path )A 346 254 :M -.167(in )A 359 254 :M f2_12 sf .663<50D5>A f0_12 sf .351 .035( )J 376 254 :M (has )S 396 254 :M -.084(this )A 417 254 :M .186 .019(property. )J 465 254 :M (If )S 477 254 :M -.33(the)A 59 272 :M .031 .003(path between X)J f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf ( and X)S f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .012 .001( is not in )J f2_12 sf <50D5>S f0_12 sf .018 .002(, but was added in step \(1\) of )J 373 272 :M -.22(the )A 391 272 :M -.146(formation )A 441 272 :M (of )S 455 272 :M f2_12 sf .918(P)A f0_12 sf .683 .068(, )J 471 272 :M -.22(then)A 59 290 :M -.143(the path between )A 142 290 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 159 290 :M -.109(and )A 180 290 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 197 290 :M (is )S 209 290 :M -.326(a )A 218 290 :M -.196(correlated )A 268 290 :M -.061(error )A 295 290 :M -.163(trek )A 317 290 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 334 290 :M f1_12 sf .4A f0_12 sf .096 .01( )J 351 290 :M .876(X)A f0_7 sf 0 3 rm .354(v)A 0 -3 rm f0_12 sf .552 .055(, )J 372 290 :M -.065(which )A 405 290 :M -.234(clearly )A 440 290 :M -.108(d-connects)A 59 308 :M .077(X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .11 .011( and X)J f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .108 .011( given )J f2_12 sf .072(Z)A f0_12 sf .053(\\{X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .052(,X)A f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .14 .014(}. If the path between X)J f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .11 .011( and X)J f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .08 .008( is not in )J 387 308 :M f2_12 sf .875<50D5>A f0_12 sf .843 .084(, )J 408 308 :M -.111(but )A 427 308 :M .264 .026(was )J 450 308 :M -.13(added )A 482 308 :M -.334(in)A 59 326 :M .154 .015(step \(2\) of the formation of )J f2_12 sf .058<50D5>A f0_12 sf .141 .014(, then X)J f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf .092 .009( = X)J f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf .116 .012(, X)J f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf .092 .009( = X)J f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf .129 .013(, and the path )J 385 326 :M -.139(between )A 428 326 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 445 326 :M -.109(and )A 466 326 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 483 326 :M (is)S 59 344 :M -.326(a )A 68 344 :M -.205(directed )A 109 344 :M -.165(path )A 133 344 :M -.08(from )A 160 344 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 176 344 :M -.167(to )A 189 344 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 206 344 :M -.249(that )A 227 344 :M (does )S 253 344 :M -.111(not )A 272 344 :M -.189(contain )A 310 344 :M -.109(any )A 331 344 :M -.219(member )A 373 344 :M (of )S 387 344 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 403 344 :M -.128(Hence )A 437 344 :M -.22(the )A 456 344 :M -.165(path )A 481 344 :M (d-)S 59 362 :M -.049(connects X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf -.048( and X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf -.042( given )A f2_12 sf -.073(Z)A f0_12 sf -.043(. Similarly, if the path between )A 342 362 :M -.165(path )A 366 362 :M -.139(between )A 409 362 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 426 362 :M -.109(and )A 447 362 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 464 362 :M (is )S 476 362 :M -.167(not)A 59 380 :M -.167(in )A 72 380 :M f2_12 sf .875<50D5>A f0_12 sf .843 .084(, )J 93 380 :M -.111(but )A 112 380 :M .264 .026(was )J 135 380 :M -.13(added )A 167 380 :M -.167(in )A 180 380 :M -.082(step )A 203 380 :M (\(3\) )S 221 380 :M (of )S 235 380 :M -.22(the )A 253 380 :M -.146(formation )A 303 380 :M .387(of)A f2_12 sf .232 .023( )J 318 380 :M .918(P)A f0_12 sf .683 .068(, )J 334 380 :M -.165(then )A 358 380 :M .478(X)A f0_7 sf 0 3 rm .193(u)A 0 -3 rm f0_12 sf .166 .017( )J 375 380 :M .211 .021(= )J 386 380 :M .876(X)A f0_7 sf 0 3 rm .354(b)A 0 -3 rm f0_12 sf .552 .055(, )J 407 380 :M .478(X)A f0_7 sf 0 3 rm .193(v)A 0 -3 rm f0_12 sf .166 .017( )J 424 380 :M .211 .021(= )J 435 380 :M .876(X)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 456 380 :M -.109(and )A 477 380 :M -.33(the)A 59 398 :M -.053(path between X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf -.053( and X)A f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf -.047( is a directed path from X)A f0_7 sf 0 3 rm (b)S 0 -3 rm f0_12 sf -.048( to X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.048( that )A 349 398 :M (does )S 375 398 :M -.111(not )A 394 398 :M -.189(contain )A 432 398 :M -.109(any )A 453 398 :M -.263(member)A 59 416 :M (of )S f2_12 sf (Z)S f0_12 sf -.005(. Hence the path d-connects X)A f0_7 sf 0 3 rm (u)S 0 -3 rm f0_12 sf ( and X)S f0_7 sf 0 3 rm (v)S 0 -3 rm f0_12 sf ( given )S f2_12 sf (Z)S f0_12 sf (.)S 77 434 :M -.072(We will now show that every )A 220 434 :M -.163(vertex )A 253 434 :M -.249(that )A 274 434 :M -.052(occurs )A 309 434 :M (as )S 323 434 :M -.326(a )A 332 434 :M -.206(collider )A 371 434 :M -.167(in )A 384 434 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 397 434 :M (has )S 417 434 :M -.326(a )A 426 434 :M -.131(descendant )A 482 434 :M -.334(in)A 59 452 :M f2_12 sf -.144(Z)A f0_12 sf -.083(, and every vertex that occurs as a non-collider in )A f2_12 sf -.168(Q)A f0_12 sf -.084( is )A 326 452 :M -.111(not )A 345 452 :M -.167(in )A 358 452 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 374 452 :M -.13(Every )A 406 452 :M -.163(vertex )A 439 452 :M -.249(that )A 460 452 :M -.062(occurs)A 59 470 :M (as )S 73 470 :M -.326(a )A 82 470 :M -.206(collider )A 121 470 :M -.167(in )A 134 470 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 147 470 :M (is )S 160 470 :M -.163(an )A 176 470 :M -.122(ancestor )A 220 470 :M (of )S 235 470 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 252 470 :M -.139(because )A 294 470 :M (steps )S 323 470 :M (\(2\) )S 342 470 :M -.109(and )A 364 470 :M (\(3\) )S 383 470 :M -.167(in )A 397 470 :M -.22(the )A 416 470 :M -.146(formation )A 467 470 :M (of )S 482 470 :M f2_12 sf (Q)S 59 488 :M f0_12 sf -.114(removed all occurrences of colliders that were not ancestors )A 344 488 :M (of )S 358 488 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 374 488 :M -.13(Every )A 406 488 :M -.163(vertex )A 439 488 :M -.249(that )A 460 488 :M -.062(occurs)A 59 506 :M (as )S 73 506 :M -.326(a )A 82 506 :M -.137(non-collider )A 143 506 :M -.167(in )A 156 506 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 174 506 :M -.109(and )A 195 506 :M (as )S 209 506 :M -.326(a )A 218 506 :M -.137(non-collider )A 279 506 :M -.167(in )A 292 506 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 305 506 :M (is )S 317 506 :M -.111(not )A 336 506 :M -.167(in )A 350 506 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 367 506 :M -.139(because )A 409 506 :M -.129(every )A 440 506 :M -.163(vertex )A 474 506 :M -.331(that)A 59 524 :M -.068(occurs as a non-collider in )A f2_12 sf -.138(Q)A f0_12 sf -.06( is not in )A f2_12 sf -.118(Z)A f0_12 sf -.077(. The only )A 300 524 :M -.163(vertices )A 340 524 :M -.249(that )A 361 524 :M -.22(may )A 385 524 :M -.129(occur )A 415 524 :M (as )S 429 524 :M -.109(non-colliders)A 59 542 :M -.167(in )A 72 542 :M f2_12 sf -.253(Q)A f0_12 sf ( )S 85 542 :M -.111(but )A 104 542 :M -.111(not )A 123 542 :M -.167(in )A 136 542 :M f2_12 sf .275<51D5>A f0_12 sf .124 .012( )J 154 542 :M -.215(are )A 172 542 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 188 542 :M -.109(and )A 209 542 :M .876(X)A f0_7 sf 0 3 rm .354(b)A 0 -3 rm f0_12 sf .552 .055(. )J 230 542 :M .135(X)A f0_7 sf 0 3 rm (a)S 0 -3 rm f0_12 sf ( )S 246 542 :M (is )S 258 542 :M -.111(not )A 277 542 :M -.167(in )A 290 542 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 306 542 :M -.139(because )A 347 542 :M -.219(either )A 377 542 :M -.334(it )A 387 542 :M (is )S 399 542 :M -.197(equal )A 428 542 :M -.167(to )A 441 542 :M .478(X)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 459 542 :M (or )S 474 542 :M 1.049(X)A f0_7 sf 0 3 rm .424(2)A 0 -3 rm f0_12 sf (,)S 59 560 :M -.049(neither of which is in )A f2_12 sf -.085(Z)A f0_12 sf -.042(, or it is )A 211 560 :M -.111(not )A 230 560 :M -.163(an )A 245 560 :M -.122(ancestor )A 288 560 :M (of )S 302 560 :M f2_12 sf (Z)S f0_12 sf ( )S 314 560 :M (by )S 330 560 :M -.024(construction. )A 396 560 :M -.066(Similarly, )A 447 560 :M .478(X)A f0_7 sf 0 3 rm .193(b)A 0 -3 rm f0_12 sf .166 .017( )J 464 560 :M (is )S 476 560 :M -.167(not)A 59 578 :M .269 .027(in )J f2_12 sf .227(Z)A f0_12 sf (.)S 77 596 :M -.105(Hence )A f2_12 sf -.175(Q)A f0_12 sf -.086( is a sequence of paths that satisfy )A 283 596 :M -.097(properties )A 334 596 :M .486 .049(\(i\), )J 353 596 :M .23 .023(\(ii\), )J 375 596 :M -.109(and )A 396 596 :M -.198(\(iii\) )A 417 596 :M (of )S 431 596 :M -.33(Lemma )A 470 596 :M .833 .083(8. )J 484 596 :M -.327(It)A 59 614 :M -.053(follows from Lemma 8 that X)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.066A f0_12 sf -.055( X and X)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.066A f0_12 sf -.05( Y are d-connected given )A f2_12 sf -.08(Z)A f0_12 sf -.049( in G\(M\). )A f1_12 sf <5C>S 77 644 :M f2_12 sf 1.01 .101(Theorem 1)J f0_12 sf .295 .03(: If M is )J 179 644 :M -.326(a )A 188 644 :M .237 .024(SEM, )J 220 644 :M -.109(and )A 241 644 :M .629 .063({X} )J 266 644 :M -.109(and )A 287 644 :M -.061({Y} )A 311 644 :M -.215(are )A 329 644 :M -.117(d-separated )A 387 644 :M -.132(given )A 417 644 :M f2_12 sf (Z)S f0_12 sf ( )S 429 644 :M -.167(in )A 442 644 :M .171(G\(M\),)A 59 661 :M .307 .031(then )J f1_12 sf .128(r)A f0_12 sf .106(\(X,Y.)A f2_12 sf .156(Z)A f0_12 sf .151 .015(\) = 0 in )J f1_12 sf .138(S)A f0_12 sf .141(\(M\).)A endp %%Page: 29 29 %%BeginPageSetup initializepage (peter; page: 29 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (29)S gR gS 0 0 552 730 rC 77 56 :M f2_12 sf .019(Proof.)A f0_12 sf .059 .006( By Lemma 6 and Lemma 9 there is a SEM M\325\(M,X,Y,)J f2_12 sf (Z)S f0_12 sf (\) )S 393 56 :M -.083(with )A 418 56 :M -.22(the )A 436 56 :M -.206(marginal )A 481 56 :M (of)S 59 74 :M f1_12 sf .165(S)A f0_12 sf .138(\(M\325\(M,X,Y,)A f2_12 sf .186(Z)A f0_12 sf .192 .019(\)\) = )J f1_12 sf .165(S)A f0_12 sf .372 .037(\(M\), and )J 211 74 :M .629 .063({X} )J 236 74 :M -.109(and )A 257 74 :M -.061({Y} )A 281 74 :M -.117(d-separated )A 339 74 :M -.132(given )A 369 74 :M f2_12 sf (Z)S f0_12 sf ( )S 381 74 :M -.167(in )A 394 74 :M .121(G\(M\325\(M,X,Y,)A f2_12 sf .157(Z)A f0_12 sf .179 .018(\)\) )J 484 74 :M (=)S 59 92 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\). )J 177 92 :M -.139(Because )A 220 92 :M .358(G)A f0_7 sf 0 3 rm .125(Construct)A 0 -3 rm f0_12 sf .238(\(G\(M\),X,Y,)A f2_12 sf .331(Z)A f0_12 sf .263 .026(\) )J 335 92 :M (is )S 348 92 :M -.22(the )A 367 92 :M -.205(directed )A 409 92 :M -.064(graph )A 441 92 :M (of )S 456 92 :M -.326(a )A 466 92 :M -.331(latent)A 59 110 :M -.087(variable model M\325\(M,X,Y,)A f2_12 sf -.129(Z)A f0_12 sf -.075(\) with correlation matrix that has marginal )A f1_12 sf -.114(S)A f0_12 sf -.091(\(M\), no )A 445 110 :M -.218(correlated)A 59 128 :M .604 .06(errors, )J 95 128 :M -.109(and )A 116 128 :M .306 .031(X )J 129 128 :M -.109(and )A 150 128 :M -.663(Y )A 162 128 :M -.215(are )A 180 128 :M -.117(d-separated )A 239 128 :M -.132(given )A 270 128 :M f2_12 sf (Z)S f0_12 sf ( )S 283 128 :M -.167(in )A 297 128 :M .426(G)A f0_7 sf 0 3 rm .149(Construct)A 0 -3 rm f0_12 sf .284(\(G\(M\),X,Y,)A f2_12 sf .394(Z)A f0_12 sf .41 .041(\), )J 416 128 :M -.334(it )A 427 128 :M (follows )S 468 128 :M -.107(from)A 59 146 :M .084 .008(Lemma 2 that )J f1_12 sf (r)S f0_12 sf .028(\(X,Y.)A f2_12 sf (Z)S f0_12 sf .04 .004(\) = 0 in )J f1_12 sf (S)S f0_12 sf (. )S f1_12 sf <5C>S 77 170 :M f2_12 sf 1.973 .197(Theorem )J 131 170 :M 1.672 .167(2: )J 147 170 :M f0_12 sf (If )S 159 170 :M -.667(M )A 173 170 :M (is )S 185 170 :M -.326(a )A 194 170 :M .237 .024(SEM, )J 226 170 :M -.109(and )A 247 170 :M .25({X)A f0_7 sf 0 3 rm .068(i)A 0 -3 rm f0_12 sf .276 .028(} )J 274 170 :M -.109(and )A 295 170 :M .25({X)A f0_7 sf 0 3 rm .068(j)A 0 -3 rm f0_12 sf .276 .028(} )J 322 170 :M -.215(are )A 340 170 :M -.148(d-connected )A 402 170 :M -.132(given )A 433 170 :M f2_12 sf (Z)S f0_12 sf ( )S 446 170 :M -.167(in )A 460 170 :M .171(G\(M\),)A 59 188 :M .449 .045(then )J f1_12 sf .187(r)A f0_12 sf .18(\(X)A f0_7 sf 0 3 rm .055(i)A 0 -3 rm f0_12 sf .166(,X)A f0_7 sf 0 3 rm .055(j)A 0 -3 rm f0_12 sf .085(.)A f2_12 sf .228(Z)A f0_12 sf .166 .017(\) )J f1_12 sf .187A f0_12 sf .192 .019( 0 in )J f1_12 sf .202(S)A f0_12 sf .205(\(M\).)A 77 206 :M f2_12 sf .815(Proof.)A f0_12 sf .463 .046( )J 119 206 :M .197 .02(Suppose )J 165 206 :M -.249(that )A 187 206 :M .25({X)A f0_7 sf 0 3 rm .068(i)A 0 -3 rm f0_12 sf .276 .028(} )J 215 206 :M -.109(and )A 237 206 :M .25({X)A f0_7 sf 0 3 rm .068(j)A 0 -3 rm f0_12 sf .276 .028(} )J 265 206 :M -.215(are )A 284 206 :M -.148(d-connected )A 346 206 :M -.132(given )A 377 206 :M f2_12 sf (Z)S f0_12 sf ( )S 390 206 :M -.167(in )A 404 206 :M 1.114 .111(G, )J 422 206 :M -.109(and )A 444 206 :M -.22(the )A 463 206 :M -.109(set )A 481 206 :M (of)S 59 224 :M -.039(vertices in G is )A f2_12 sf -.075(V)A f0_12 sf -.043(. Form a graph Transform\(G\) with vertices )A f2_12 sf -.07(T)A f0_12 sf ( )S f1_12 sf -.08A f0_12 sf ( )S 373 224 :M f2_12 sf .25(V)A f0_12 sf .087 .009( )J 386 224 :M -.167(in )A 399 224 :M -.22(the )A 417 224 :M -.073(following )A 467 224 :M .337(way.)A 59 242 :M -.044(For a pair of vertices X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf -.05( and X)A f0_7 sf 0 3 rm -.052(m)A 0 -3 rm f0_12 sf -.036( in )A f2_12 sf -.082(V)A f0_12 sf -.043(, there is a directed edge X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.112A f0_12 sf -.055( X)A f0_7 sf 0 3 rm -.052(m)A 0 -3 rm f0_12 sf ( )S 400 242 :M -.167(in )A 413 242 :M -.024(Transform\(G\) )A 484 242 :M -.327(if)A 59 260 :M -.052(and only if there is a directed edge X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.132A f0_12 sf -.065( X)A f0_7 sf 0 3 rm -.06(m)A 0 -3 rm f0_12 sf -.056( in G. )A 300 260 :M .258 .026(For )J 321 260 :M -.163(vertices )A 361 260 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 378 260 :M -.109(and )A 399 260 :M -.055(X)A f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf ( )S 417 260 :M -.167(in )A 430 260 :M f2_12 sf .79(V)A f0_12 sf .497 .05(, )J 447 260 :M -.196(there )A 474 260 :M (is )S 486 260 :M (a)S 59 278 :M -.163(vertex )A 92 278 :M .198(T\(X)A f0_7 sf 0 3 rm .104(k)A 0 -3 rm f0_12 sf .173(,X)A f0_7 sf 0 3 rm .162(m)A 0 -3 rm f0_12 sf .189 .019(\) )J 142 278 :M -.167(in )A 155 278 :M f2_12 sf .171(T)A f0_12 sf .356 .036(,and )J 188 278 :M -.064(edges )A 219 278 :M -.055(X)A f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf ( )S 237 278 :M f1_12 sf .126A f0_12 sf ( )S 253 278 :M .198(T\(X)A f0_7 sf 0 3 rm .104(k)A 0 -3 rm f0_12 sf .173(,X)A f0_7 sf 0 3 rm .162(m)A 0 -3 rm f0_12 sf .189 .019(\) )J 304 278 :M f1_12 sf .126A f0_12 sf ( )S 321 278 :M .478(X)A f0_7 sf 0 3 rm .193(k)A 0 -3 rm f0_12 sf .166 .017( )J 339 278 :M -.164(if )A 351 278 :M -.109(and )A 373 278 :M -.164(if )A 385 278 :M -.109(and )A 407 278 :M -.083(only )A 433 278 :M -.164(if )A 445 278 :M -.196(there )A 473 278 :M (is )S 486 278 :M (a)S 59 296 :M -.057(double-headed arrow X)A f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.132A f0_12 sf -.061( X)A f0_7 sf 0 3 rm -.057(m)A 0 -3 rm f0_12 sf -.05( in G. \(For convenience in writing equations, for each latent)A 59 314 :M (variable T\(X)S f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf (,X)S f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf (\) in Transform\(G\), we will also refer to it as T\(X)S f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf (,X)S f0_7 sf 0 3 rm (k)S 0 -3 rm f0_12 sf (\).\))S 77 332 :M .111 .011(For {X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .03(,X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf (} )S f1_12 sf S f0_12 sf ( )S f2_12 sf (Z)S f0_12 sf ( )S f1_12 sf S f0_12 sf ( )S f2_12 sf (V)S f0_12 sf .059 .006(, if {X)J f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf .081 .008(} and {X)J f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf .096 .01(} are d-connected given )J f2_12 sf (Z)S f0_12 sf .061 .006( in G, then they )J 463 332 :M -.215(are )A 481 332 :M (d-)S 59 350 :M -.182(connected )A 111 350 :M -.132(given )A 142 350 :M f2_12 sf (Z)S f0_12 sf ( )S 155 350 :M -.167(in )A 169 350 :M .31 .031(Transform\(G\). )J 245 350 :M (By )S 264 350 :M -.33(Lemma )A 304 350 :M (3 )S 315 350 :M -.196(there )A 343 350 :M (is )S 356 350 :M -.326(a )A 366 350 :M -.223(SEM )A 396 350 :M .261 .026(M\325, )J 420 350 :M -.083(with )A 447 350 :M -.062(G\(M\325\) )A 484 350 :M (=)S 59 368 :M 1.097 .11(Transform\(G\(M\)\), and )J f1_12 sf .264(r)A f0_12 sf .253(\(X)A f0_7 sf 0 3 rm .078(i)A 0 -3 rm f0_12 sf .234(,X)A f0_7 sf 0 3 rm .078(j)A 0 -3 rm f0_12 sf .12(.)A f2_12 sf .321(Z)A f0_12 sf .683 .068(\) )J cF f1_12 sf .068A sf .683 .068J 77 386 :M -.029(Let )A f2_12 sf -.037(Double)A f0_12 sf -.039(\(X)A f0_7 sf 0 3 rm (i)S 0 -3 rm f0_12 sf -.028(\) be the set of vertices X)A f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf -.029( in G such that there is an edge X)A f0_7 sf 0 3 rm (m)S 0 -3 rm f0_12 sf ( )S f1_12 sf -.077A f0_12 sf -.036( X)A f0_7 sf 0 3 rm (j)S 0 -3 rm f0_12 sf -.031( in )A 478 386 :M 1.337(G.)A 59 404 :M .231 .023(In M\325,)J 163 425 242 27 rC 405 452 :M psb currentpoint pse 163 425 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7744 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (X) -4 346 sh (a) 2550 346 sh (X) 3008 346 sh (b) 4984 346 sh (T\(X) 5329 346 sh (,) 6085 346 sh (X) 6222 346 sh (\)) 6745 346 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 7409 346 sh 224 ns (i) 295 442 sh (im) 2747 442 sh (m) 3309 442 sh (ij) 5194 442 sh (i) 5989 442 sh (m) 6523 442 sh (i) 7529 442 sh (X) 3905 719 sh (X) 5201 719 sh (X) 809 719 sh (X) 2153 719 sh 160 ns (m) 4093 775 sh (i) 5388 776 sh (m) 997 775 sh (i) 2340 776 sh 384 /Symbol f1 (=) 493 346 sh (+) 3605 346 sh (+) 6947 346 sh 224 ns (\316) 4269 719 sh (\316) 1173 719 sh 576 ns (\345) 4523 433 sh (\345) 1451 433 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 7233 346 sh 224 /Times-Bold f1 (Double) 4410 719 sh (Parents) 1313 719 sh 224 /MT_Times-Roman f1 (\() 5105 719 sh (\)) 5485 719 sh (\() 2057 719 sh (\)) 2437 719 sh end MTsave restore pse gR gS 0 0 552 730 rC 50 425 1 27 rF 59 467 :M f0_12 sf -.034(Now define)A 213 470 141 27 rC 354 497 :M psb currentpoint pse 213 470 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4512 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) -16 346 sh (e) 4019 346 sh 224 /Times-Roman f1 (i) 177 442 sh (ij) 1980 442 sh (i) 2775 442 sh (m) 3309 442 sh (i) 4315 442 sh (X) 691 719 sh (X) 1987 719 sh 384 ns (b) 1770 346 sh (T\(X) 2115 346 sh (,) 2871 346 sh (X) 3008 346 sh (\)) 3531 346 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 4195 346 sh 160 ns (m) 879 775 sh (i) 2174 776 sh 384 /Symbol f1 (=) 375 346 sh (+) 3733 346 sh 224 ns (\316) 1055 719 sh 576 ns (\345) 1309 433 sh 224 /Times-Bold f1 (Double) 1196 719 sh 224 /MT_Times-Roman f1 (\() 1891 719 sh (\)) 2271 719 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 512 :M f0_12 sf -.104(It follows then that)A 229 515 110 27 rC 339 542 :M psb currentpoint pse 229 515 :M psb /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3520 div 864 3 -1 roll exch div scale currentpoint translate 64 38 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (X) -4 346 sh (a) 1901 346 sh (X) 2235 346 sh 224 ns (i) 295 442 sh (ij) 2098 442 sh (m) 2536 442 sh (i) 3311 442 sh (X) 809 719 sh (X) 2153 719 sh 160 ns (m) 997 775 sh (i) 2340 776 sh 384 /Symbol f1 (=) 493 346 sh (+) 2832 346 sh 224 ns (\316) 1173 719 sh 576 ns (\345) 1451 433 sh /f2 {ff matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) 3118 346 sh 224 /Times-Bold f1 (Parents) 1313 719 sh 224 /Times-Roman f1 (\() 2057 719 sh (\)) 2437 719 sh end MTsave restore pse gR gS 0 0 552 730 rC 59 557 :M f0_12 sf .392 .039(is a SEM M, with G\(M\) = G, and )J f1_12 sf .191(r)A f0_12 sf .184(\(X)A f0_7 sf 0 3 rm .057(i)A 0 -3 rm f0_12 sf .17(,X)A f0_7 sf 0 3 rm .057(j)A 0 -3 rm f0_12 sf .087(.)A f2_12 sf .233(Z)A f0_12 sf .308 .031(\) )J cF f1_12 sf .031A sf .308 .031(\3120. )J f1_12 sf <5C>S 77 593 :M f0_12 sf -.326(We )A 97 593 :M -.166(will )A 119 593 :M -.064(prove )A 150 593 :M -.187(Theorem )A 196 593 :M (5 )S 207 593 :M -.106(before )A 242 593 :M -.187(Theorem )A 289 593 :M (3 )S 300 593 :M -.139(because )A 342 593 :M (we )S 361 593 :M -.166(will )A 384 593 :M (use )S 405 593 :M -.187(Theorem )A 452 593 :M (5 )S 463 593 :M -.167(in )A 477 593 :M -.33(the)A 59 611 :M -.016(proof of Theorem 3.)A 77 629 :M f2_12 sf 1.973 .197(Theorem )J 131 629 :M 1.672 .167(5: )J 147 629 :M f0_12 sf (If )S 159 629 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 176 629 :M -.109(and )A 197 629 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 214 629 :M -.215(are )A 232 629 :M -.165(path )A 256 629 :M -.123(diagrams )A 303 629 :M -.249(that )A 325 629 :M -.215(are )A 344 629 :M -.196(covariance )A 399 629 :M -.198(equivalent )A 452 629 :M -.08(over )A 478 629 :M f2_12 sf .504(O)A f0_12 sf (,)S 59 647 :M -.064(then G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.063( and G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.056( are d-separation equivalent over )A f2_12 sf -.111(O)A f0_12 sf (.)S 77 665 :M f2_12 sf .815(Proof.)A f0_12 sf .463 .046( )J 119 665 :M .197 .02(Suppose )J 165 665 :M -.249(that )A 187 665 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 205 665 :M -.109(and )A 227 665 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 245 665 :M -.215(are )A 265 665 :M -.111(not )A 286 665 :M -.108(d-separation )A 350 665 :M -.198(equivalent )A 404 665 :M -.08(over )A 431 665 :M f2_12 sf .405(O)A f0_12 sf .237 .024(. )J 450 665 :M .056(Suppose)A 59 683 :M -.065(without loss of generality that there is some {X}, {Y} and )A f2_12 sf -.109(Z)A f0_12 sf -.072( included )A 393 683 :M -.167(in )A 406 683 :M f2_12 sf .405(O)A f0_12 sf .237 .024(, )J 423 683 :M (such )S 449 683 :M -.249(that )A 470 683 :M .409({X})A endp %%Page: 30 30 %%BeginPageSetup initializepage (peter; page: 30 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (30)S gR gS 0 0 552 730 rC 59 56 :M f0_12 sf -.062(and {Y} are d-connected given )A f2_12 sf -.097(Z)A f0_12 sf -.058( in G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.056(, but not in )A 300 56 :M .876(G)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(. )J 321 56 :M (By )S 339 56 :M -.187(Theorem )A 385 56 :M .833 .083(2, )J 399 56 :M -.196(there )A 426 56 :M (is )S 438 56 :M -.082(some )A 467 56 :M -.334(SEM)A 59 74 :M -.667(M )A 73 74 :M -.083(with )A 98 74 :M -.079(G\(M\) )A 129 74 :M .211 .021(= )J 140 74 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 157 74 :M (such )S 183 74 :M -.249(that )A 204 74 :M f1_12 sf .282(r)A f0_12 sf .234(\(X,Y.)A f2_12 sf .343(Z)A f0_12 sf .273 .027(\) )J 256 74 :M f1_12 sf 1.347A f0_12 sf 2.359 .236(\3120. )J 283 74 :M (By )S 302 74 :M -.187(Theorem )A 349 74 :M .833 .083(1, )J 364 74 :M -.196(there )A 392 74 :M (is )S 405 74 :M (no )S 422 74 :M -.223(SEM )A 451 74 :M -.33<4DD520>A 470 74 :M -.11(with)A 59 92 :M .277 .028(G\(M\325\) = G)J f0_7 sf 0 3 rm .053(2)A 0 -3 rm f0_12 sf .188 .019(, in which )J f1_12 sf .1(r)A f0_12 sf .083(\(X,Y.)A f2_12 sf .122(Z)A f0_12 sf .089 .009(\) )J f1_12 sf .1A f0_12 sf .282 .028(\3120. Hence G)J f1_7 sf 0 3 rm .053(1)A 0 -3 rm f0_12 sf .188 .019( and G)J f0_7 sf 0 3 rm .053(2)A 0 -3 rm f0_12 sf .243 .024( are not covariance )J 418 92 :M -.198(equivalent )A 470 92 :M -.106(over)A 59 110 :M f2_12 sf .114(O)A f0_12 sf .061 .006(. )J f1_12 sf <5C>S 77 128 :M f0_12 sf .136 .014(Let )J f2_12 sf .057(Ancestors)A f2_7 sf 0 -5 rm (*)S 0 5 rm f0_12 sf .162 .016(\(X,G\) be the set of ancestors of X, exccluding X, in directed )J 447 128 :M -.064(graph )A 478 128 :M 1.337(G,)A 59 146 :M .246 .025(and )J f2_12 sf .099(Descendants)A f2_7 sf 0 -5 rm .059(*)A 0 5 rm f0_12 sf .275 .027(\(X,G\) be the set of descendants of X excluding X in G.)J 77 164 :M f2_12 sf 1.122 .112(Lemma )J 122 164 :M .563(10:)A f0_12 sf .317 .032( )J 144 164 :M (In )S 158 164 :M -.326(a )A 167 164 :M -.205(directed )A 208 164 :M -.282(acyclic )A 244 164 :M -.064(graph )A 275 164 :M 1.114 .111(G, )J 292 164 :M -.164(if )A 303 164 :M .306 .031(X )J 316 164 :M -.109(and )A 337 164 :M -.663(Y )A 349 164 :M -.215(are )A 367 164 :M -.111(not )A 387 164 :M -.108(adjacent, )A 434 164 :M -.663(Y )A 447 164 :M (is )S 460 164 :M -.111(not )A 480 164 :M -.326(an)A 59 182 :M -.122(ancestor )A 102 182 :M (of )S 116 182 :M 1.114 .111(X, )J 134 182 :M f2_12 sf .363(Ancestors)A f2_7 sf 0 -5 rm .226(*)A 0 5 rm f0_12 sf 1.862 .186(\(Y,G\)\\{X} )J 252 182 :M f1_12 sf .33A f0_12 sf .116 .012( )J 266 182 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(, )J 283 182 :M -.109(and )A 305 182 :M f2_12 sf (Z)S f0_12 sf ( )S 318 182 :M f1_12 sf -.161A f0_12 sf ( )S 332 182 :M f2_12 sf .424(Descendants)A f2_7 sf 0 -5 rm .255(*)A 0 5 rm f0_12 sf 1.523 .152(\(Y,G\) )J 440 182 :M .211 .021(= )J 452 182 :M f1_12 sf .699A f0_12 sf .386 .039(, )J 471 182 :M -.22(then)A 59 200 :M -.036({X} and {Y} are d-separated given )A f2_12 sf -.056(Z)A f0_12 sf (.)S 77 218 :M f2_12 sf .815(Proof.)A f0_12 sf .463 .046( )J 118 218 :M .197 .02(Suppose )J 163 218 :M -.249(that )A 184 218 :M .306 .031(X )J 197 218 :M -.109(and )A 218 218 :M -.663(Y )A 230 218 :M -.215(are )A 248 218 :M -.111(not )A 267 218 :M -.108(adjacent, )A 313 218 :M -.111(but )A 332 218 :M -.196(there )A 359 218 :M (is )S 371 218 :M -.326(a )A 380 218 :M -.165(path )A 404 218 :M .306 .031(U )J 418 218 :M -.249(that )A 440 218 :M -.108(d-connects)A 59 236 :M -.015({X} and {Y} given )A f2_12 sf (Z)S f0_12 sf -.014(. Suppose that U contains )A 289 236 :M -.163(an )A 304 236 :M -.163(edge )A 330 236 :M -.663(A )A 342 236 :M f1_12 sf .126A f0_12 sf ( )S 358 236 :M -.663(Y )A 370 236 :M -.109(and )A 391 236 :M -.663(A )A 403 236 :M f1_12 sf .405 .04J 416 236 :M f2_12 sf .569(Z)A f0_12 sf .388 .039(. )J 432 236 :M -.131(Since )A 462 236 :M -.663(A )A 474 236 :M (is )S 486 236 :M (a)S 59 254 :M -.137(non-collider )A 120 254 :M (on )S 136 254 :M .306 .031(U )J 149 254 :M .688 .069(\(A)J cF f1_12 sf .069A sf .688 .069(X\), )J 189 254 :M -.334(it )A 200 254 :M (follows )S 241 254 :M -.249(that )A 263 254 :M .306 .031(U )J 277 254 :M (does )S 304 254 :M -.111(not )A 324 254 :M -.145(d-connect )A 375 254 :M .629 .063({X} )J 401 254 :M -.109(and )A 423 254 :M -.061({Y} )A 448 254 :M -.132(given )A 479 254 :M f2_12 sf .724(Z)A f0_12 sf (,)S 59 272 :M -.122(contrary )A 102 272 :M -.167(to )A 115 272 :M .32 .032(hypothesis. )J 174 272 :M .197 .02(Suppose )J 219 272 :M -.165(then )A 243 272 :M -.249(that )A 264 272 :M .306 .031(U )J 277 272 :M -.123(contains )A 320 272 :M -.163(an )A 335 272 :M -.163(edge )A 361 272 :M -.663(A )A 373 272 :M f1_12 sf .126A f0_12 sf ( )S 390 272 :M .281 .028(Y. )J 407 272 :M -.164(It )A 419 272 :M (follows )S 460 272 :M -.249(that )A 482 272 :M (U)S 59 290 :M -.071(contains a collider, because by hypothesis, Y is not an ancestor )A 361 290 :M (of )S 375 290 :M 1.114 .111(X. )J 392 290 :M -.33(Let )A 411 290 :M (C )S 423 290 :M -.163(be )A 438 290 :M -.22(the )A 456 290 :M -.235(collider)A 59 308 :M (on )S 83 308 :M .306 .031(U )J 104 308 :M -.093(closest )A 148 308 :M -.167(to )A 169 308 :M .281 .028(Y. )J 193 308 :M (C )S 213 308 :M (is )S 233 308 :M -.326(a )A 250 308 :M -.131(descendant )A 314 308 :M (of )S 336 308 :M .281 .028(Y, )J 360 308 :M .277 .028(so )J 383 308 :M f2_12 sf .453(Descendants)A f0_12 sf .431(\(C,G\))A f1_12 sf S 59 326 :M f2_12 sf .326(Descendants)A f2_7 sf 0 -5 rm .196(*)A 0 5 rm f0_12 sf 1.154 .115(\(Y,G\). Hence )J f2_12 sf .326(Descendants)A f0_12 sf 1.074 .107(\(C,G\) )J f1_12 sf .517A f0_12 sf .153 .015( )J f2_12 sf .449(Z)A f0_12 sf .311 .031( = )J f1_12 sf .554A f0_12 sf .459 .046(, so )J 369 326 :M -.197(again )A 398 326 :M -.167(in )A 411 326 :M -.084(this )A 432 326 :M -.161(case )A 456 326 :M .306 .031(U )J 469 326 :M (does)S 59 344 :M -.037(not d-connect {X} and {Y} given )A f2_12 sf -.057(Z)A f0_12 sf (. )S f1_12 sf <5C>S 77 362 :M f2_12 sf 1.973 .197(Theorem )J 132 362 :M .387(3:)A f0_12 sf .232 .023( )J 148 362 :M (If )S 161 362 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 179 362 :M -.109(and )A 201 362 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 219 362 :M -.215(are )A 238 362 :M -.205(directed )A 280 362 :M -.282(acyclic )A 318 362 :M .596 .06(graphs, )J 360 362 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 379 362 :M -.109(and )A 402 362 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 421 362 :M -.215(are )A 441 362 :M -.218(covariance)A 59 380 :M -.092(equivalent if and only if G)A f0_7 sf 0 3 rm -.069(1)A 0 -3 rm f0_12 sf -.105( and G)A f0_7 sf 0 3 rm -.069(2)A 0 -3 rm f0_12 sf -.097( are d-separation equivalent)A 77 398 :M f2_12 sf .815(Proof.)A f0_12 sf .463 .046( )J 120 398 :M (By )S 140 398 :M -.187(Theorem )A 188 398 :M .833 .083(5, )J 204 398 :M -.164(if )A 217 398 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 236 398 :M -.109(and )A 260 398 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 280 398 :M -.215(are )A 301 398 :M -.196(covariance )A 358 398 :M -.198(equivalent )A 413 398 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 433 398 :M -.109(and )A 457 398 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 477 398 :M -.323(are)A 59 416 :M -.099(d-separation equivalent.)A 77 434 :M .197 .02(Suppose )J 122 434 :M -.249(that )A 143 434 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 160 434 :M -.109(and )A 182 434 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 200 434 :M -.215(are )A 219 434 :M -.108(d-separation )A 282 434 :M -.089(equivalent, )A 339 434 :M -.109(and )A 361 434 :M -.667(M )A 376 434 :M (is )S 389 434 :M -.326(a )A 399 434 :M -.223(SEM )A 428 434 :M -.083(with )A 454 434 :M -.234(directed)A 59 452 :M -.063(acyclic graph G\(M\) = G)A f0_7 sf 0 3 rm (1)S 0 -3 rm f0_12 sf -.062(. We can form a SEM M\325\325 where G\(M\325\325\) is a subgraph of )A 454 452 :M .596(G)A f0_7 sf 0 3 rm (2)S 0 -3 rm 467 450 :M f0_5 sf (8)S 470 452 :M f0_12 sf ( )S 474 452 :M -.163(and)A 59 470 :M f1_12 sf (S)S f0_12 sf <284DD5D529203D>S f1_12 sf ( S)S f0_12 sf .01 .001(\(M\) in the following way.)J 77 488 :M -.195(Order )A 108 488 :M -.22(the )A 126 488 :M -.145(variables )A 172 488 :M -.167(in )A 185 488 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 202 488 :M .277 .028(so )J 217 488 :M -.249(that )A 238 488 :M .306 .031(X )J 251 488 :M -.131(comes )A 285 488 :M -.106(before )A 319 488 :M -.663(Y )A 331 488 :M -.167(in )A 344 488 :M -.22(the )A 362 488 :M -.081(ordering )A 406 488 :M -.083(only )A 431 488 :M -.164(if )A 442 488 :M .306 .031(X )J 455 488 :M (is )S 467 488 :M -.111(not )A 486 488 :M (a)S 59 506 :M -.062(descendant of Y. Form a directed acyclic graph G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.058(\325 that has G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.063( as a subgraph by )A 443 506 :M -.143(putting )A 480 506 :M -.326(an)A 59 524 :M -.092(edge between X )A 139 524 :M -.109(and )A 160 524 :M -.663(Y )A 172 524 :M -.164(if )A 183 524 :M -.109(and )A 204 524 :M -.083(only )A 229 524 :M -.164(if )A 240 524 :M .306 .031(X )J 253 524 :M -.121(precedes )A 298 524 :M -.663(Y )A 310 524 :M -.167(in )A 323 524 :M -.22(the )A 341 524 :M .186 .019(ordering. )J 389 524 :M (In )S 403 524 :M -.109(any )A 424 524 :M -.223(SEM )A 452 524 :M -.33<4DD520>A 470 524 :M -.11(with)A 59 542 :M -.064(graph )A 90 542 :M -.062(G\(M\325\) )A 125 542 :M .211 .021(= )J 136 542 :M .721(G)A f0_7 sf 0 3 rm .291(2)A 0 -3 rm f0_12 sf .693 .069J 161 542 :M -.22(the )A 179 542 :M -.061(error )A 206 542 :M -.247(term )A 232 542 :M (for )S 251 542 :M -.326(a )A 261 542 :M -.205(variable )A 303 542 :M -.663(Y )A 316 542 :M (is )S 329 542 :M -.15(independent )A 391 542 :M (of )S 406 542 :M -.22(the )A 425 542 :M -.092(parents )A 464 542 :M (of )S 479 542 :M .337(Y.)A 59 560 :M -.098(Hence if X is a parent of Y in )A 202 560 :M .432 .043(G\(M\325\), )J 241 560 :M -.22(the )A 259 560 :M -.031(regression )A 312 560 :M -.208(coefficient )A 365 560 :M (of )S 379 560 :M .306 .031(X )J 392 560 :M (when )S 422 560 :M -.663(Y )A 434 560 :M (is )S 446 560 :M -.037(regressed)A 59 578 :M -.095(on its parents in G\(M\325\) using )A f1_12 sf -.142(S)A f0_12 sf -.092(\(M\) is equal to the linear coefficient of X in the )A 433 578 :M -.165(equation )A 477 578 :M (for)S 59 596 :M .281 .028(Y. )J 75 596 :M -.165(Remove )A 118 596 :M -.163(an )A 133 596 :M -.163(edge )A 159 596 :M -.08(from )A 186 596 :M .306 .031(X )J 199 596 :M -.167(to )A 212 596 :M -.663(Y )A 224 596 :M -.062(G\(M\325\) )A 259 596 :M -.164(if )A 270 596 :M -.109(and )A 292 596 :M -.083(only )A 318 596 :M -.164(if )A 330 596 :M -.22(the )A 349 596 :M -.219(linear )A 380 596 :M -.208(coefficient )A 434 596 :M (of )S 449 596 :M .306 .031(X )J 463 596 :M -.167(in )A 477 596 :M -.33(the)A 59 614 :M -.165(equation )A 103 614 :M (for )S 121 614 :M -.663(Y )A 133 614 :M (is )S 145 614 :M .833 .083(0. )J 159 614 :M -.219(The )A 181 614 :M -.109(result )A 212 614 :M (is )S 225 614 :M -.326(a )A 235 614 :M -.223(SEM )A 264 614 :M .248 .025<4DD5D52C20>J 291 614 :M -.062(where )A 325 614 :M f1_12 sf .05(S)A f0_12 sf .139 .014<284DD5D52920>J 364 614 :M .211 .021(= )J 376 614 :M f1_12 sf .278(S)A f0_12 sf .69 .069(\(M\), )J 411 614 :M -.109(and )A 433 614 :M -.05<47284DD5D52920>A 473 614 :M (is )S 486 614 :M (a)S 59 632 :M .49 .049(subgraph of G)J f0_7 sf 0 3 rm .082(2)A 0 -3 rm f0_12 sf .165A 59 674 :M ( )S 59 671.48 -.48 .48 203.48 671 .48 59 671 @a 59 684 :M f0_8 sf (8)S 63 687 :M f0_10 sf .204 .02( Note this includes the possibility that G\(S\325\325\) = G)J f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf (.)S endp %%Page: 31 31 %%BeginPageSetup initializepage (peter; page: 31 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (31)S gR gS 0 0 552 730 rC 77 56 :M f0_12 sf -.035(Suppose X and Y are not adjacent in G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.032(. In G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.032(, either X is not an ancestor of )A 445 56 :M -.663(Y )A 457 56 :M (or )S 471 56 :M -.663(Y )A 483 56 :M (is)S 59 74 :M -.111(not )A 78 74 :M -.163(an )A 93 74 :M -.122(ancestor )A 136 74 :M (of )S 150 74 :M (X; )S 166 74 :M .2 .02(suppose )J 209 74 :M -.095(without )A 249 74 :M .237 .024(loss )J 272 74 :M (of )S 286 74 :M -.197(generality )A 336 74 :M -.249(that )A 357 74 :M -.663(Y )A 370 74 :M (is )S 383 74 :M -.111(not )A 403 74 :M -.163(an )A 419 74 :M -.122(ancestor )A 463 74 :M (of )S 478 74 :M 1.337(X.)A 59 92 :M -.003(Then X and Y are d-separated given )A f2_12 sf (Parents)S f0_12 sf (\(Y,G)S f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf (\) in G)S f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf (.By Lemma )S 395 92 :M .769 .077(10, )J 415 92 :M -.167(in )A 428 92 :M .876(G)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(, )J 449 92 :M .629 .063({X} )J 474 92 :M -.163(and)A 59 110 :M -.061({Y} )A 83 110 :M -.215(are )A 101 110 :M -.117(d-separated )A 159 110 :M -.132(given )A 189 110 :M f2_12 sf .338(Parents)A f0_12 sf .366(\(Y,G)A f0_7 sf 0 3 rm .211(2)A 0 -3 rm f0_12 sf .647 .065J 276 110 :M -.139(because )A 318 110 :M f2_12 sf .279(Parents)A f0_12 sf .302(\(Y,G)A f0_7 sf 0 3 rm .174(2)A 0 -3 rm f0_12 sf .455 .045J 402 110 :M -.123(contains )A 446 110 :M -.331(all )A 462 110 :M (of )S 477 110 :M -.33(the)A 59 128 :M -.025(ancestors of Y in G)A f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf -.026(, and no descendants of )A 271 128 :M -.663(Y )A 283 128 :M -.167(in )A 296 128 :M .876(G)A f0_7 sf 0 3 rm .354(2)A 0 -3 rm f0_12 sf .552 .055(. )J 317 128 :M -.139(Because )A 360 128 :M .478(G)A f0_7 sf 0 3 rm .193(1)A 0 -3 rm f0_12 sf .166 .017( )J 377 128 :M -.109(and )A 398 128 :M .478(G)A f0_7 sf 0 3 rm .193(2)A 0 -3 rm f0_12 sf .166 .017( )J 415 128 :M -.215(are )A 433 128 :M -.118(d-separation)A 59 146 :M -.089(equivalent, )A 115 146 :M .629 .063({X} )J 140 146 :M -.109(and )A 161 146 :M -.061({Y} )A 186 146 :M -.215(are )A 205 146 :M -.117(d-separated )A 264 146 :M -.132(given )A 295 146 :M f2_12 sf .279(Parents)A f0_12 sf .302(\(Y,G)A f0_7 sf 0 3 rm .174(2)A 0 -3 rm f0_12 sf .455 .045J 379 146 :M -.167(in )A 393 146 :M .876(G)A f0_7 sf 0 3 rm .354(1)A 0 -3 rm f0_12 sf .552 .055(. )J 415 146 :M (By )S 434 146 :M -.187(Theorem )A 481 146 :M 1(1,)A 59 163 :M f1_12 sf .273(r\()A f0_12 sf .301(X,Y.)A f2_12 sf .29(Parents)A f0_12 sf .313(\(Y,G)A f0_7 sf 0 3 rm .18(2)A 0 -3 rm f0_12 sf .594 .059J 182 163 :M .211 .021(= )J 193 163 :M (0 )S 203 163 :M -.167(in )A 216 163 :M f1_12 sf .278(S)A f0_12 sf .69 .069(\(M\). )J 250 163 :M -.128(Hence )A 284 163 :M -.22(the )A 302 163 :M -.031(regression )A 356 163 :M -.208(coefficient )A 410 163 :M (of )S 425 163 :M .306 .031(X )J 439 163 :M (when )S 470 163 :M -.663(Y )A 483 163 :M (is)S 59 181 :M -.033(regressed )A 108 181 :M (on )S 124 181 :M f2_12 sf .279(Parents)A f0_12 sf .302(\(Y,G)A f0_7 sf 0 3 rm .174(2)A 0 -3 rm f0_12 sf .455 .045J 207 181 :M (using )S 237 181 :M f1_12 sf .278(S)A f0_12 sf .69 .069(\(M\), )J 271 181 :M (is )S 283 181 :M -.197(equal )A 312 181 :M -.167(to )A 325 181 :M .833 .083(0. )J 339 181 :M -.164(It )A 350 181 :M (follows )S 390 181 :M -.249(that )A 411 181 :M -.196(there )A 439 181 :M (is )S 452 181 :M (no )S 469 181 :M -.217(edge)A 59 199 :M .015 .002(between X and Y in G\(M\325\325\). Hence )J f1_12 sf (S)S f0_12 sf <284DD5D529203D20>S f1_12 sf (S)S f0_12 sf .015 .001(\(M\), and G\(M\325\325\) is a subgraph of G)J f0_7 sf 0 3 rm (2)S 0 -3 rm f0_12 sf (. )S f1_12 sf <5C>S 247 236 :M f2_12 sf .605(Bibliography)A 77 254 :M f0_12 sf .342 .034(Andersson, )J 136 254 :M 1.792 .179(S., )J 155 254 :M -.082(Madigan, )A 204 254 :M 1.028 .103(D., )J 224 254 :M -.109(and )A 245 254 :M (Perlman, )S 292 254 :M .277 .028(M. )J 310 254 :M (\(1995\) )S 346 254 :M -.663(A )A 358 254 :M -.205(Characterization )A 439 254 :M (of )S 454 254 :M -.197(Markov)A 59 272 :M -.13(Equivalence Classes for Acyclic Digraphs, Technical Report )A 347 272 :M .714 .071(287, )J 373 272 :M -.264(Department )A 431 272 :M (of )S 445 272 :M -.033(Statistics,)A 59 290 :M -.026(University of Washington.)A 77 308 :M -.04(Blalock, )A 121 308 :M 1.798 .18(H., )J 142 308 :M .667 .067(1961, )J 174 308 :M -.11(Causal )A 210 308 :M -.095(Inferences )A 263 308 :M -.167(in )A 276 308 :M -.153(Nonexperimental )A 361 308 :M (Research, )S 413 308 :M .523 .052(\(W. )J 437 308 :M .562 .056(W. )J 457 308 :M (Norton)S 59 326 :M .493 .049(and Co., New York\).)J 77 344 :M -.019(Bollen, K., 1989, Structural Equations with Latent Variables. \(Wiley, New York\).)A 77 362 :M -.029(Chickering, )A 138 362 :M .281 .028(D. )J 155 362 :M (\(1995\) )S 193 362 :M -.663(A )A 207 362 :M -.143(Transformational )A 294 362 :M -.205(Characterization )A 376 362 :M (of )S 392 362 :M -.198(Equivalent )A 448 362 :M -.14(Bayesian)A 59 380 :M (Network )S 105 380 :M .169 .017(Structures, )J 161 380 :M -.058(Proceedings )A 223 380 :M (of )S 237 380 :M -.22(the )A 255 380 :M -.206(Eleventh )A 300 380 :M -.13(Conference )A 359 380 :M (on )S 376 380 :M -.149(Uncertainty )A 436 380 :M -.167(in )A 450 380 :M -.33(Artificial)A 59 398 :M -.126(Intelligence, )A 122 398 :M -.125(Philippe )A 166 398 :M -.045(Besnard )A 210 398 :M -.109(and )A 232 398 :M -.131(Steve )A 263 398 :M .229 .023(Hanks )J 299 398 :M 1.186 .119(\(Eds.\), )J 338 398 :M -.164(Morgan )A 381 398 :M -.08(Kaufmann )A 437 398 :M .101(Publishers,)A 59 416 :M .278 .028(Inc., San Mateo, CA.)J 77 434 :M .329 .033(Frydenberg, )J 141 434 :M 1.025 .103(M., )J 164 434 :M .667 .067(1990, )J 197 434 :M -.219(The )A 220 434 :M -.197(chain )A 251 434 :M -.064(graph )A 284 434 :M -.164(Markov )A 327 434 :M -.085(property,)A f5_12 sf -.1(Scandinvaian )A 440 434 :M -.332(Journal )A 481 434 :M .666(of)A 59 452 :M .14(Statistics,)A f0_12 sf .091 .009( )J f2_12 sf .2(17)A f0_12 sf .859 .086(, 333-353.)J 77 470 :M .21 .021(Geiger, )J 118 470 :M 1.028 .103(D., )J 139 470 :M -.109(and )A 161 470 :M .219 .022(Pearl, )J 194 470 :M 1.795 .179(J., )J 212 470 :M .667 .067(1988, )J 245 470 :M -.236(Logical )A 285 470 :M -.109(and )A 307 470 :M -.241(Algorithmic )A 368 470 :M -.097(properties )A 420 470 :M (of )S 436 470 :M -.167(Conditional)A 59 488 :M -.048(Independence. )A 135 488 :M -.256(Technical )A 187 488 :M -.11(Report )A 226 488 :M .668 .067(R-97, )J 261 488 :M -.148(Cognitive )A 314 488 :M (Systems )S 361 488 :M -.028(Laboratory, )A 424 488 :M -.065(University )A 481 488 :M (of)S 59 506 :M -.042(California, Los Angeles.)A 77 524 :M .83 .083(C. )J 95 524 :M .375 .038(Glymour, )J 148 524 :M .83 .083(R. )J 166 524 :M .184 .018(Scheines, )J 218 524 :M 1.107 .111(P. )J 235 524 :M .375 .037(Spirtes, )J 279 524 :M -.109(and )A 303 524 :M 1.114 .111(K. )J 323 524 :M -.131(Kelly )A 356 524 :M (\(1987\) )S 395 524 :M -.149(Discovering )A 459 524 :M -.131(Causal)A 59 542 :M -.116(Structure: Artificial Intelligence, Philosophy and Statistical Modeling, Academic Press, )A 473 542 :M (San)S 59 560 :M (Diego, CA.)S 77 578 :M .83 .083(C. )J 93 578 :M .375 .038(Glymour, )J 144 578 :M 1.107 .111(P. )J 159 578 :M .375 .037(Spirtes, )J 200 578 :M -.109(and )A 221 578 :M .83 .083(R. )J 237 578 :M -.081(Scheines )A 283 578 :M .596 .06(\(1994\), )J 323 578 :M (In )S 337 578 :M -.197(Place )A 367 578 :M (of )S 382 578 :M -.032(Regression )A 440 578 :M -.109(\(in )A 458 578 :M -.164(Patrick)A 59 596 :M (Suppes: )S 103 596 :M -.198(Scientific )A 153 596 :M .307 .031(Philosopher, )J 220 596 :M -.083(Paul )A 248 596 :M (Humphreys )S 311 596 :M (\(editor\), )S 357 596 :M (Vol. )S 385 596 :M .833 .083(1, )J 402 596 :M (Kluwer )S 445 596 :M -.377(Academic)A 59 614 :M .143 .014(Publishers, Dordrecht, Holland.)J 77 632 :M .173 .017(Goldberger, )J 139 632 :M 1.028 .103(A., )J 159 632 :M -.045(Duncan, )A 204 632 :M .281 .028(O. )J 221 632 :M 1.189 .119(\(eds.\), )J 258 632 :M .667 .067(1973, )J 291 632 :M -.131(Structural )A 342 632 :M -.165(Equation )A 389 632 :M -.166(Models )A 429 632 :M -.167(in )A 443 632 :M -.22(the )A 462 632 :M -.198(Social)A 59 650 :M .056 .006(Sciences \(Seminar Press, New York\).)J 77 668 :M -.034(Haavelmo, )A 137 668 :M 1.284 .128(T., )J 160 668 :M .667 .067(1943, )J 196 668 :M -.219(The )A 222 668 :M -.211(statistical )A 273 668 :M -.194(implications )A 338 668 :M (of )S 356 668 :M -.326(a )A 369 668 :M -.055(system )A 410 668 :M (of )S 429 668 :M -.12(simultaneous)A 59 686 :M .252 .025(equations, )J f5_12 sf .057(Econometrica)A f0_12 sf .051 .005(, )J f2_12 sf .061(11)A f0_12 sf .197 .02(, 1-12.)J endp %%Page: 32 32 %%BeginPageSetup initializepage (peter; page: 32 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (32)S gR gS 0 0 552 730 rC 77 56 :M f0_12 sf (Kiiveri, )S 119 56 :M 1.114 .111(H. )J 137 56 :M -.109(and )A 159 56 :M .423 .042(Speed, )J 197 56 :M 1.483 .148(T.,1982, )J 245 56 :M -.131(Structural )A 296 56 :M -.082(analysis )A 339 56 :M (of )S 354 56 :M -.248(multivariate )A 414 56 :M -.264(data: )A 441 56 :M -.663(A )A 455 56 :M .06(review,)A 59 74 :M f5_12 sf .355 .035(Sociological Methodology)J f0_12 sf .205 .021(, Leinhardt, S. \(ed.\). Jossey-Bass, San Francisco.)J 77 92 :M .568 .057(Kiiveri, H., Speed, T., and )J 212 92 :M (Carlin, )S 249 92 :M 1.795 .179(J., )J 266 92 :M .667 .067(1984, )J 298 92 :M -.108(Recursive )A 349 92 :M -.163(causal )A 382 92 :M .199 .02(models, )J 424 92 :M f5_12 sf -.332(Journal )A 463 92 :M .555 .055(of )J 477 92 :M -.33(the)A 59 110 :M -.047(Australian Mathematical Society)A f0_12 sf -.027(, )A f2_12 sf -.055(36)A f0_12 sf -.048(, 30-52.)A 77 128 :M .598 .06(Koster, )J 118 128 :M 1.795 .179(J., )J 136 128 :M (\(1995\) )S 173 128 :M -.164(Markov )A 215 128 :M -.064(Properties )A 268 128 :M (of )S 283 128 :M -.049(Non-Recursive )A 360 128 :M -.11(Causal )A 397 128 :M (Models, )S 442 128 :M -.165(Annals )A 481 128 :M (of)S 59 146 :M (Statistics, November 1995.)S 77 164 :M -.018(Lee, S., and Hershberger, S. \(1990\). A simple rule for generating equivalent models )A 482 164 :M -.334(in)A 59 182 :M -.081(covariance structure modeling. Multivariate Behavioral Research, 25, 313-334.)A 77 200 :M -.097(Lauritzen, )A 129 200 :M 1.792 .179(S., )J 148 200 :M (Dawid, )S 187 200 :M 1.028 .103(A., )J 207 200 :M .21 .021(Larsen, )J 247 200 :M 1.535 .154(B., )J 267 200 :M -.092(Leimer, )A 309 200 :M 1.854 .185(H.,1990, )J 359 200 :M -.135(Independence )A 429 200 :M -.097(properties )A 481 200 :M (of)S 59 218 :M .412 .041(directed Markov fields, )J f5_12 sf .109(Networks)A f0_12 sf .097 .01(, )J f2_12 sf .116(20)A f0_12 sf .498 .05(, 491-505.)J 77 236 :M -.087(Meek, C. \(1995\) Causal inference and causal explanation with )A 375 236 :M -.065(background )A 435 236 :M .039(knowledge,)A 59 254 :M f5_12 sf -.18(Proceedings )A 121 254 :M .555 .055(of )J 135 254 :M -.22(the )A 153 254 :M -.122(Eleventh )A 198 254 :M -.131(Conference )A 256 254 :M (on )S 272 254 :M -.21(Uncertainty )A 331 254 :M -.167(in )A 344 254 :M -.233(Artificial )A 389 254 :M -.123(Intelligence)A f0_12 sf -.156(, )A 452 254 :M -.143(Philippe)A 59 272 :M -.045(Besnard )A 102 272 :M -.109(and )A 123 272 :M -.131(Steve )A 153 272 :M .229 .023(Hanks )J 188 272 :M 1.186 .119(\(Eds.\), )J 226 272 :M -.164(Morgan )A 268 272 :M -.08(Kaufmann )A 323 272 :M .48 .048(Publishers, )J 382 272 :M 1.12 .112(Inc., )J 410 272 :M (San )S 433 272 :M -.109(Mateo, )A 471 272 :M .167(CA,)A 59 290 :M .955 .096(pp. 403-410.)J 77 308 :M .219 .022(Pearl, )J 110 308 :M 1.795 .179(J., )J 128 308 :M (\(1986\) )S 165 308 :M .781 .078(Fusion, )J 207 308 :M -.026(propagation, )A 272 308 :M -.109(and )A 294 308 :M -.089(structuring )A 350 308 :M -.167(in )A 364 308 :M -.219(belief )A 396 308 :M .535 .053(networks, )J 450 308 :M f5_12 sf -.259(Artificial)A 59 326 :M .057(Intelligence)A f0_12 sf ( )S f2_12 sf .072(29)A f0_12 sf .286 .029(, 241-88.)J 77 344 :M .219 .022(Pearl, )J 109 344 :M 1.795 .179(J., )J 126 344 :M .596 .06(\(1988\). )J 166 344 :M f5_12 sf -.282(Probabilistic )A 229 344 :M -.073(Reasoning )A 283 344 :M -.167(in )A 297 344 :M -.211(Intelligent )A 349 344 :M .372(Systems)A f0_12 sf .374 .037(, )J 398 344 :M -.14(\(Morgan )A 444 344 :M -.14(Kaufman:)A 59 362 :M -.022(San Mateo, CA\).)A 77 380 :M .219 .022(Pearl, )J 111 380 :M 1.111 .111(J. )J 126 380 :M -.109(and )A 149 380 :M -.107(Verma, )A 190 380 :M .558 .056(T. )J 207 380 :M .596 .06(\(1991\). )J 249 380 :M -.663(A )A 263 380 :M -.109(theory )A 299 380 :M (of )S 315 380 :M -.121(inferred )A 358 380 :M -.031(causation, )A 412 380 :M -.167(in )A 427 380 :M f5_12 sf -.232(Principles )A 481 380 :M .666(of)A 59 398 :M .529 .053(Knowledge )J 121 398 :M -.189(Representation )A 198 398 :M -.333(and )A 222 398 :M -.165(Reasoning: )A 283 398 :M -.18(Proceedings )A 349 398 :M .555 .055(of )J 367 398 :M -.22(the )A 389 398 :M .217 .022(Second )J 432 398 :M -.36(International)A 59 416 :M -.051(Conference)A f0_12 sf -.052( \(Morgan Kaufmann, San Mateo, CA\).)A 77 434 :M .623 .062(Pearl, J. \(1997\). Graphs, )J 202 434 :M -.147(Causality )A 250 434 :M -.109(and )A 271 434 :M -.131(Structural )A 321 434 :M -.165(Equation )A 367 434 :M (Models, )S 410 434 :M -.256(Technical )A 459 434 :M -.131(Report)A 59 452 :M -.039(R-253, Cognitive Science Laboratory, UCLA.)A 77 470 :M (Raftery, )S 122 470 :M .281 .028(A. )J 140 470 :M .596 .06(\(1995\), )J 182 470 :M -.094(Baysian )A 226 470 :M -.265(Model )A 262 470 :M -.183(Selection )A 311 470 :M -.167(in )A 326 470 :M -.165(social )A 359 470 :M (research, )S 408 470 :M -.167(in )A 424 470 :M (Marsden, )S 476 470 :M .337(ed.)A 59 488 :M -.092(Sociological Methodology, Blackwells, Cambridge MA.)A 77 506 :M .166 .017(Richardson, )J 142 506 :M .558 .056(T. )J 160 506 :M (\(1994\) )S 199 506 :M -.064(Properties )A 254 506 :M (of )S 271 506 :M -.221(Cyclic )A 308 506 :M -.145(Graphical )A 361 506 :M (Models, )S 407 506 :M .478 .048(Master's )J 456 506 :M .113(Thesis,)A 59 524 :M -.101(一本道无码.)A 77 542 :M -.072(Richardson, T. \(1996a\). A Polynomial Algorithm for Deciding )A 378 542 :M -.209(Equivalence )A 439 542 :M -.167(in )A 452 542 :M -.329(Directed)A 59 560 :M -.107(Cyclic Graphical Models. Technical Report PHIL-63, Department of )A 387 560 :M .473 .047(Philosophy, )J 449 560 :M -.187(Carnegie)A 59 578 :M -.077(Mellon University.)A 77 596 :M .166 .017(Richardson, )J 141 596 :M .558 .056(T. )J 158 596 :M (\(1996b\) )S 203 596 :M -.663(A )A 218 596 :M -.146(Discovery )A 273 596 :M -.221(Algorithm )A 328 596 :M (for )S 349 596 :M -.293(Dirtected )A 398 596 :M -.221(Cyclic )A 435 596 :M .794 .079(Graphs. )J 481 596 :M (In)S 59 614 :M f5_12 sf -.145(Uncertainty in Artificial Intelligence: Proceedings of the Twelfth )A 365 614 :M -.131(Conference )A 423 614 :M f0_12 sf .71 .071(\(F.Jensen )J 474 614 :M -.163(and)A 59 632 :M .322 .032(E.Horvitz, eds.\), 462-469, Morgan Kaufmann, San Francisco.)J 77 650 :M -.066(Richardson, T. \(1996c\) Feedback Models: Interpretation and Discovery. )A 423 650 :M 1.111 .111(Ph.D. )J 456 650 :M .113(Thesis,)A 59 668 :M -.041(Dept. of Philosophy, Carnegie-Mellon University.)A endp %%Page: 33 33 %%BeginPageSetup initializepage (peter; page: 33 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (33)S gR gS 0 0 552 730 rC 77 56 :M f0_12 sf .184 .018(Scheines, )J 127 56 :M .83 .083(R. )J 143 56 :M (\(1994\) )S 180 56 :M (Causation, )S 236 56 :M -.063(Indistinguishability, )A 336 56 :M -.109(and )A 358 56 :M .325 .032(Regression, )J 420 56 :M -.167(in )A 434 56 :M -.046(Sofstat )A 472 56 :M .502('93:)A 59 74 :M -.164(Advances )A 109 74 :M -.167(in )A 122 74 :M -.199(Statistial )A 166 74 :M -.038(Software )A 213 74 :M .833 .083(4, )J 227 74 :M (Frank )S 260 74 :M -.124(Faulbaum )A 312 74 :M (\(editor\), )S 356 74 :M (Gustav )S 395 74 :M -.045(Fischer )A 435 74 :M -.092(Verlag, )A 475 74 :M .5(pp.)A 59 92 :M .201(89-98.)A 77 110 :M .375 .037(Spirtes, )J 119 110 :M 1.107 .111(P. )J 135 110 :M -.109(and )A 158 110 :M .375 .038(Glymour, )J 211 110 :M 1.535 .154(C., )J 233 110 :M .667 .067(1990, )J 267 110 :M -.11(Causal )A 305 110 :M -.109(Structure )A 354 110 :M -.199(Among )A 395 110 :M -.163(Measured )A 447 110 :M -.246(Variables)A 59 128 :M -.051(Preserved with Unmeasured )A 197 128 :M -.097(Variables. )A 249 128 :M -.256(Technical )A 298 128 :M -.11(Report )A 334 128 :M (一本道无码-LCL-90-5, )S 421 128 :M -.13(Laboratory )A 477 128 :M (for)S 59 146 :M -.099(Computational Linguistics, 一本道无码.)A 77 164 :M .375 .037(Spirtes, )J 118 164 :M 1.792 .179(P., )J 137 164 :M -.109(and )A 158 164 :M .375 .038(Glymour, )J 209 164 :M .561 .056(C.\(1991\) )J 257 164 :M -.331(An )A 275 164 :M -.184(algorithm )A 325 164 :M (for )S 344 164 :M -.08(fast )A 366 164 :M -.121(recovery )A 412 164 :M (of )S 427 164 :M (sparse )S 462 164 :M -.196(causal)A 59 182 :M .322 .032(graphs, )J f5_12 sf .348 .035(Social Science Computer Review, )J f2_12 sf .092(9)A f0_12 sf .332 .033(, 62-72.)J 77 200 :M .375 .037(Spirtes, )J 118 200 :M 1.792 .179(P., )J 137 200 :M -.107(Verma, )A 176 200 :M .558 .056(T. )J 192 200 :M (\(1992\) )S 229 200 :M -.209(Equivalence )A 291 200 :M (of )S 306 200 :M -.11(Causal )A 343 200 :M -.166(Models )A 383 200 :M -.083(with )A 409 200 :M -.275(Latent )A 443 200 :M -.108(Variables.)A 59 218 :M -.076(Technical Report 一本道无码-PHIL-33, Department of Philosophy, Carnegie )A 400 218 :M -.277(Mellon )A 437 218 :M .035(University,)A 59 236 :M .155 .016(October, 1992.)J 77 254 :M .576 .058(Spirtes, P., Glymour, C., and Scheines, )J 274 254 :M .561 .056(R.\(1993\) )J 322 254 :M f5_12 sf -.134(Causation, )A 377 254 :M -.15(Prediction, )A 433 254 :M -.333(and )A 454 254 :M .104(Search)A f2_12 sf (,)S 59 272 :M f0_12 sf -.061(\(Springer-Verlag Lecture Notes in Statistics 81, New York\).)A 77 290 :M .375 .037(Spirtes, )J 118 290 :M 1.792 .179(P., )J 137 290 :M (\(1995\) )S 173 290 :M -.288(Directed )A 216 290 :M -.221(Cyclic )A 250 290 :M -.145(Graphical )A 300 290 :M -.141(Representation )A 374 290 :M (of )S 388 290 :M -.122(Feedback )A 438 290 :M (Models, )S 482 290 :M -.334(in)A 59 308 :M -.056(Proceedings)A f5_12 sf ( )S 121 308 :M .555 .055(of )J 135 308 :M -.22(the )A 154 308 :M -.122(Eleventh )A 200 308 :M -.131(Conference )A 259 308 :M (on )S 276 308 :M -.21(Uncertainty )A 336 308 :M -.167(in )A 350 308 :M -.233(Artificial )A 396 308 :M -.123(Intelligence)A f0_12 sf -.156(, )A 459 308 :M .518 .052(ed. )J 479 308 :M (by)S 59 326 :M -.007(Philippe Besnard and Steve Hanks, Morgan Kaufmann Publishers, Inc., San Mateo.)A 77 344 :M .631 .063(Spirtes, P., Richardson, T., Meek, C., Scheines, R., and Glymour, C., \(1996\). Using)J 59 362 :M -.164(D-separation )A 125 362 :M -.167(to )A 140 362 :M -.257(Calculate )A 189 362 :M -.162(Zero )A 217 362 :M -.188(Partial )A 254 362 :M -.109(Correlations )A 319 362 :M -.167(in )A 335 362 :M -.218(Linear )A 372 362 :M -.166(Models )A 414 362 :M -.083(with )A 442 362 :M -.182(Correlated)A 59 380 :M -.044(Errors, Technical Report 一本道无码-72-Phil.)A 77 398 :M .375 .037(Spirtes, )J 122 398 :M 1.792 .179(P., )J 146 398 :M -.109(and )A 172 398 :M .166 .017(Richardson, )J 239 398 :M .558 .056(T. )J 259 398 :M .596 .06(\(1996\), )J 304 398 :M -.663(A )A 321 398 :M -.133(Polynomial )A 384 398 :M -.331(Time )A 417 398 :M -.221(Algorithm )A 474 398 :M .168(For)A 59 416 :M -.24(Determining )A 122 416 :M -.33(DAG )A 152 416 :M -.209(Equivalence )A 214 416 :M -.167(in )A 228 416 :M -.22(the )A 247 416 :M -.079(Presence )A 294 416 :M (of )S 309 416 :M -.275(Latent )A 344 416 :M -.219(Variables )A 394 416 :M -.109(and )A 417 416 :M -.183(Selection )A 466 416 :M .167(Bias,)A 59 434 :M -.117(Proceedings of the 6th International Workshop on Artificial Intelligence and Statistics.)A 77 452 :M -.046(Stelzl, )A 112 452 :M .839 .084(I. )J 125 452 :M (\(1986\) )S 162 452 :M -.083(Changing )A 213 452 :M -.163(causal )A 247 452 :M -.146(relations )A 292 452 :M -.055(withou )A 330 452 :M -.123(changing )A 378 452 :M -.22(the )A 397 452 :M -.249(fit: )A 415 452 :M -.083(Some )A 448 452 :M -.064(rules )A 477 452 :M (for)S 59 470 :M -.083(generating equivalent LISREL-models. Multivariate Behavioral Research, 21, 309-331.)A 77 488 :M -.097(Whittaker, )A 131 488 :M 1.852 .185(J.,1990, )J 176 488 :M f5_12 sf -.369(Graphical )A 226 488 :M -.053(Models )A 265 488 :M -.167(in )A 278 488 :M -.046(Applied )A 319 488 :M -.332(Multivariate )A 380 488 :M -.133(Statistics )A 427 488 :M f0_12 sf -.045(\(Wiley, )A 468 488 :M .174(New)A 59 506 :M .07(York\).)A 77 524 :M .491 .049(Wright, S. \(1934\). )J 171 524 :M -.219(The )A 193 524 :M -.166(method )A 232 524 :M (of )S 246 524 :M -.165(path )A 270 524 :M -.074(coefficients, )A 332 524 :M f5_12 sf -.055(Annals )A 369 524 :M .555 .055(of )J 383 524 :M -.387(Mathematical )A 449 524 :M -.148(Statistics)A 59 542 :M f2_12 sf .328(5)A f0_12 sf 1.41 .141(, 161-215.)J endp %%Page: 34 34 %%BeginPageSetup initializepage (peter; page: 34 of 34)setjob %%EndPageSetup gS 0 0 552 730 rC 479 695 12 30 rC 479 722 :M f0_12 sf (34)S gR gS 0 0 552 730 rC 257 56 :M f2_12 sf .57(Appendix)A 77 74 :M f0_12 sf -.12(d-separation equivalence holds )A 226 74 :M -.139(between )A 269 74 :M (two )S 291 74 :M -.165(path )A 315 74 :M -.123(diagrams )A 362 74 :M -.095(without )A 402 74 :M -.196(correlated )A 452 74 :M (errors )S 484 74 :M -.327(if)A 59 92 :M -.109(and )A 82 92 :M -.083(only )A 109 92 :M -.164(if )A 122 92 :M -.165(they )A 148 92 :M -.163(have )A 176 92 :M -.22(the )A 197 92 :M -.163(same )A 228 92 :M -.208(adjacencies )A 288 92 :M -.109(and )A 312 92 :M -.22(the )A 333 92 :M -.163(same )A 364 92 :M -.099(unshielded )A 422 92 :M -.031(colliders. )A 473 92 :M -.328(The)A 59 110 :M -.073(following )A 109 110 :M -.164(examples )A 157 110 :M .479 .048(show )J 187 110 :M -.249(that )A 209 110 :M -.084(this )A 231 110 :M -.166(simple )A 267 110 :M -.163(rule )A 290 110 :M (does )S 317 110 :M -.111(not )A 337 110 :M .246 .025(work )J 367 110 :M (when )S 398 110 :M -.196(there )A 426 110 :M -.215(are )A 445 110 :M -.218(correlated)A 59 128 :M -.029(errors. Consider the following path diagrams.)A 95 218 :M f2_12 sf 1.052 .105( Path Diagram G)J 239 218 :M 1.094 .109( Path Diagram G\325)J 247 236 :M 3.34 .334(Figure 15)J 59 260 :M f0_12 sf -.084(Both )A 86 260 :M (of )S 100 260 :M -.131(these )A 128 260 :M -.165(path )A 152 260 :M -.123(diagrams )A 199 260 :M -.189(contain )A 237 260 :M -.22(the )A 255 260 :M -.163(same )A 283 260 :M -.208(adjacencies )A 340 260 :M -.109(and )A 362 260 :M -.163(same )A 391 260 :M -.099(unshielded )A 447 260 :M -.035(colliders.)A 59 278 :M .571 .057(However, )J 112 278 :M -.167(in )A 126 278 :M -.083(Path )A 152 278 :M -.282(Diagram )A 197 278 :M 1.033 .103(G\325, )J 219 278 :M .306 .031(X )J 233 278 :M (is )S 246 278 :M -.117(d-separated )A 305 278 :M -.08(from )A 333 278 :M -.326(W )A 349 278 :M -.132(given )A 380 278 :M -.22(the )A 399 278 :M -.199(empty )A 433 278 :M -.165(set; )A 455 278 :M -.167(in )A 470 278 :M -.11(Path)A 59 296 :M -.086(Diagram G it is not.)A 227 185 16 23 rC 227 200 :M (W)S gR gS 198 185 16 23 rC 198 200 :M f0_12 sf (Z)S gR gS 168 185 16 23 rC 168 200 :M f0_12 sf (Y)S gR gS 138 185 16 23 rC 138 200 :M f0_12 sf (X)S gR gS 0 0 552 730 rC 149 196.75 -.75 .75 160.75 196 .75 149 196 @a np 158 194 :M 158 198 :L 166 196 :L 158 194 :L .75 lw eofill -.75 -.75 158.75 198.75 .75 .75 158 194 @b -.75 -.75 158.75 198.75 .75 .75 166 196 @b 158 194.75 -.75 .75 166.75 196 .75 158 194 @a 178 196.75 -.75 .75 189.75 196 .75 178 196 @a np 187 194 :M 187 198 :L 195 196 :L 187 194 :L eofill -.75 -.75 187.75 198.75 .75 .75 187 194 @b -.75 -.75 187.75 198.75 .75 .75 195 196 @b 187 194.75 -.75 .75 195.75 196 .75 187 194 @a 208 196.75 -.75 .75 219.75 196 .75 208 196 @a np 217 194 :M 217 198 :L 225 196 :L 217 194 :L eofill -.75 -.75 217.75 198.75 .75 .75 217 194 @b -.75 -.75 217.75 198.75 .75 .75 225 196 @b 217 194.75 -.75 .75 225.75 196 .75 217 194 @a -180 -90 80 36 212.5 180.5 @n -.75 -.75 171.75 182.75 .75 .75 171 180 @b np 174 179 :M 170 178 :L 170 186 :L 174 179 :L eofill 170 178.75 -.75 .75 174.75 179 .75 170 178 @a -.75 -.75 170.75 186.75 .75 .75 170 178 @b -.75 -.75 170.75 186.75 .75 .75 174 179 @b -.75 -.75 233.75 182.75 .75 .75 233 181 @b np 235 180 :M 230 180 :L 234 188 :L 235 180 :L eofill 230 180.75 -.75 .75 235.75 180 .75 230 180 @a 230 180.75 -.75 .75 234.75 188 .75 230 180 @a -.75 -.75 234.75 188.75 .75 .75 235 180 @b -90 0 50 40 208.5 182.5 @n 379 181 16 23 rC 379 196 :M f0_12 sf (W)S gR gS 350 181 16 23 rC 350 196 :M f0_12 sf (Z)S gR gS 320 181 16 23 rC 320 196 :M f0_12 sf (Y)S gR gS 290 181 16 23 rC 290 196 :M f0_12 sf (X)S gR gS 0 0 552 730 rC 301 192.75 -.75 .75 312.75 192 .75 301 192 @a np 310 190 :M 310 194 :L 318 192 :L 310 190 :L .75 lw eofill -.75 -.75 310.75 194.75 .75 .75 310 190 @b -.75 -.75 310.75 194.75 .75 .75 318 192 @b 310 190.75 -.75 .75 318.75 192 .75 310 190 @a 330 192.75 -.75 .75 341.75 192 .75 330 192 @a np 339 190 :M 339 194 :L 347 192 :L 339 190 :L eofill -.75 -.75 339.75 194.75 .75 .75 339 190 @b -.75 -.75 339.75 194.75 .75 .75 347 192 @b 339 190.75 -.75 .75 347.75 192 .75 339 190 @a -180 -90 80 36 364.5 176.5 @n -.75 -.75 323.75 178.75 .75 .75 323 176 @b np 326 175 :M 322 174 :L 322 182 :L 326 175 :L eofill 322 174.75 -.75 .75 326.75 175 .75 322 174 @a -.75 -.75 322.75 182.75 .75 .75 322 174 @b -.75 -.75 322.75 182.75 .75 .75 326 175 @b -.75 -.75 385.75 178.75 .75 .75 385 177 @b np 387 176 :M 382 176 :L 386 184 :L 387 176 :L eofill 382 176.75 -.75 .75 387.75 176 .75 382 176 @a 382 176.75 -.75 .75 386.75 184 .75 382 176 @a -.75 -.75 386.75 184.75 .75 .75 387 176 @b -90 0 50 40 360.5 178.5 @n 364 192.75 -.75 .75 372.75 192 .75 364 192 @a np 370 190 :M 370 194 :L 378 192 :L 370 190 :L eofill -.75 -.75 370.75 194.75 .75 .75 370 190 @b -.75 -.75 370.75 194.75 .75 .75 378 192 @b 370 190.75 -.75 .75 378.75 192 .75 370 190 @a np 366 194 :M 366 190 :L 358 192 :L 366 194 :L eofill -.75 -.75 366.75 194.75 .75 .75 366 190 @b -.75 -.75 358.75 192.75 .75 .75 366 190 @b 358 192.75 -.75 .75 366.75 194 .75 358 192 @a endp %%Trailer end %%EOF