%!PS-Adobe-3.0 %%Title: (Microsoft Word - mbr.20) %%Creator: (Microsoft Word: LaserWriter 8 8.3.4) %%CreationDate: (1:57 AM Friday, January 17, 1997) %%For: (peter) %%Pages: 53 %%DocumentFonts: Times-Bold Symbol Times-Roman Times-Italic TimesNewRomanPSMT TimesNewRomanPS-BoldMT Courier Courier-Bold %%DocumentNeededFonts: Times-Bold Symbol Times-Roman Times-Italic TimesNewRomanPSMT TimesNewRomanPS-BoldMT Courier Courier-Bold %%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 - mbr.20)def /Creator(Microsoft Word: LaserWriter 8 8.3.4)def /CreationDate(1:57 AM Friday, January 17, 1997)def /For(peter)def /Pages 53 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 220 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. 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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. 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%%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-Bold %%IncludeFont: Symbol %%IncludeFont: Times-Roman %%IncludeFont: Times-Italic %%IncludeFont: TimesNewRomanPSMT %%IncludeFont: TimesNewRomanPS-BoldMT %%IncludeFont: Courier %%IncludeFont: Courier-Bold /f0_1/Times-Bold :mre /f0_12 f0_1 12 scf /f0_10 f0_1 10 scf /f0_7 f0_1 7 scf /f1_1/Symbol :bsr 240/apple pd :esr /f1_14 f1_1 14 scf /f1_12 f1_1 12 scf /f1_10 f1_1 10 scf /f1_7 f1_1 7 scf /f2_1 f1_1 def /f2_12 f2_1 12 scf /f2_7 f2_1 7 scf /f3_1/Times-Roman :mre /f3_12 f3_1 12 scf /f3_11 f3_1 11 scf /f3_10 f3_1 10 scf /f3_9 f3_1 9 scf /f3_7 f3_1 7 scf /f3_6 f3_1 6 scf /f4_1/Times-Italic :mre /f4_14 f4_1 14 scf /f4_12 f4_1 12 scf /f5_1 f1_1 :mi /f5_12 f5_1 12 scf /f6_1 f3_1 :v def /f7_1 f3_1 1.087 scf /f7_12 f7_1 12 scf /f7_7 f7_1 7 scf /f8_1/TimesNewRomanPSMT :mre /f8_12 f8_1 12 scf /f8_10 f8_1 10 scf /f8_8 f8_1 8 scf /f9_1/TimesNewRomanPS-BoldMT :mre /f9_12 f9_1 12 scf /f9_8 f9_1 8 scf /f10_1/Courier :mre /f10_12 f10_1 12 scf /f11_1/Courier-Bold :mre /f11_12 f11_1 12 scf /Courier findfont[10 0 0 -10 0 0]:mf setfont %%EndSetup %%Page: 1 1 %%BeginPageSetup initializepage (peter; page: 1 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 227 54 :M f0_12 sf (The TETRAD Project:)S 150 70 :M (Constraint Based )S 242 70 :M (Aids to Causal Model Specification)S 158 102 :M f3_12 sf (Richard Scheines, Peter Spirtes, Clark Glymour,)S 171 118 :M (Christopher Meek and Thomas Richardson)S 151 134 :M (Department of Philosophy, 一本道无码)S 418 129 :M f3_7 sf (1)S 95 166 :M f3_12 sf 1.761 .176(The statistical community has brought logical rigor and mathematical)J 95 182 :M .271 .027(precision to the problem of using data to make inferences about a model\325s)J 95 198 :M 1.301 .13(parameter values. The TETRAD project, and related work in computer)J 95 214 :M 1.512 .151(science and statistics, aims to apply those standards to the problem of)J 95 230 :M .11 .011(using data and background knowledge to make inferences about a model\325s)J 95 246 :M 2.483 .248(specification. We begin by drawing the analogy between parameter)J 95 262 :M 1.786 .179(estimation and model specification search. We then describe how the)J 95 278 :M .334 .033(specification of a structural equation model entails familiar constraints on)J 95 294 :M 1.974 .197(the covariance matrix for all admissible values of its parameters; we)J 95 310 :M .992 .099(survey results on the equivalence of structural equation models, and we)J 95 326 :M 1.116 .112(discuss search strategies for model specification. We end by presenting)J 95 342 :M (several algorithms that are implemented in the TETRAD II program.)S 240 386 :M f4_12 sf (1)S 246 386 :M (.)S 249 386 :M 4 0 rm ( )S 256 386 :M (Motivation)S 81 414 :M f3_12 sf .235 .024(A principal aim of many sciences is to model causal systems well enough to provide)J 59 430 :M .912 .091(sound insight into their structures and mechanisms, and to provide reliable predictions)J 59 446 :M .154 .015(about the effects of policy interventions. In order to succeed in that aim, a model must be)J 59 462 :M 1.148 .115(specified at least approximately correctly. Unfortunately, this is not an easy problem.)J 59 478 :M 1.947 .195(When some of the causes are unknown and/or unobserved, there are an infinity of)J 59 494 :M .157 .016(possible causal models and it is not obvious how to go about constructing plausible ones.)J 59 510 :M .306 .031(To make matters worse, there may be many models that are compatible with background)J 59 526 :M (knowledge and the data, but which lead to entirely different causal conclusions.)S 81 542 :M 1.48 .148(The process of statistical modeling is typically divided into at least two distinct)J 59 558 :M .235 .024(phases: a model specification phase in which a model \(with free parameters\) is specified,)J 59 574 :M .743 .074(and a parameter estimation and statistical testing phase in which the free parameters of)J 59 590 :M .065 .007(the specified model are estimated and various hypotheses are put to a statistical test. Both)J 59 614 :M ( )S 59 611.48 -.48 .48 203.48 611 .48 59 611 @a 81 623 :M f3_6 sf (1)S 84 627 :M f3_10 sf .529 .053( Research for this paper was supported by the National Science Foundation through grant 9102169)J 59 639 :M 1.11 .111(and the Navy Personnel Research and Development Center and the Office of Naval Research through)J 59 651 :M 1.199 .12(grants N0014-93-0568 #N00014-95-1-0684. Correspondence should be directed to Richard Scheines,)J 59 663 :M 4.802 .48(Department of Philosophy, 一本道无码, Pittsburgh, PA 15213. E-mail:)J 59 675 :M (RS2L@andrew.cmu.edu.)S endp %%Page: 2 2 %%BeginPageSetup initializepage (peter; page: 2 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 261 709 28 16 rC 283 722 :M f3_12 sf (2)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf 1.779 .178(model specification and parameter estimation can fruitfully be thought of as search)J 59 70 :M .357 .036(problems. Parameter estimation can be thought of as a search through a large space for a)J 59 86 :M .951 .095(particular vector of values that satisfies a given set of constraints. For example, it is a)J 59 102 :M .641 .064(search problem to find a vector of parameter values that maximize the likelihood of the)J 59 118 :M 1.188 .119(data given the model. Statisticians have fruitfully investigated a number of parameter)J 59 134 :M .839 .084(estimation problems under a variety of background assumptions, e.g., normal and non-)J 59 150 :M .678 .068(normal distributional assumptions, recursive or non-recursive models, etc. Even though)J 59 166 :M .985 .099(model specification affects parameter estimation \(Kaplan, 1988\) and predictions about)J 59 182 :M .325 .033(the effects of adopting different policies \(Strotz & Wold, 1960; Spirtes, et al., 1993\), the)J 59 198 :M 1.963 .196(great bulk of theoretical attention on statistical causal models has been devoted to)J 59 214 :M .325 .033(estimating their parameters, to developing statistics for testing individual parameters and)J 59 230 :M 2.715 .272(overall model fit \(Bollen and Long, 1993\), and to techniques for making minor)J 59 246 :M .206 .021(respecifications of models that fit data poorly \(Kaplan, 1990; Saris, et al., 1987; Sorbom,)J 59 262 :M (1989; Bentler, 1986; J)S 166 262 :M (\232reskog and S)S 234 262 :M (\232rbom, 1993\).)S 81 278 :M .356 .036(We believe the problem of model specification is in many respects analogous to the)J 59 294 :M 2.429 .243(problem of parameter estimation, and that the same kind of rigor brought to the)J 59 310 :M .981 .098(development and evaluation of algorithms for parameter estimation can and should be)J 59 326 :M .205 .021(applied to algorithms for model specification. In a model specification problem a class of)J 59 342 :M .809 .081(models is searched, and various models assessed. Our concern is with structuring such)J 59 358 :M .127 .013(searches so as to have guarantees of reliability analogous to those available for parameter)J 59 374 :M (estimators.)S 81 390 :M .56 .056(This is )J 118 390 :M f4_12 sf .207(not)A f3_12 sf .495 .049( to say that it is our intention to )J 293 390 :M f4_12 sf (replace)S 329 390 :M f3_12 sf .366 .037( well-founded theoretical sources)J 59 406 :M .448 .045(of model specification with automatic procedures. Where theory and domain knowledge)J 59 422 :M .11 .011(provide justified constraints on model specification, those constraints should be used, and)J 59 438 :M 1.217 .122(one of the important desiderata for model search procedures is that they make use of)J 59 454 :M .909 .091(whatever domain knowledge is available. But rarely, if ever, is social scientific theory)J 59 470 :M .259 .026(and background knowledge sufficiently well confirmed and suff)J 370 470 :M .292 .029(iciently strong to entail a)J 59 486 :M .439 .044(unique model specification. There are typically many theoretically plausible alternatives)J 59 502 :M .871 .087(to a given model, some of which support wholly different causal conclusions and thus)J 59 518 :M (lead to different conclusions about which policies should be adopted.)S 81 534 :M .794 .079(Just as in the case of parameter estimation, the results we will present here do not)J 59 550 :M .148 .015(free one from having to make assumptions; instead, they make rigorous and explicit what)J 59 566 :M .46 .046(can and cannot be learned about the world if one is willing to make certain assumptions)J 59 582 :M 2.14 .214(and not others. If, for example, one is willing to assume that causal relations are)J 59 598 :M .366 .037(approximately linear and additive, that there is no feedback, that error terms are i.i.d and)J 59 614 :M .41 .041(uncorrelated, and that the Causal Independence and Faithfulness assumptions \(explained)J 59 630 :M .805 .081(in detail in sections )J 160 630 :M .806 .081(3.1 and 3.2\) are satisfied, then quite a lot can be learned about the)J 59 646 :M 3.241 .324(causal structure underlying one\325s data. If one is only willing to make weaker)J 59 662 :M .536 .054(assumptions, then less can be learned, although what can be learned may still be useful.)J 59 678 :M .716 .072(Our aim is to make precise exactly what can and cannot be learned in each context. As)J endp %%Page: 3 3 %%BeginPageSetup initializepage (peter; page: 3 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 261 709 28 16 rC 283 722 :M f3_12 sf (3)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .589 .059(with proofs of properties of estimators, the results are mathematical, not moral: they do)J 59 70 :M (not say what assumptions ought to be made.)S 59 98 :M f4_12 sf (1)S 65 98 :M (.)S 68 98 :M (1)S 74 98 :M ( )S 81 98 :M (Structural Equation Models and Directed Graphs)S 81 120 :M f3_12 sf .779 .078(Many of the results and procedures we will describe are very general and apply to)J 59 136 :M .159 .016(models of categorical as well as continuous data, but for the sake of concreteness we will)J 59 152 :M .923 .092(illustrate them with linear structural equation models \(hereafter, SEMs\) \(Bollen, 1989;)J 59 168 :M 1.566 .157(James, Mulaik, and Brett, 1982\). SEMs include linear regression models \(Weisberg,)J 59 184 :M .352 .035(1985\), path analytic models \(Wright, 1934\), factor analytic models \(Lawley & Maxwell,)J 59 200 :M 1.809 .181(1971\), panel models \(Blalock, 1985; Wheaton, et al., 1977\), simultaneous equation)J 59 216 :M .591 .059(models \(Goldberger & Duncan, 1973\), MIMIC models \(Bye, et al., 1985\), and multiple)J 59 232 :M .137 .014(indicator models \(Sullivan, et. al., 1979\). This section introduces the terminology we will)J 59 248 :M (use throughout the rest of the paper.)S 81 264 :M .535 .054(The variables in a SEM can be divided into two sets, the \322error variables\323 or \322error)J 59 280 :M .468 .047(terms,\323 and the substantive variables. Corresponding to each substantive variable X)J f3_7 sf 0 3 rm (i)S 0 -3 rm 470 280 :M f3_12 sf .717 .072( is a)J 59 296 :M .467 .047(linear equation with X)J f3_7 sf 0 3 rm (i)S 0 -3 rm 171 296 :M f3_12 sf .562 .056( on the left hand side of the equation, and the direct causes of X)J 489 299 :M f3_7 sf (i)S 59 312 :M f3_12 sf .398 .04(plus the error term )J f1_12 sf .137(e)A f7_7 sf 0 3 rm .055(i)A 0 -3 rm f3_12 sf .375 .038( on the right hand side of the equation. )J 353 312 :M .359 .036(Since we have no interest in)J 59 328 :M .789 .079(first moments, without loss of generality each variable can be expressed )J 422 328 :M .791 .079(as a deviation)J 59 344 :M (from its mean.)S 81 360 :M .6 .06(Consider, for example, two SEMs S)J 259 363 :M f3_7 sf (1)S 263 360 :M f3_12 sf .787 .079( and S)J 295 363 :M f3_7 sf (2)S 299 360 :M f3_12 sf .574 .057( over )J f0_12 sf (X)S 337 360 :M f3_12 sf .697 .07( = {X)J f3_7 sf 0 3 rm (1)S 0 -3 rm 370 360 :M f3_12 sf .82 .082(, X)J 386 363 :M f3_7 sf (2)S 390 360 :M f3_12 sf .82 .082(, X)J 406 363 :M f3_7 sf (3)S 410 360 :M f3_12 sf .656 .066(}, where in both)J 59 376 :M .37 .037(SEMs X)J 101 379 :M f3_7 sf (1)S 105 376 :M f3_12 sf .388 .039( is a direct cause of X)J f3_7 sf 0 3 rm (2)S 0 -3 rm 216 376 :M f3_12 sf .385 .039( and X)J f3_7 sf 0 3 rm (2)S 0 -3 rm 253 376 :M f3_12 sf .388 .039( is a direct cause of X)J f3_7 sf 0 3 rm (3)S 0 -3 rm 364 376 :M f3_12 sf .303 .03(. The structural equations)J 488 371 :M f3_7 sf (2)S 59 392 :M f3_12 sf (in Figure 1 are common to both S)S 220 395 :M f3_7 sf (1)S 224 392 :M f3_12 sf ( and S)S 254 395 :M f3_7 sf (2)S 258 392 :M f3_12 sf (.)S 230 424 :M f7_12 sf (X)S f7_7 sf 0 3 rm (1 )S 0 -3 rm f7_12 sf (= )S f1_12 sf (e)S f7_7 sf 0 3 rm (1)S 0 -3 rm 230 441 :M f7_12 sf -.396(X)A f7_7 sf 0 3 rm (2)S 0 -3 rm 243 441 :M f7_12 sf ( = )S f1_12 sf (b)S 265 444 :M f7_7 sf (1)S 270 441 :M f7_12 sf ( X)S f7_7 sf 0 3 rm (1)S 0 -3 rm 287 441 :M f7_12 sf ( + )S f1_12 sf (e)S f7_7 sf 0 3 rm (2)S 0 -3 rm 230 458 :M f7_12 sf -.396(X)A f7_7 sf 0 3 rm (3)S 0 -3 rm 243 458 :M f7_12 sf ( = )S f1_12 sf (b)S 265 461 :M f7_7 sf (2)S 270 458 :M f7_12 sf ( X)S f7_7 sf 0 3 rm (2)S 0 -3 rm 287 458 :M f7_12 sf ( + )S f1_12 sf (e)S f7_7 sf 0 3 rm (3)S 0 -3 rm 145 487 :M f0_12 sf (Figure )S 182 487 :M (1: Structural Equations for SEMs S)S 365 490 :M f0_7 sf (1)S 369 487 :M f0_12 sf ( and S)S f0_7 sf 0 3 rm (2)S 0 -3 rm 59 515 :M f3_12 sf .108 .011(In these equations, )J 153 515 :M f1_12 sf (b)S 160 518 :M f3_7 sf .048 .005(1 )J f3_12 sf 0 -3 rm .159 .016(and )J 0 3 rm f1_12 sf 0 -3 rm (b)S 0 3 rm 192 518 :M f3_7 sf (2)S 196 515 :M f3_12 sf .109 .011( are free parameters ranging over real values, and )J f1_12 sf (e)S f3_7 sf 0 3 rm (1)S 0 -3 rm 446 515 :M f3_12 sf .111(,)A f3_7 sf 0 3 rm .065 .007( )J 0 -3 rm 452 515 :M f1_12 sf (e)S f3_7 sf 0 3 rm (2 )S 0 -3 rm f3_12 sf .129 .013(and )J f1_12 sf (e)S f3_7 sf 0 3 rm (3)S 0 -3 rm 59 531 :M f3_12 sf 1.827 .183(are unmeasured random variables called error terms. Suppose that )J 407 531 :M f1_12 sf (e)S f3_7 sf 0 3 rm (1)S 0 -3 rm 416 531 :M f3_12 sf 1.975(,)A f3_7 sf 0 3 rm 1.152 .115( )J 0 -3 rm 425 531 :M f1_12 sf .832(e)A f3_7 sf 0 3 rm .691 .069(2 )J 0 -3 rm f3_12 sf 2.292 .229(and )J f1_12 sf .832(e)A f3_7 sf 0 3 rm (3)S 0 -3 rm 470 531 :M f3_12 sf 2.405 .24( are)J 59 547 :M 1.042 .104(distributed as multivariate normal. In S)J 261 550 :M f3_7 sf (1)S 265 547 :M f3_12 sf 1.075 .108( we will assume that the correlation between)J 59 563 :M .138 .014(each pair of distinct error terms is fixed at zero. The free parameters of S)J f3_7 sf 0 3 rm (1 )S 0 -3 rm f3_12 sf .116 .012(are )J 434 563 :M f2_12 sf .18 .018(q )J 444 563 :M f3_12 sf .165 .017(= <)J 461 563 :M f2_12 sf (b)S 468 563 :M f3_12 sf .043 .004(, )J f0_12 sf .063(P)A f3_12 sf .084(>,)A 59 579 :M .519 .052(where )J f2_12 sf .224 .022(b )J 102 579 :M f3_12 sf .326 .033(is the set of linear coefficients {)J f1_12 sf (b)S 265 582 :M f3_7 sf (1)S 269 579 :M f3_12 sf .49 .049(, )J 276 579 :M f1_12 sf (b)S 283 582 :M f3_7 sf (2)S 287 579 :M f3_12 sf .302 .03(} and )J f0_12 sf .198(P)A f3_12 sf .38 .038( is the set of variances of the error)J 59 595 :M .148 .015(terms. We will use )J 153 595 :M f2_12 sf (S)S f0_7 sf 0 3 rm (S1)S 0 -3 rm f3_12 sf <28>S 171 595 :M f2_12 sf (q)S f2_7 sf 0 3 rm (1)S 0 -3 rm 181 595 :M f3_12 sf .133 .013(\) to denote the covariance matrix parameterized by the vector )J 481 595 :M f2_12 sf (q)S f2_7 sf 0 3 rm (1)S 0 -3 rm 59 611 :M f3_12 sf .216 .022(for model S)J 117 614 :M f3_7 sf (1)S 121 611 :M f3_12 sf .203 .02(, and occasionally leave out the model subscript if the context makes it clear)J 59 648 :M ( )S 59 645.48 -.48 .48 203.48 645 .48 59 645 @a 81 657 :M f3_6 sf (2)S 84 661 :M f3_10 sf .365 .037( We realize that it is slightly unconventional to write the trivial equation for the exogenous variable)J 59 673 :M (X)S f3_9 sf 0 2 rm (1)S 0 -2 rm 71 673 :M f3_10 sf .367 .037( in terms of its error, but this serves to give the error terms a unified and special status as providing all)J 59 687 :M (the exogenous source of stochastic variation for the system.)S endp %%Page: 4 4 %%BeginPageSetup initializepage (peter; page: 4 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 261 709 28 16 rC 283 722 :M f3_12 sf (4)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf 2.322 .232(which model is being referred to. If all the pairs of error terms in a SEM S are)J 59 70 :M (uncorrelated, we say S is a SEM with )S 242 70 :M f0_12 sf (uncorrelated errors)S 343 70 :M f3_12 sf (.)S 81 86 :M .06 .006(Let S)J 107 89 :M f7_7 sf (2)S 112 86 :M f3_12 sf .126 .013( contain the same structural equations as S)J f7_7 sf 0 3 rm (1)S 0 -3 rm 321 86 :M f3_12 sf .065 .006(, but in S)J 365 89 :M f7_7 sf (2)S 370 86 :M f3_12 sf .052 .005( allow the errors between)J 59 102 :M f7_12 sf -.396(X)A f7_7 sf 0 3 rm (2)S 0 -3 rm 72 102 :M f3_12 sf .348 .035( and )J f7_12 sf .352(X)A f7_7 sf 0 3 rm (3)S 0 -3 rm 110 102 :M f3_12 sf .616 .062( to be correlated, i.e., make the correlation between the errors of )J 431 102 :M f7_12 sf -.396(X)A f7_7 sf 0 3 rm (2)S 0 -3 rm 444 102 :M f3_12 sf .348 .035( and )J f7_12 sf .352(X)A f7_7 sf 0 3 rm (3)S 0 -3 rm 482 102 :M f3_12 sf .873 .087( a)J 59 118 :M .326 .033(free parameter, instead of fixing it at zero, as in S)J f7_7 sf 0 3 rm (1)S 0 -3 rm 305 118 :M f3_12 sf .337 .034(. In S)J 332 121 :M f7_7 sf (2)S 337 121 :M f3_7 sf ( )S f3_12 sf 0 -3 rm .329 .033(the free parameters are )J 0 3 rm 454 118 :M f2_12 sf .204(q)A f3_12 sf .277 .028( = <)J 481 118 :M f2_12 sf (b)S 488 118 :M f3_12 sf (,)S 59 134 :M f0_12 sf .156<50D5>A f3_12 sf .429 .043(>, where )J f2_12 sf .241 .024(b )J 127 134 :M f3_12 sf .388 .039(is the set of linear coefficients {)J 284 134 :M f1_12 sf (b)S 291 137 :M f3_7 sf (1)S 295 134 :M f3_12 sf (,)S f1_12 sf (b)S 305 137 :M f3_7 sf (2)S 309 134 :M f3_12 sf .345 .034(} and )J f0_12 sf .174<50D5>A f3_12 sf .422 .042( is the set of variances of the)J 59 150 :M .718 .072(error terms and the correlation between )J f1_12 sf .197(e)A f3_7 sf 0 3 rm .164 .016(2 )J 0 -3 rm f3_12 sf .543 .054(and )J f1_12 sf .197(e)A f3_7 sf 0 3 rm (3)S 0 -3 rm 298 150 :M f3_12 sf .695 .07(. If the correlations between any of the)J 59 166 :M (error terms in a SEM are not fixed at zero, we will call it a SEM with )S f0_12 sf (correlated errors)S 482 166 :M f3_12 sf (.)S 485 161 :M f3_7 sf (3)S 81 182 :M f3_12 sf .699 .07(It is possible to associate with each SEM with uncorrelated errors a directed graph)J 59 198 :M .242 .024(that represents the causal structure of the model and the form of the linear equations. For)J 59 214 :M .206 .021(example, the directed graph associated with the substantive variables in S)J 416 217 :M f3_7 sf (1)S 420 214 :M f3_12 sf .247 .025( is X)J f3_7 sf 0 3 rm (1)S 0 -3 rm 447 214 :M f1_12 sf S 459 214 :M f3_12 sf .325 .032( X)J 472 217 :M f3_7 sf (2)S 476 214 :M f3_12 sf .066 .007( )J f1_12 sf S 59 230 :M f3_12 sf (X)S 68 233 :M f3_7 sf (3)S 72 230 :M f3_12 sf .565 .056(, because X)J 130 233 :M f3_7 sf (1)S 134 230 :M f3_12 sf .554 .055( is the only substantive variable that occurs on the right hand side of the)J 59 246 :M (equation for X)S 129 249 :M f3_7 sf (2)S 133 246 :M f3_12 sf (, and X)S f3_7 sf 0 3 rm (2)S 0 -3 rm 172 246 :M f3_12 sf .009 .001( is the only substantive variable that appears on the right hand side)J 59 262 :M .578 .058(of the equation for X)J 164 265 :M f3_7 sf (3)S 168 262 :M f3_12 sf .565 .057(. We generally do not include error terms in our path diagrams of)J 59 278 :M 1.003 .1(SEMs unless the errors are correlated. We enclose measured variables in boxes, latent)J 59 294 :M (variables in circles, and leave error variables unenclosed.)S 180 317 19 15 rC 180 329 :M f8_12 sf (X)S gR gS 189 323 14 12 rC 189 332 :M f8_10 sf (1)S gR gS 272 317 19 15 rC 272 329 :M f8_12 sf (X)S gR gS 281 323 14 12 rC 281 332 :M f8_10 sf (2)S gR gS 358 317 19 15 rC 358 329 :M f8_12 sf (X)S gR gS 367 323 14 12 rC 367 332 :M f8_10 sf (3)S gR gS 274 360 14 15 rC 274 373 :M f1_12 sf (e)S gR gS 280 367 15 12 rC 280 376 :M f8_10 sf (2)S gR gS 358 359 14 15 rC 358 372 :M f1_12 sf (e)S gR gS 364 366 14 12 rC 364 375 :M f8_10 sf (3)S gR gS 169 313 212 82 rC 201 326.75 -.75 .75 256.75 326 .75 201 326 @a np 254 322 :M 254 330 :L 262 326 :L 254 322 :L .75 lw eofill -.75 -.75 254.75 330.75 .75 .75 254 322 @b -.75 -.75 254.75 330.75 .75 .75 262 326 @b 254 322.75 -.75 .75 262.75 326 .75 254 322 @a 294 325.75 -.75 .75 343.75 325 .75 294 325 @a np 341 321 :M 341 328 :L 349 325 :L 341 321 :L eofill -.75 -.75 341.75 328.75 .75 .75 341 321 @b -.75 -.75 341.75 328.75 .75 .75 349 325 @b 341 321.75 -.75 .75 349.75 325 .75 341 321 @a -.75 -.75 280.75 357.75 .75 .75 280 346 @b np 277 348 :M 284 348 :L 280 340 :L 277 348 :L eofill 277 348.75 -.75 .75 284.75 348 .75 277 348 @a 280 340.75 -.75 .75 284.75 348 .75 280 340 @a -.75 -.75 277.75 348.75 .75 .75 280 340 @b -.75 -.75 366.75 355.75 .75 .75 366 342 @b np 362 344 :M 369 344 :L 366 336 :L 362 344 :L eofill 362 344.75 -.75 .75 369.75 344 .75 362 344 @a 366 336.75 -.75 .75 369.75 344 .75 366 336 @a -.75 -.75 362.75 344.75 .75 .75 366 336 @b 90 180 80 26 322.5 380.5 @n 0 90 78 26 321.5 380.5 @n 175.5 315.5 23 19 rS 268.5 317.5 23 19 rS 354.5 317.5 23 19 rS gR gS 0 0 552 730 rC 172 420 :M f0_12 sf (Figure )S 209 420 :M (2. SEM S)S 257 423 :M f0_7 sf (2)S 261 420 :M f0_12 sf ( with)S 287 420 :M ( correlated errors)S 81 448 :M f3_12 sf .102 .01(The typical path diagram that would be given for S)J 328 451 :M f3_7 sf (2)S 332 448 :M f3_12 sf .114 .011( is shown in Figure )J 428 448 :M .109 .011(2. This is not)J 59 464 :M .621 .062(strictly a directed graph because of the curved line between error terms )J f1_12 sf .184(e)A f3_7 sf 0 3 rm .152 .015(2 )J 0 -3 rm f3_12 sf .506 .051(and )J f1_12 sf .184(e)A f3_7 sf 0 3 rm (3)S 0 -3 rm 455 464 :M f3_12 sf .597 .06(, which)J 59 480 :M 1.298 .13(indicates that )J f1_12 sf .359(e)A f3_7 sf 0 3 rm .298 .03(2 )J 0 -3 rm f3_12 sf .99 .099(and )J f1_12 sf .359(e)A f3_7 sf 0 3 rm (3)S 0 -3 rm 172 480 :M f3_12 sf 1.132 .113( are correlated. It is generally accepted that correlation is to be)J 59 496 :M .282 .028(explained by some form of causal connection. Accordingly, if )J 364 496 :M f1_12 sf .161(e)A f3_7 sf 0 3 rm .134 .013(2 )J 0 -3 rm f3_12 sf .477 .048(and )J 395 496 :M f1_12 sf (e)S f3_7 sf 0 3 rm (3)S 0 -3 rm 404 496 :M f3_12 sf .301 .03( are correlated we)J 59 512 :M .112 .011(will assume that either )J 171 512 :M f1_12 sf (e)S f3_7 sf 0 3 rm (2 )S 0 -3 rm f3_12 sf .133 .013(causes )J f1_12 sf (e)S f3_7 sf 0 3 rm (3)S 0 -3 rm 224 512 :M f3_12 sf .052 .005(, )J f1_12 sf .055(e)A f3_7 sf 0 3 rm .045 .005(3 )J 0 -3 rm f3_12 sf .223 .022(causes )J 275 512 :M f1_12 sf (e)S f3_7 sf 0 3 rm (2)S 0 -3 rm 284 512 :M f3_12 sf .12 .012(, some latent variable causes both )J f1_12 sf (e)S f3_7 sf 0 3 rm (2 )S 0 -3 rm f3_12 sf .101 .01(and )J f1_12 sf (e)S f3_7 sf 0 3 rm (3)S 0 -3 rm 488 512 :M f3_12 sf (,)S 59 528 :M 3.189 .319(or some combination of these. In other words, curved lines are an ambiguous)J 59 544 :M 1.113 .111(representation of a causal connection. In section )J 307 544 :M 1.207 .121(3.1, for each SEM S with correlated)J 59 560 :M .173 .017(errors we will show how to construct a directed acyclic graph G with latent variables that)J 59 576 :M (represents important causal and statistical features of S.)S 81 592 :M 1.079 .108(Finally, a directed graph is )J 221 592 :M f0_12 sf (acyclic)S 256 592 :M f3_12 sf 1.106 .111( if it contains no directed path from a variable)J 59 608 :M (back to itself. A SEM is said to be )S f0_12 sf (recursive)S f3_12 sf ( \(an RSEM\) if its directed graph is acyclic.)S 59 674 :M ( )S 59 671.48 -.48 .48 203.48 671 .48 59 671 @a 81 683 :M f3_6 sf (3)S 84 687 :M f3_10 sf (We do not consider SEMs with other sorts of constraints on the parameters, e.g., equality constraints.)S endp %%Page: 5 5 %%BeginPageSetup initializepage (peter; page: 5 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 261 709 28 16 rC 283 722 :M f3_12 sf (5)S gR gS 0 0 552 730 rC 59 54 :M f4_12 sf (1)S 65 54 :M (.)S 68 54 :M (2)S 74 54 :M ( )S 81 54 :M (Causal Structure: Predicting the Effects of Manipulations)S 81 76 :M f3_12 sf .371 .037(In this section we will consider the causal interpretation of RSEMs \(Strotz & Wold,)J 59 92 :M .895 .09(1960\). The causal interpretation of non-recursive SEMs is not well understood and we)J 59 108 :M .377 .038(will not discuss it here. Consider the following hypothetical situation. For each member)J 59 124 :M .048 .005(of a population it is recorded how many cigarettes they have smoked in the last month \(S\))J 59 140 :M .474 .047(and how yellow their fingers are \(Yf\). Let us suppose that the correct causal description)J 59 156 :M (of this system is given by RSEM M in )S 246 156 :M (Figure 3.)S 253 190 :M (Yf = a S + )S f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm 272 206 :M f3_12 sf (S = )S f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 114 222 :M f1_12 sf (r)S 121 222 :M f3_12 sf <28>S 125 222 :M f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f1_12 sf (,)S f3_12 sf ( )S f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 152 222 :M f3_12 sf (\) = 0, mean\()S 214 222 :M f1_12 sf (e)S f3_7 sf 0 3 rm (Y)S 0 -3 rm f3_12 sf (\) = 0, mean\( )S 286 222 :M f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f3_12 sf (\) = 0, var\()S f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 355 222 :M f3_12 sf (\) = 1, var\()S f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f3_12 sf (\) = 1 - a)S f3_7 sf 0 -5 rm (2)S 0 5 rm .75 lw 226 241 97 26 rC 297.5 241.5 24 24 rS 299 243 21 21 rC gS 1.01 1 scale 295.921 255 :M f3_12 sf ( )S gR gS 1.01 1 scale 298.89 255 :M f8_12 sf (Y)S gR gS 1.01 1 scale 306.807 255 :M f8_12 sf (f)S gR gR .75 lw gS 226 241 97 26 rC 226.5 241.5 16 21 rS 228 243 13 18 rC gS 1.01 1 scale 225.652 255 :M f3_12 sf ( )S gR gS 1.01 1 scale 228.621 255 :M f8_12 sf (S)S gR gR gS 226 241 97 26 rC 246 251.75 -.75 .75 289.75 251 .75 246 251 @a np 289 247 :M 289 255 :L 292 251 :L 289 247 :L eofill -.75 -.75 289.75 255.75 .75 .75 289 247 @b -.75 -.75 289.75 255.75 .75 .75 292 251 @b 289 247.75 -.75 .75 292.75 251 .75 289 247 @a gR gS 0 0 552 730 rC 226 292 :M f0_12 sf (Figure )S 263 292 :M (3. RSEM M)S 81 320 :M f3_12 sf .695 .07(The graph of this RSEM is asymmetric because it contains an arrow from S to Yf,)J 59 336 :M (but not from Yf to S. What consequences does the causal asymmetry have?)S 81 352 :M .416 .042(The causal asymmetry is reflected in the predictions that M makes about the effects)J 59 368 :M 1.82 .182(of interventions on the values of the random variables S and Yf respectively.)J 463 363 :M f3_7 sf (4)S 466 368 :M f3_12 sf 2.287 .229( The)J 59 384 :M .352 .035(prediction of the effect of an intervention on a system is a counterfactual prediction, that)J 59 400 :M .023 .002(is, it is a prediction not about the existing population, but about a population that does not)J 59 416 :M 1.16 .116(exist and might never exist. Of course, how an actual intervention would affect other)J 59 432 :M .087 .009(variables would depend upon )J 204 432 :M f4_12 sf .031(how)A f3_12 sf .085 .008( we intervened in the system. We will consider theories)J 59 448 :M .633 .063(that predict the effects of a kind of ideal intervention in which the only variables in the)J 59 464 :M .112 .011(system affected )J 138 464 :M f4_12 sf (directly)S 175 464 :M f3_12 sf .123 .012( are those that we manipulate by setting their value. For example,)J 59 480 :M 1.007 .101(suppose we intervene ideally in the population originally described by M to eliminate)J 59 496 :M .986 .099(smoking, i.e., we set S = 0. Then the new causal system that is the result of this ideal)J 59 512 :M .064 .006(intervention would be described by model M)J 276 515 :M f3_7 sf (S)S 280 512 :M f3_12 sf .072 .007( \(Figure 4\), in which the only change is that)J 59 528 :M 1.026 .103(the equation for S in M is replaced by a new equation in which all of the coefficients)J 59 544 :M 1.189 .119(relating S to other variables \(in this case just )J f1_12 sf .403(e)A f3_7 sf 0 3 rm (S)S 0 -3 rm 301 544 :M f3_12 sf 1.422 .142(\) are set to 0, and S is set equal to a)J 59 560 :M (constant.)S 102 555 :M f3_7 sf (5)S 59 614 :M f3_12 sf ( )S 59 611.48 -.48 .48 203.48 611 .48 59 611 @a 81 623 :M f3_6 sf (4)S 84 627 :M f3_10 sf .413 .041( The causal asymmetry is also reflected in the quite different statistical relationships between )J 467 627 :M f1_10 sf .129(e)A f3_6 sf 0 2 rm .069(s)A 0 -2 rm f3_10 sf .384 .038( and)J 59 639 :M .126 .013(Yf on the one hand, and )J 159 639 :M f1_10 sf (e)S f3_6 sf 0 2 rm (Yf)S 0 -2 rm f3_10 sf .103 .01( and S on the other hand. From the Causal Independence assumption introduced)J 59 651 :M 1.106 .111(in section 3, it follows that in M, )J f1_10 sf .422(e)A f3_6 sf 0 2 rm .304(Yf)A 0 -2 rm f3_10 sf 1.181 .118( and S are uncorrelated, but it does not entail that )J 434 651 :M f1_10 sf .436(e)A f3_6 sf 0 2 rm .331(S)A 0 -2 rm f3_10 sf 1.11 .111( and Yf are)J 59 663 :M (uncorrelated.)S 81 671 :M f3_6 sf (5)S 84 675 :M f3_10 sf .181 .018( In this example, the constant is zero, but we could of course just as easily have an ideal intervention)J 59 687 :M (which set S to any other value.)S endp %%Page: 6 6 %%BeginPageSetup initializepage (peter; page: 6 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 261 709 28 16 rC 283 722 :M f3_12 sf (6)S gR gS 0 0 552 730 rC 253 54 :M f3_12 sf (Yf = a S + )S f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm 273 70 :M f3_12 sf (S = 0)S 115 86 :M f1_12 sf (r)S 122 86 :M f3_12 sf <28>S 126 86 :M f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f1_12 sf (,)S f3_12 sf ( )S f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 153 86 :M f3_12 sf (\) = 0, mean\()S 215 86 :M f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 224 86 :M f3_12 sf (\) = 0, mean\( )S 286 86 :M f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f3_12 sf (\) = 0, var\()S f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 355 86 :M f3_12 sf (\) = 0, var\()S f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f3_12 sf (\) = 1 - a)S f3_7 sf 0 -5 rm (2)S 0 5 rm .75 lw 226 105 97 26 rC 297.5 105.5 24 24 rS 299 107 21 21 rC gS 1.01 1 scale 295.921 119 :M f3_12 sf ( )S gR gS 1.01 1 scale 298.89 119 :M f8_12 sf (Y)S gR gS 1.01 1 scale 306.807 119 :M f8_12 sf (f)S gR gR .75 lw gS 226 105 97 26 rC 226.5 105.5 16 21 rS 228 107 13 18 rC gS 1.01 1 scale 225.652 119 :M f3_12 sf ( )S gR gS 1.01 1 scale 228.621 119 :M f8_12 sf (S)S gR gR gS 226 105 97 26 rC 246 115.75 -.75 .75 289.75 115 .75 246 115 @a np 289 111 :M 289 119 :L 292 115 :L 289 111 :L eofill -.75 -.75 289.75 119.75 .75 .75 289 111 @b -.75 -.75 289.75 119.75 .75 .75 292 115 @b 289 111.75 -.75 .75 292.75 115 .75 289 111 @a gR gS 0 0 552 730 rC 224 156 :M f0_12 sf (Figure )S 261 156 :M (4. RSEM M)S 322 159 :M f0_7 sf (S)S 81 200 :M f3_12 sf .206 .021(If, for an arbitrary parameterization of M)J f3_7 sf 0 3 rm (S)S 0 -3 rm 284 200 :M f3_12 sf .298 .03(, f)J 295 203 :M f3_7 sf (S)S 299 200 :M f3_12 sf .225 .022( is the density function according to M)J f3_7 sf 0 3 rm (S)S 0 -3 rm 59 216 :M f3_12 sf .188 .019(\(and similarly for M and f)J f3_7 sf 0 3 rm .071(M)A 0 -3 rm f3_12 sf .164 .016(\) then the effect on Yf of ideally intervening to set S to 0, i.e.,)J 59 232 :M (f)S 63 235 :M f3_7 sf (S)S 67 232 :M f3_12 sf (\(Yf | S = 0\), is equal to the density, in RSEM M, of Yf conditional on S = 0, i.e., f)S 463 235 :M f3_7 sf (M)S f3_12 sf 0 -3 rm (\(Yf |)S 0 3 rm 59 248 :M (S = 0\).)S 81 264 :M (M)S 92 267 :M f3_7 sf (S)S 96 264 :M f3_12 sf .948 .095( is a correct theory of the results of the kind of ideal intervention in which the)J 59 280 :M 1.945 .194(only variable in the system that is directly affected is S. Of course, whether some)J 59 296 :M .837 .084(particular course of action is an ideal intervention of this kind is an empirical question)J 59 312 :M .525 .053(that is outside the scope of M or M)J f3_7 sf 0 3 rm (S)S 0 -3 rm 237 312 :M f3_12 sf .515 .051(. For example, we could try and force people not to)J 59 328 :M -.002(smoke \(in such a way that the only variable directly affected is smoking\) by passing a law)A 59 344 :M .336 .034(against smoking. If the law is effective, and it does not directly affect the values of other)J 59 360 :M (variables, then M)S 143 363 :M f3_7 sf (S)S 147 360 :M f3_12 sf ( is the correct description of the new system; otherwise it is not.)S 81 376 :M .197 .02(Suppose now that we were to perform an ideal intervention on yellowed fingers \(Yf\))J 59 392 :M 1.37 .137(in the system described by M, i.e. we were to intervene in such a way that the only)J 59 408 :M .774 .077(change to the system is that the equation for Yf in M is replaced by a new equation in)J 59 424 :M .224 .022(which all of the coefficients relating Yf to other variables are set to 0, and Yf is set equal)J 59 440 :M .125 .012(to a constant. Call this new theory M)J f3_7 sf 0 3 rm .028(Yf)A 0 -3 rm f3_12 sf .117 .012(. Note that we have removed the edge from S to Yf)J 59 456 :M .61 .061(in the graph of M)J f3_7 sf 0 3 rm .153(Yf)A 0 -3 rm f3_12 sf .718 .072(, because variations in S no longer cause variations in Yf and hence)J 59 472 :M (the coefficient of S in the equation for Yf is set to zero.)S 270 504 :M (Yf = 0)S 272 520 :M (S = )S f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 115 536 :M f1_12 sf (r)S 122 536 :M f3_12 sf <28>S 126 536 :M f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f1_12 sf (,)S f3_12 sf ( )S f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 153 536 :M f3_12 sf (\) = 0, mean\()S 215 536 :M f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 224 536 :M f3_12 sf (\) = 0, mean\( )S 286 536 :M f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f3_12 sf (\) = 0, var\()S f1_12 sf (e)S f3_7 sf 0 3 rm (S)S 0 -3 rm 355 536 :M f3_12 sf (\) = 0, var\()S f1_12 sf (e)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f3_12 sf (\) = 1 - a)S f3_7 sf 0 -5 rm (2)S 0 5 rm 226 555 97 26 rC 297.5 555.5 24 24 rS 299 557 21 21 rC gS 1.01 1 scale 295.921 569 :M f3_12 sf ( )S gR gS 1.01 1 scale 298.89 569 :M f8_12 sf (Y)S gR gS 1.01 1 scale 306.807 569 :M f8_12 sf (f)S gR gR gS 226 555 97 26 rC 226.5 555.5 16 21 rS 228 557 13 18 rC gS 1.01 1 scale 225.652 569 :M f3_12 sf ( )S gR gS 1.01 1 scale 228.621 569 :M f8_12 sf (S)S gR gR gS 0 0 552 730 rC 222 606 :M f0_12 sf (Figure )S 259 606 :M (5. RSEM M)S 320 609 :M f0_7 sf (Yf)S 81 634 :M f3_12 sf 1.768 .177(One consequence of the causal asymmetry in M is that if we perform an ideal)J 59 650 :M .523 .052(intervention on S in M, then f)J 206 653 :M f3_7 sf .139(M)A f3_12 sf 0 -3 rm .453 .045(\(Yf\) changes, but when we perform an ideal intervention)J 0 3 rm 59 666 :M .061 .006(on Yf in M, f)J f3_7 sf 0 3 rm ( M)S 0 -3 rm f3_12 sf .077 .008(\(S\) does not change \(because Yf is not a cause of S\). It is also important to)J 59 682 :M .175 .017(note that the distribution of smoking )J 239 682 :M f4_12 sf (conditional)S 294 682 :M f3_12 sf .189 .019( on Yf = 0, i.e., f)J f3_7 sf 0 3 rm .127(M)A 0 -3 rm f3_12 sf .19 .019(\(S | Yf = 0\), is )J f4_12 sf .105(not)A f3_12 sf .278 .028( the)J endp %%Page: 7 7 %%BeginPageSetup initializepage (peter; page: 7 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 261 709 28 16 rC 283 722 :M f3_12 sf (7)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .027 .003(same as the distribution of S when an ideal intervention is performed on Yf, i.e., f)J 452 57 :M f3_7 sf (Yf)S f3_12 sf 0 -3 rm .027 .003(\(S | Yf)J 0 3 rm 59 70 :M (= 0\). In the case of any ideal intervention on Yf, f)S f3_7 sf 0 3 rm (Yf)S 0 -3 rm f3_12 sf (\(S | Yf = c\) = f)S 375 73 :M f3_7 sf (Yf)S f3_12 sf 0 -3 rm (\(S\).)S 0 3 rm 81 86 :M 1.098 .11(Given an RSEM M in which all parameters are identified, and information about)J 59 102 :M .412 .041(how an intervention affects a given variable in the system, predicting the effects of ideal)J 59 118 :M 1.382 .138(interventions is easy; one can simply make the suitable changes to M in the manner)J 59 134 :M .182 .018(described above. The problem is much more difficult if M is only partially specified \(e.g.)J 59 150 :M .306 .031(the directions of only some of the arrows in the graph of M are known\), or if M contains)J 59 166 :M 2.056 .206(latent variables and not all of the parameters are identifiable. A general theory of)J 59 182 :M 1.861 .186(representing interventions in causal systems \(not limited to RSEMs\) in a graphical)J 59 198 :M .477 .048(framework, and of predicting the effects of interventions for a partially specified model)J 59 214 :M 1.001 .1(is presented in Spirtes, et al. 1993, chapter 7. Examples of making predictions from a)J 59 230 :M .79 .079(partially specified model will be presented in section 5. Robins \(1986\) made important)J 59 246 :M .53 .053(advances on the problem of predicting the effects of interventions in models with latent)J 59 262 :M 1.077 .108(variables. Pearl \(1995\) gives a more general solution of the problem of predicting the)J 59 278 :M .041 .004(effects of interventions in models with latent variables using the graphical representations)J 59 294 :M (of interventions presented in Spirtes, et al. 1993.)S 59 322 :M f4_12 sf (1)S 65 322 :M (.)S 68 322 :M (3)S 74 322 :M ( )S 81 322 :M (Parameter Estimation and SEM Specification)S 81 344 :M f3_12 sf .14 .014(Having clarified the objects under discussion, we now return to our analogy between)J 59 360 :M 1.027 .103(parameter estimation and model specification search. Ideally, there are four properties)J 59 376 :M (that an estimation procedure should have:)S 95 392 :M f1_12 sf S 100 392 :M .716 .072( )J 100 392 :M f3_12 sf .716 .072( )J 104 392 :M f4_12 sf (Identification)S 169 392 :M f3_12 sf .439 .044(. An estimation procedure should be able to determine whether or)J 104 408 :M .805 .081(not a parameter is identifiable, i.e. determine whether or not there is a unique)J 104 424 :M (estimate that satisfies the given constraints.)S 95 440 :M f1_12 sf S 100 440 :M .346 .035( )J 100 440 :M f3_12 sf ( )S f4_12 sf .033(Consistency)A 161 440 :M f3_12 sf .229 .023(. If a parameter is identified, it should be the case that as the sample)J 104 456 :M 1.554 .155(size grows without limit, the probability approaches one that the difference)J 104 472 :M (between the true value and the estimated value approaches zero.)S 95 488 :M f1_12 sf S 100 488 :M 3.661 .366( )J 100 488 :M f3_12 sf 3.661 .366( )J 107 488 :M f4_12 sf 1.307 .131(Error Probabilities)J 204 488 :M f3_12 sf 2.202 .22(. The sampling distribution of the estimator should be)J 104 504 :M (known.)S 95 520 :M f1_12 sf S 100 520 :M .233 .023( )J 100 520 :M f3_12 sf .233 .023( )J 104 520 :M f4_12 sf .077 .008(Practical Reliability)J 202 520 :M f3_12 sf .141 .014(. The estimation procedure should be reliable on samples of)J 104 536 :M 1.565 .157(realistic size, and relatively robust against small violations of the operative)J 104 552 :M (assumptions.)S 81 584 :M 1.077 .108(We will examine each of these desiderata in more detail, and point out analogies)J 59 600 :M 3.387 .339(\(and some disanalogies\) between parameter estimation and model specification)J 59 616 :M .645 .064(procedures. To see how the analogy extends to model specification procedures, we will)J 59 632 :M .866 .087(consider for the sake of concreteness properties of the PC algorithm, which is a model)J 59 648 :M 2.673 .267(specification search procedure implemented in the Build module of TETRAD II)J 59 664 :M 1.116 .112(\(described in sections 5.1.1 and )J 222 664 :M 1.228 .123(8.2\). For the points we make in this section, it is not)J 59 680 :M 1.288 .129(necessary to know the details of the PC algorithm. The only features relevant to this)J endp %%Page: 8 8 %%BeginPageSetup initializepage (peter; page: 8 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 261 709 28 16 rC 283 722 :M f3_12 sf (8)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .341 .034(section are that it takes as input: 1\) a sample covariance matrix \(under the assumption of)J 59 70 :M .659 .066(multivariate normality, and 2\) background knowledge, and it outputs a graphical object)J 59 86 :M .488 .049(called a )J f4_12 sf .205(pattern)A 136 86 :M f3_12 sf .803 .08( that represents a class of RSEMs without latent variables or correlated)J 59 102 :M .766 .077(errors that are statistically equivalent \(in a sense we make precise in section 4 below\).)J 59 118 :M 1.784 .178(Again, for concreteness, we will use maximum likelihood \(ML\) estimation and the)J 59 134 :M (algorithms that implement it as the example of a parameter estimator.)S 59 162 :M f4_12 sf (1)S 65 162 :M (.)S 68 162 :M (3)S 74 162 :M (.)S 77 162 :M (1)S 83 162 :M ( )S 90 162 :M (Identifiability)S 81 181 :M f3_12 sf .282 .028(If there is a unique ML estimate of a parameter in a SEM, then the parameter is said)J 59 197 :M .907 .091(to be identifiable. When a parameter is not identifiable, it has more than one value for)J 59 213 :M 1.227 .123(which the likelihood of the data is maximal given the model. Although many special)J 59 229 :M .051 .005(cases have been solved \(e.g., see Becker, et al., 1994\), necessary and sufficient conditions)J 59 245 :M (for SEM parameter identifiability are not known.)S 81 261 :M 2.328 .233(In the case of SEM specification procedures there is a problem analogous to)J 59 277 :M .827 .083(parameter non-identifiability. There are many pairs of RSEMs R)J f3_7 sf 0 3 rm (1)S 0 -3 rm 385 277 :M f3_12 sf 1.028 .103( and R)J f3_7 sf 0 3 rm (2)S 0 -3 rm 423 277 :M f3_12 sf 1.036 .104( that have the)J 59 293 :M 1.493 .149(same set of measured variables, and no latent variables or correlated errors, that are)J 59 309 :M f0_12 sf .025 .002(covariance equivalent)J 171 309 :M f3_12 sf .046 .005( in the following sense: for every parameterization )J 417 309 :M f2_12 sf (q)S f0_7 sf 0 3 rm (i)S 0 -3 rm 425 309 :M f3_12 sf .052 .005( of R)J f3_7 sf 0 3 rm (1)S 0 -3 rm 453 309 :M f3_12 sf .054 .005( there is)J 59 325 :M .875 .088(a parameterization )J f2_12 sf .232(q)A f0_7 sf 0 3 rm .138 .014(j )J 0 -3 rm 165 325 :M f3_12 sf .775 .077(of R)J f3_7 sf 0 3 rm (2)S 0 -3 rm 191 325 :M f3_12 sf .949 .095( such that )J 244 325 :M f2_12 sf (S)S f0_7 sf 0 3 rm (R1)S 0 -3 rm 260 325 :M f3_12 sf <28>S 264 325 :M f2_12 sf (q)S f0_7 sf 0 3 rm (i)S 0 -3 rm 272 325 :M f3_12 sf .546 .055(\) = )J f2_12 sf .555(S)A f0_7 sf 0 3 rm .668(R2)A 0 -3 rm 307 325 :M f3_12 sf <28>S 311 325 :M f2_12 sf .29(q)A f0_7 sf 0 3 rm .108(j)A 0 -3 rm f3_12 sf .769 .077(\), and vice versa. When R)J f3_7 sf 0 3 rm (1)S 0 -3 rm 453 325 :M f3_12 sf 1.012 .101( and R)J 487 328 :M f3_7 sf (2)S 59 341 :M f3_12 sf 2.362 .236(have no latent variables or correlated errors, then covariance equivalence has the)J 59 357 :M .15 .015(following consequence: for any covariance matrix over the measured variables, if R)J f3_7 sf 0 3 rm (1)S 0 -3 rm 471 357 :M f3_12 sf .197 .02( and)J 59 373 :M (R)S f3_7 sf 0 3 rm (2)S 0 -3 rm 71 373 :M f3_12 sf 1.975 .198( are both parameterized by the respective ML estimates of their free parameters)J 59 389 :M f2_12 sf (S)S f0_7 sf 0 3 rm (R1)S 0 -3 rm 75 389 :M f3_12 sf <28>S 79 389 :M f2_12 sf .112(q)A f0_7 sf 0 3 rm .101(ML)A 0 -3 rm f3_12 sf .188 .019(\) and )J f2_12 sf .127(S)A f0_7 sf 0 3 rm .153(R2)A 0 -3 rm 140 389 :M f3_12 sf <28>S 144 389 :M f2_12 sf .142(q)A f0_7 sf 0 3 rm .128(ML)A 0 -3 rm f3_12 sf .328 .033(\), then the p-values of the )J 291 389 :M f1_12 sf (c)S 298 384 :M f3_7 sf (2)S 302 389 :M f3_12 sf .352 .035( likelihood ratio test for R)J 428 392 :M f3_7 sf (1)S 432 389 :M f3_12 sf .375 .037( and R)J f3_7 sf 0 3 rm (2)S 0 -3 rm 468 389 :M f3_12 sf .387 .039( will)J 59 405 :M .035 .004(be identical. Thus the data cannot help us distinguish between R)J 368 408 :M f3_7 sf (1)S 372 405 :M f3_12 sf .04 .004( and R)J f3_7 sf 0 3 rm (2)S 0 -3 rm 407 405 :M f3_12 sf .043 .004(. This is a kind of)J 59 421 :M (\322causal underidentification.\323)S 81 437 :M 1.667 .167(The slogan that \322correlation is not causation\323 expresses the idea that from data)J 59 453 :M -.002(including only the existence of a single significant correlation between variables A and B,)A 59 469 :M .405 .04(the causal structure governing A and B is underidentified. That is, a correlation between)J 59 485 :M .321 .032(two variables A and B could be produced by A causing B, B causing A, a latent variable)J 59 501 :M .425 .042(that causes both A and B, or some combination of these. But just as a single example of)J 59 517 :M .04 .004(an underidentified SEM does not show that parameters are always underidentified, or that)J 59 533 :M .053 .005(parameter estimation is always impossible or useless, the existence of a single example of)J 59 549 :M .319 .032(covariance equivalent SEMs does not show that specification search for SEMs is always)J 59 565 :M (impossible or useless.)S 81 581 :M 2.292 .229(For some SEMs, certain parameters may be identifiable while others are not.)J 59 597 :M 2.491 .249(Similarly, certain features of an RSEM R might be common to every R\325 that is)J 59 613 :M .766 .077(covariance equivalent to R. We will show examples in which a covariance equivalence)J 59 629 :M .09 .009(class of RSEMs )J f4_12 sf .032(all)A 152 629 :M f3_12 sf .1 .01( share the feature that some variable A is a \(possibly indirect\) cause of)J 59 645 :M 2.348 .235(B; we will show other examples in which )J f4_12 sf .814(none)A f3_12 sf 2.386 .239( of the members of a covariance)J 59 661 :M .36 .036(equivalence class is A \(even indirectly\) a cause of B. As explained in detail in section 4,)J 59 677 :M 1.021 .102(for various special cases, necessary and sufficient, or necessary conditions for various)J endp %%Page: 9 9 %%BeginPageSetup initializepage (peter; page: 9 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 261 709 28 16 rC 283 722 :M f3_12 sf (9)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf 2.354 .235(kinds of statistical equivalence are known. Because of the problem of covariance)J 59 70 :M .754 .075(equivalence, the output of our algorithms will generally not be a single RSEM. Instead)J 59 86 :M 1.997 .2(the output will be an object that represents a )J f4_12 sf .764(class)A 328 86 :M f3_12 sf 2.166 .217( of RSEMs consistent with the)J 59 102 :M .773 .077(assumptions made and which marks those features shared by all of the members of the)J 59 118 :M (RSEMs output.)S 81 134 :M 1.208 .121(By outputting a representation of covariance equivalence class of RSEMs, rather)J 59 150 :M 1.349 .135(than a single SEM, the PC algorithm addresses the problem that there may be many)J 59 166 :M .351 .035(different structural equation models that are compatible with background knowledge and)J 59 182 :M .22 .022(fit the data equally well \(as measured by a p-value, for example\). However, it may be the)J 59 198 :M .151 .015(case that there are SEMs which are not covariance equivalent, but nonetheless fit the data)J 59 214 :M f4_12 sf (almost)S 91 214 :M f3_12 sf 1.568 .157( equally well; ideally an algorithm should output all such models, rather than)J 59 230 :M .188 .019(simply choose the best. This problem could be addressed by outputting multiple patterns,)J 59 246 :M .862 .086(rather than a single pattern. Devising an algorithm \(or modifying the PC algorithm\) to)J 59 262 :M 1.958 .196(output representations of all models that fit the data well and are compatible with)J 59 278 :M (background knowledge is an important area of future research.)S 59 306 :M f4_12 sf (1)S 65 306 :M (.)S 68 306 :M (3)S 74 306 :M (.)S 77 306 :M (2)S 83 306 :M ( )S 90 306 :M (Consistency and Correctness)S 81 325 :M f3_12 sf .354 .035(A SEM parameter estimation algorithm takes as input a sample covariance matrix )J 484 325 :M f0_12 sf (S)S 59 341 :M f3_12 sf 1.893 .189(and distributional assumptions about the population from which )J 396 341 :M f0_12 sf (S)S 403 341 :M f3_12 sf 2.291 .229( was drawn, and)J 59 357 :M 1.28 .128(produces as output an estimate )J 220 357 :M f2_12 sf .812(q)A f3_7 sf 0 3 rm .952 .095(est )J 0 -3 rm 238 357 :M f3_12 sf 1.205 .12(of the population parameters )J 387 357 :M f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 404 357 :M f3_12 sf 1.335 .134(. If the measured)J 59 374 :M .024 .002(variables are indeed multivariate normal, and the specified model holds in the population,)J 59 390 :M .564 .056(then a)J 89 390 :M .508 .051(symptotically, as the sample size goes to infinity, the sampling distribution of )J 474 390 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 59 407 :M f3_12 sf .783 .078(goes to N\()J 111 407 :M f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 128 407 :M f3_12 sf .947 .095(, J)J 141 402 :M f3_7 sf (-1)S 147 407 :M f3_12 sf <28>S 151 407 :M f2_12 sf .238(q)A f3_12 sf .654 .065(\)\), where J\()J f2_12 sf .238(q)A f3_12 sf .685 .068(\) is the Fisher information matrix \(cf. Tanner, 1993, p.)J 59 424 :M (16\).)S 78 424 :M .864 .086( So )J 99 424 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 116 427 :M .994 .099( )J 119 424 :M f3_12 sf .22 .022(is a )J f0_12 sf .16(consistent)A 191 424 :M f3_12 sf .643 .064( estimator in the sense that as the sample size grows without)J 59 441 :M (bound the difference between )S f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 221 444 :M ( )S 223 441 :M f3_12 sf (and )S f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 260 444 :M ( )S 262 441 :M f3_12 sf (will, with probability 1, converge to zero.)S 81 458 :M .088 .009(The PC algorithm takes as input a sample covariance matrix )J f0_12 sf (S)S 381 458 :M f3_12 sf .094 .009(, the assumption that )J 484 458 :M f0_12 sf (S)S 59 474 :M f3_12 sf 1.346 .135(is drawn from a multivariate normal population described by an RSEM R)J f3_7 sf 0 3 rm .359(pop)A 0 -3 rm 447 474 :M f3_12 sf 1.671 .167( with no)J 59 490 :M .296 .03(latent variables or correlated errors, and produces as output a )J 360 490 :M f0_12 sf (pattern)S 398 490 :M f3_12 sf .302 .03( which represents a)J 59 506 :M .992 .099(class of RSEMs that are covariance equivalent to R)J 319 509 :M f3_7 sf .587 .059(pop )J f3_12 sf 0 -3 rm 1.242 .124(\(see section )J 0 3 rm 395 506 :M (4\).)S 408 501 :M f3_7 sf (6)S 411 506 :M f3_12 sf .754 .075( Let )J f0_12 sf .971(M)A f0_7 sf 0 3 rm .4(PC)A 0 -3 rm f3_12 sf 1.055 .105( be the)J 59 522 :M .803 .08(pattern output by the PC algorithm, and )J f0_12 sf .517(M)A f0_7 sf 0 3 rm .172(pop)A 0 -3 rm f3_12 sf .735 .074( be the pattern that represents the class of)J 59 538 :M 1.008 .101(RSEMs covariance equivalent to R)J 235 541 :M f3_7 sf (pop)S 246 538 :M f3_12 sf 1.167 .117(. Since there is no obvious metric to express the)J 59 554 :M 2.345 .235(difference between )J f0_12 sf 1.095(M)A f0_7 sf 0 3 rm .893 .089(PC )J 0 -3 rm 187 554 :M f3_12 sf 1.817 .182(and )J f0_12 sf 1.417(M)A f0_7 sf 0 3 rm .471(pop)A 0 -3 rm f3_12 sf 2.267 .227(, we will not follow the analogy with parameter)J 59 570 :M .106 .011(estimation and say that the PC algorithm is consistent. We can, however, state and prove)J 59 586 :M .329 .033(a closely related property which we call correctness. The PC algorithm is )J f0_12 sf .415 .042(correct )J f3_12 sf .339 .034(in the)J 59 602 :M .655 .065(following sense: if the Causal Independence and Faithfulness assumptions are satisfied,)J 59 618 :M .842 .084(then, as the sample grows without bound, the probability that )J 370 618 :M f0_12 sf .527(M)A f0_7 sf 0 3 rm .396 .04(PC )J 0 -3 rm f3_12 sf (=)S 400 621 :M f0_7 sf .165 .017( )J f0_12 sf 0 -3 rm 1.177(M)A 0 3 rm f0_7 sf 1.042 .104(pop )J 429 618 :M f3_12 sf .645 .065(converges to)J 59 634 :M (one.)S 79 629 :M f3_7 sf (7)S 59 650 :M f3_12 sf ( )S 59 647.48 -.48 .48 203.48 647 .48 59 647 @a 81 659 :M f3_6 sf (6)S 84 663 :M f3_10 sf .57 .057( In some cases the input to the PC algorithm is not consistent with the assumptions made. In these)J 59 675 :M (cases it is possible that the output of the PC algorithm is not strictly a pattern.)S 81 683 :M f3_6 sf (7)S 84 687 :M f3_10 sf .758 .076( The PC algorithm performs a series of statistical tests of zero partial correlations; the asymptotic)J endp %%Page: 10 10 %%BeginPageSetup initializepage (peter; page: 10 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (10)S gR gS 0 0 552 730 rC 59 54 :M f4_12 sf (1)S 65 54 :M (.)S 68 54 :M (3)S 74 54 :M (.)S 77 54 :M (3)S 83 54 :M ( )S 90 54 :M (Sampling Distribution)S 81 73 :M f3_12 sf .901 .09(In a SEM, to estimate the sampling distribution of )J 337 73 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 354 76 :M .086 .009( )J f3_12 sf 0 -3 rm 1.063 .106(on finite samples, we have)J 0 3 rm 59 89 :M .229 .023(two choices. First, if the sample size is reasonably large we can use )J f2_12 sf .088(q)A f3_7 sf 0 3 rm .147(ML)A 0 -3 rm 407 89 :M f3_12 sf .26 .026( as an estimate of)J 59 105 :M f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 76 105 :M f3_12 sf 1.272 .127(, and then use the asymptotic theory described above \()J f2_12 sf .439(q)A f3_7 sf 0 3 rm .738(ML)A 0 -3 rm 373 108 :M 2.165 .217( )J 377 105 :M f3_12 sf 1.968 .197(~ )J 389 105 :M (N\()S 402 105 :M f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 419 105 :M f3_12 sf 1.804 .18(, J)J 432 100 :M f3_7 sf (-1)S 438 105 :M f3_12 sf <28>S 442 105 :M f2_12 sf (q)S f3_12 sf <2929>S 456 105 :M 1.665 .167(\) as an)J 59 121 :M .68 .068(estimate of the sampling distribution of )J 258 121 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 275 121 :M f3_12 sf .679 .068(. Second, we can approximate the sampling)J 59 137 :M .435 .044(distribution of )J f2_12 sf .137(q)A f3_7 sf 0 3 rm .224 .022(ML )J 0 -3 rm 150 137 :M f3_12 sf .387 .039(empirically by Monte Carlo techniques \(Boomsma, 1982\). We can do)J 59 153 :M 1.572 .157(this by assuming )J 150 153 :M f2_12 sf (S)S f3_12 sf <28>S 161 153 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 178 153 :M f3_12 sf <29>S 182 153 :M 2.191 .219( = )J 200 153 :M f2_12 sf (S)S f3_12 sf <28>S 211 153 :M f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 228 153 :M f3_12 sf <29>S 232 153 :M 1.487 .149(. We can then repeatedly sample from )J f2_12 sf .61(S)A f3_12 sf <28>S 445 153 :M f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 462 153 :M f3_12 sf <29>S 466 153 :M 1.721 .172(, and)J 59 169 :M .16 .016(calculate the ML estimate for each sample \(Figure 6\). Although for small N the sampling)J 59 185 :M .187 .019(distribution of )J 131 185 :M f2_12 sf .069(q)A f3_7 sf 0 3 rm .104 .01(ML )J 0 -3 rm f3_12 sf .207 .021(is not multivariate normal \(Boomsma, 1982\), it can be still be usefully)J 59 201 :M (summarized by the standard deviation \(standard errors\) and the mean.)S 282 320 51 23 rC 282 333 :M ( )S 285 333 :M ( )S 288 333 :M f2_12 sf (.)S 292 333 :M ( )S 295 333 :M (.)S 299 333 :M ( )S 302 333 :M (.)S 306 333 :M ( )S 309 333 :M (.)S 313 333 :M ( )S 316 333 :M (.)S 320 333 :M ( )S 323 333 :M (.)S 282 350 :M (.)S gR gS 350 320 51 23 rC 350 333 :M f3_12 sf ( )S 353 333 :M ( )S 356 333 :M f2_12 sf (q)S 363 336 :M f0_7 sf (M)S 370 336 :M (L)S 375 333 :M f3_12 sf <28>S 379 333 :M f0_12 sf (S)S 387 336 :M f0_7 sf (n)S 391 333 :M f3_12 sf <29>S gR gS 214 320 51 23 rC 214 333 :M f3_12 sf ( )S 217 333 :M ( )S 220 333 :M f2_12 sf (q)S 227 336 :M f0_7 sf (M)S 234 336 :M (L)S 239 333 :M f3_12 sf <28>S 243 333 :M f0_12 sf (S)S 251 336 :M f0_7 sf (2)S 255 333 :M f3_12 sf <29>S gR gS 149 321 51 23 rC 149 334 :M f3_12 sf ( )S 152 334 :M ( )S 155 334 :M f2_12 sf (q)S 162 337 :M f0_7 sf (M)S 169 337 :M (L)S 174 334 :M f3_12 sf <28>S 178 334 :M f0_12 sf (S)S 186 337 :M f0_7 sf (1)S 190 334 :M f3_12 sf <29>S gR gS 352 281 31 20 rC 352 290 :M f3_12 sf ( )S 355 290 :M ( )S 358 290 :M f0_12 sf (S)S 366 293 :M f0_7 sf (n)S gR gS 222 281 31 20 rC 222 290 :M f3_12 sf ( )S 225 290 :M ( )S 228 290 :M f0_12 sf (S)S 236 293 :M f0_7 sf (2)S gR gS 160 282 31 20 rC 160 291 :M f3_12 sf ( )S 163 291 :M ( )S 166 291 :M f0_12 sf (S)S 174 294 :M f0_7 sf (1)S gR gS 283 282 51 23 rC 283 295 :M f3_12 sf ( )S 286 295 :M ( )S 289 295 :M f2_12 sf (.)S 293 295 :M ( )S 296 295 :M (.)S 300 295 :M ( )S 303 295 :M (.)S 307 295 :M ( )S 310 295 :M (.)S 314 295 :M ( )S 317 295 :M (.)S 321 295 :M ( )S 324 295 :M (.)S 283 312 :M (.)S gR gS 252 221 45 24 rC 252 234 :M f3_12 sf ( )S 255 234 :M ( )S 258 234 :M f2_12 sf (S)S 266 234 :M f3_12 sf <28>S 270 234 :M f2_12 sf (q)S 277 237 :M f0_7 sf (p)S 281 237 :M (o)S 285 237 :M (p)S 289 234 :M f3_12 sf <29>S gR gS 148 220 254 125 rC -.75 -.75 189.75 282.75 .75 .75 257 243 @b -.75 -.75 244.75 279.75 .75 .75 269 244 @b 299 240.75 -.75 .75 344.75 279 .75 299 240 @a -.75 -.75 177.75 319.75 .75 .75 177 303 @b -.75 -.75 239.75 317.75 .75 .75 239 301 @b -.75 -.75 369.75 320.75 .75 .75 369 301 @b gR gS 0 0 552 730 rC 86 370 :M f0_12 sf (Figure )S 123 370 :M (6. Monte Carlo approximation of the sampling distribution for )S f2_12 sf (q)S f0_7 sf 0 3 rm (ML)S 0 -3 rm 81 398 :M f3_12 sf .202 .02(On samples from a given model with specified parameters, the sampling distribution)J 59 414 :M .062 .006(of )J f0_12 sf .106(M)A f0_7 sf 0 3 rm .044(PC)A 0 -3 rm f3_12 sf .16 .016( is well defined. However, )J 227 414 :M f0_12 sf .101(M)A f0_7 sf 0 3 rm .042(PC)A 0 -3 rm f3_12 sf .13 .013( is not a vector of real valued parameters as )J f2_12 sf .056(q)A f3_7 sf 0 3 rm .094(ML)A 0 -3 rm 477 414 :M f3_12 sf .171 .017( is,)J 59 430 :M .94 .094(but rather a graphical object \(see section )J 267 430 :M .86 .086(5.1.1\) that represents an equivalence class of)J 59 446 :M 2.493 .249(RSEMs. Hence )J 144 446 :M f0_12 sf 1.471(M)A f0_7 sf 0 3 rm 1.106 .111(PC )J 0 -3 rm f3_12 sf 2.482 .248(is a categorical variable with no meaningful ordering of the)J 59 462 :M .506 .051(categories. Thus the variance and mean are not very useful summaries of features of the)J 59 478 :M .118 .012(distribution. We do not know how to calculate an analytic approximation of the sampling)J 59 494 :M .322 .032(distribution for )J 136 494 :M f0_12 sf .207(M)A f0_7 sf 0 3 rm .085(PC)A 0 -3 rm f3_12 sf .337 .034( on finite samples. But we can apply empirical techniques parallel to)J 59 510 :M 1.781 .178(those mentioned above for ML parameter estimation. To approximate the sampling)J 59 526 :M (distribution for )S f0_12 sf (M)S f0_7 sf 0 3 rm (PC )S 0 -3 rm f3_12 sf (on finite samples, consider )S 288 526 :M (Figure 7, which is analogous to Figure 6.)S 59 662 :M ( )S 59 659.48 -.48 .48 491.48 659 .48 59 659 @a 59 675 :M f3_10 sf .367 .037(results assume that we systematically lower the significance level as the sample size increases, in order to)J 59 687 :M (decrease the probabilities of both type I and type II errors to zero.)S endp %%Page: 11 11 %%BeginPageSetup initializepage (peter; page: 11 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (11)S gR gS 149 136 63 22 rC 149 149 :M f3_12 sf ( )S 152 149 :M ( )S 155 149 :M f2_12 sf (M)S 166 151 :M f9_8 sf (P)S 171 151 :M (C)S 177 149 :M f3_12 sf <28>S 181 149 :M f0_12 sf (S)S 189 152 :M f0_7 sf (1)S 193 149 :M f3_12 sf <29>S gR gS 282 135 51 23 rC 282 148 :M f3_12 sf ( )S 285 148 :M ( )S 288 148 :M f2_12 sf (.)S 292 148 :M ( )S 295 148 :M (.)S 299 148 :M ( )S 302 148 :M (.)S 306 148 :M ( )S 309 148 :M (.)S 313 148 :M ( )S 316 148 :M (.)S 320 148 :M ( )S 323 148 :M (.)S 282 165 :M (.)S gR gS 350 135 51 23 rC 350 148 :M f3_12 sf ( )S 353 148 :M ( )S 356 148 :M f2_12 sf (M)S 367 150 :M f9_8 sf (P)S 372 150 :M (C)S 378 148 :M f3_12 sf <28>S 382 148 :M f0_12 sf (S)S 390 151 :M f0_7 sf (n)S 394 148 :M f3_12 sf <29>S gR gS 214 135 51 23 rC 214 148 :M f3_12 sf ( )S 217 148 :M ( )S 220 148 :M f2_12 sf (M)S 231 150 :M f9_8 sf (P)S 236 150 :M (C)S 242 148 :M f3_12 sf <28>S 246 148 :M f0_12 sf (S)S 254 151 :M f0_7 sf (2)S 258 148 :M f3_12 sf <29>S gR gS 278 93 51 23 rC 278 106 :M f3_12 sf ( )S 281 106 :M ( )S 284 106 :M f2_12 sf (.)S 288 106 :M ( )S 291 106 :M (.)S 295 106 :M ( )S 298 106 :M (.)S 302 106 :M ( )S 305 106 :M (.)S 309 106 :M ( )S 312 106 :M (.)S 316 106 :M ( )S 319 106 :M (.)S 278 123 :M (.)S gR gS 347 92 31 20 rC 347 101 :M f3_12 sf ( )S 350 101 :M ( )S 353 101 :M f0_12 sf (S)S 361 104 :M f0_7 sf (n)S gR gS 217 92 31 20 rC 217 101 :M f3_12 sf ( )S 220 101 :M ( )S 223 101 :M f0_12 sf (S)S 231 104 :M f0_7 sf (2)S gR gS 155 93 31 20 rC 155 102 :M f3_12 sf ( )S 158 102 :M ( )S 161 102 :M f0_12 sf (S)S 169 105 :M f0_7 sf (1)S gR gS 251 42 45 24 rC 251 55 :M f3_12 sf ( )S 254 55 :M ( )S 257 55 :M f2_12 sf (S)S 265 55 :M f3_12 sf <28>S 269 55 :M f2_12 sf (q)S 276 58 :M f0_7 sf (p)S 280 58 :M (o)S 284 58 :M (p)S 288 55 :M f3_12 sf <29>S gR gS 148 41 254 118 rC -.75 -.75 180.75 94.75 .75 .75 256 64 @b -.75 -.75 242.75 91.75 .75 .75 268 65 @b 298 61.75 -.75 .75 348.75 93 .75 298 61 @a -.75 -.75 173.75 136.75 .75 .75 173 114 @b -.75 -.75 235.75 135.75 .75 .75 235 110 @b -.75 -.75 361.75 135.75 .75 .75 361 112 @b gR gS 0 0 552 730 rC 94 184 :M f0_12 sf (Figure )S 131 184 :M (7. Monte Carlo approximation to sampling distribution for M)S 447 187 :M f0_7 sf (PC)S 81 212 :M f3_12 sf .38 .038(A slight disanalogy occurs in estimating )J 281 212 :M f2_12 sf (S)S f3_12 sf <28>S 292 212 :M f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 309 212 :M f3_12 sf <29>S 313 212 :M .373 .037(. In the maximum likelihood setting,)J 59 228 :M f2_12 sf (S)S f3_12 sf <28>S 70 228 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 87 228 :M f3_12 sf <29>S 91 228 :M .171 .017( is used as an estimate of )J 215 228 :M f2_12 sf (S)S f3_12 sf <28>S 226 228 :M f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 243 228 :M f3_12 sf <29>S 247 228 :M .172 .017(. To obtain )J 302 228 :M f2_12 sf (S)S f3_12 sf <28>S 313 228 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 330 228 :M f3_12 sf <29>S 334 228 :M .153 .015( from our sample )J f0_12 sf (S)S 427 228 :M f3_12 sf .195 .02( and )J 451 228 :M f0_12 sf .066(M)A f0_7 sf 0 3 rm .027(PC)A 0 -3 rm f3_12 sf .089 .009(, we)J 59 244 :M .362 .036(can pick an arbitrary member M)J 217 247 :M f3_7 sf (i)S 219 244 :M f3_12 sf .339 .034( of the equivalence class of RSEMs represented by )J f0_12 sf .218(M)A f0_7 sf 0 3 rm .179(PC)A 0 -3 rm 59 260 :M f3_12 sf .013 .001(and then calculate )J 149 260 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 166 260 :M f3_12 sf ( for M)S 197 263 :M f3_7 sf (i)S 199 260 :M f3_12 sf ( and )S f0_12 sf (S)S 229 260 :M f3_12 sf .011 .001(. \(The resulting covariance matrix )J f2_12 sf (S)S f0_7 sf 0 3 rm (Mi)S 0 -3 rm 411 260 :M f3_12 sf <28>S 415 260 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 432 260 :M f3_12 sf <29>S 436 260 :M .015 .002( is the same)J 59 276 :M 1.212 .121(regardless of which member M)J f3_7 sf 0 3 rm (i)S 0 -3 rm 219 276 :M f3_12 sf .688 .069( of )J f0_12 sf 1.17(M)A f0_7 sf 0 3 rm .482(PC)A 0 -3 rm f3_12 sf 1.531 .153( we choose.\) We can then use )J 419 276 :M f2_12 sf (S)S f0_7 sf 0 3 rm (Mi)S 0 -3 rm 435 276 :M f3_12 sf <28>S 439 276 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 456 276 :M f3_12 sf <29>S 460 276 :M 1.67 .167( as an)J 59 292 :M (estimate of )S 115 292 :M f2_12 sf (S)S f3_12 sf <28>S 126 292 :M f2_12 sf (q)S f3_7 sf 0 3 rm (pop)S 0 -3 rm 143 292 :M f3_12 sf <29>S 147 292 :M (.)S 59 320 :M f4_12 sf (1)S 65 320 :M (.)S 68 320 :M (3)S 74 320 :M (.)S 77 320 :M (4)S 83 320 :M ( )S 90 320 :M (Practical Reliability)S 81 339 :M f3_12 sf .314 .031(Finally, we want to know if the estimation procedure is reliable in practice. )J 452 339 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 469 339 :M f3_12 sf .357 .036( has,)J 59 355 :M .577 .058(by definition, the property that there is no )J f2_12 sf .222(q)A f3_7 sf 0 3 rm .119 .012(i )J 0 -3 rm 279 355 :M f3_12 sf cF f1_12 sf .076A sf .765 .076( )J 290 355 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 307 358 :M .841 .084( )J 310 355 :M f3_12 sf .526 .053(s.t. p\()J 338 355 :M f0_12 sf (S)S 345 355 :M f3_12 sf (|)S f2_12 sf (q)S f3_7 sf 0 3 rm (i)S 0 -3 rm 355 355 :M f3_12 sf .518 .052(\) > p\()J f0_12 sf (S)S 390 355 :M f3_12 sf (|)S f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 409 355 :M f3_12 sf .561 .056(\). On samples of)J 59 371 :M .242 .024(realistic size, however, iterative procedures that search the parameter space such as those)J 59 387 :M .113 .011(implemented in LISREL \(J\232reskog, 1993\) and EQS \(Bentler, 1995\) cannot guarantee that)J 59 403 :M 2.025 .203(they will find )J 136 403 :M f2_12 sf (q)S f3_7 sf 0 3 rm (ML)S 0 -3 rm 153 403 :M f3_12 sf 1.899 .19(. They must begin, for example, from some starting point in the)J 59 419 :M 1.501 .15(parameter space and hill climb, and the likelihood surface might have local maxima)J 59 435 :M .089 .009(\(Scheines, Hoijtink, & Boomsma, 1995\). We can investigate the practical reliability of an)J 59 451 :M 1.601 .16(ML estimation procedure at a given sample size by 1\) drawing an RSEM R from a)J 59 467 :M .723 .072(distribution over RSEMs, 2\) drawing a parameterization )J f2_12 sf .207(q)A f0_7 sf 0 3 rm (i)S 0 -3 rm 349 467 :M f3_12 sf .852 .085( from a distribution over the)J 59 483 :M .437 .044(parameters of R to give a population )J f2_12 sf .186(S)A f0_7 sf 0 3 rm .133(R)A 0 -3 rm f3_12 sf <28>S 258 483 :M f2_12 sf (q)S f3_7 sf 0 3 rm (i)S 0 -3 rm 266 483 :M f3_12 sf <29>S 270 483 :M .514 .051(, and 3\) drawing a sample )J 402 483 :M f0_12 sf (S)S 409 483 :M f3_12 sf .362 .036( from )J f2_12 sf .228(S)A f0_7 sf 0 3 rm .162(R)A 0 -3 rm f3_12 sf <28>S 455 483 :M f2_12 sf (q)S f3_7 sf 0 3 rm (i)S 0 -3 rm 463 483 :M f3_12 sf <29>S 467 483 :M .56 .056(. We)J 59 499 :M .644 .064(can then use )J f0_12 sf (S)S 131 499 :M f3_12 sf .65 .065( as the input to the implemented estimator E, finally comparing )J 449 499 :M f2_12 sf (S)S f0_7 sf 0 3 rm (R)S 0 -3 rm f3_12 sf <28>S 465 499 :M f2_12 sf (q)S f3_7 sf 0 3 rm (E)S 0 -3 rm f3_12 sf <29>S 479 502 :M f3_7 sf 1.022 .102( )J 482 499 :M f3_12 sf (to)S 59 515 :M f2_12 sf (S)S f0_7 sf 0 3 rm (R)S 0 -3 rm f3_12 sf <28>S 75 515 :M f2_12 sf (q)S f3_7 sf 0 3 rm (i)S 0 -3 rm 83 515 :M f3_12 sf <29>S 87 515 :M ( or just )S f2_12 sf (q)S f3_7 sf 0 3 rm (E )S 0 -3 rm f3_12 sf (to )S f2_12 sf (q)S f3_7 sf 0 3 rm (i)S 0 -3 rm 155 515 :M f3_12 sf (.)S 81 531 :M .389 .039(We can investigate the reliability of model specification algorithms in an analogous)J 59 547 :M .341 .034(way. The PC algorithm performs statistical tests of vanishing partial correlations, and if)J 59 563 :M 1.809 .181(it cannot reject the null hypothesis at a significance level set by the user, then the)J 59 579 :M 1.385 .139(procedure )J 113 579 :M 1.41 .141(accepts the null hypothesis)J 250 579 :M 2.393 .239(. )J 259 579 :M 1.616 .162(If the null hypothesis is wrongly accepted or)J 59 595 :M .236 .024(rejected, the output of the procedure can be incorrect. On finite samples, the reliability of)J 59 611 :M 1.077 .108(the model specification algorithms depends upon the power of the statistical tests, the)J 59 627 :M -.002(significance level used in the tests, the distribution over the models, and the parameters of)A 59 643 :M (the models.)S 81 659 :M 1.19 .119(In \(Spirtes, et al., 1993\), and in \(Scheines, et al., 1994\), we report on systematic)J 59 675 :M .18 .018(Monte Carlo simulation studies to approximate features of the sampling distribution over)J endp %%Page: 12 12 %%BeginPageSetup initializepage (peter; page: 12 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (12)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf 1.924 .192(the PC algorithm \(and a variety of our other RSEM specification algorithm\) by 1\))J 59 70 :M .607 .061(drawing an RSEM R from a distribution over RSEMs, 2\) drawing a parameterization )J 483 70 :M f2_12 sf (q)S f0_7 sf 0 3 rm (i)S 0 -3 rm 59 86 :M f3_12 sf 2.718 .272(from a distribution over the parameters for R, 3\) drawing a sample )J 432 86 :M f0_12 sf (S)S 439 86 :M f3_12 sf 3.009 .301( from the)J 59 102 :M .855 .085(multivariate normal population )J f2_12 sf .243(S)A f0_7 sf 0 3 rm .173(R)A 0 -3 rm f3_12 sf <28>S 233 102 :M f2_12 sf (q)S f3_7 sf 0 3 rm (i)S 0 -3 rm 241 102 :M f3_12 sf <29>S 245 102 :M 1.244 .124(, and then using )J f0_12 sf (S)S 338 102 :M f3_12 sf 1.295 .13( as input to the PC algorithm.)J 59 118 :M (Finally, we compare )S f0_12 sf (M)S f0_7 sf 0 3 rm (PC)S 0 -3 rm f0_12 sf ( )S f3_12 sf (to)S f0_12 sf ( M)S f0_7 sf 0 3 rm (pop)S 0 -3 rm f3_12 sf (.)S 81 134 :M 1.221 .122(These tests indicate that the PC algorithm is reliable with respect to determining)J 59 150 :M .759 .076(which variables are adjacent in the population causal graph as long as the sample sizes)J 59 166 :M .63 .063(are on the order of 500 and the population RSEM is not highly interconnected \(i.e. that)J 59 182 :M .65 .065(not everything is either a cause or an effect of everything else\). For example, at sample)J 59 198 :M .565 .056(size 500 for sparsely connected RSEMs with 50 variables, the PC algorithm incorrectly)J 59 214 :M .052 .005(hypothesized an adjacency less than once in 1,000 times such a mistake was possible, and)J 59 230 :M 1.674 .167(incorrectly omitted an adjacency approximately 10% of the time, with the accuracy)J 59 246 :M .604 .06(improving as the sample size grows \(see page 155 of Spirtes, et. al., 1993\). We should)J 59 262 :M 1.536 .154(note, however, that these simulation tests satisfied all the distributional assumptions)J 59 278 :M .196 .02(underlying the algorithm and did not allow parameter values close to 0. We have not yet)J 59 294 :M (systematically explored the effect of small violations of these or other assumptions.)S 59 322 :M f4_12 sf (1)S 65 322 :M (.)S 68 322 :M (4)S 74 322 :M ( )S 81 322 :M (Difficulty of Search)S 81 344 :M f3_12 sf 2.543 .254(In practice, model specification problems are very difficult for \(at least\) the)J 59 360 :M (following reasons:)S 82 376 :M f1_12 sf S 87 376 :M 9 .9( )J 100 376 :M f3_12 sf 1.101 .11(Data sets may fail to record variables \(confounders\) that produce associations)J 100 392 :M (among recorded variables.)S 82 408 :M f1_12 sf S 87 408 :M 10 1( )J 100 408 :M f3_12 sf .228 .023(When no limitation is placed on the number of "latent variables," the number of)J 100 424 :M (alternative SEMs may be literally infinite.)S 82 440 :M f1_12 sf S 87 440 :M 10 1( )J 100 440 :M f3_12 sf .541 .054(Many distinct SEMs may produce the same, or nearly the same distributions of)J 100 456 :M (recorded variables.)S 82 472 :M f1_12 sf S 87 472 :M 9 .9( )J 100 472 :M f3_12 sf .822 .082(Natural and social populations may be mixtures of SEMs with different causal)J 100 488 :M (graphs.)S 82 504 :M f1_12 sf S 87 504 :M 9 .9( )J 100 504 :M f3_12 sf 1.058 .106(Values of quantities recorded for some units in a data set may be missing for)J 100 520 :M (other units.)S 82 536 :M f1_12 sf S 87 536 :M 8 .8( )J 100 536 :M f3_12 sf 1.457 .146(There may be \322selection bias\323--that is, a measured variable may be causally)J 100 552 :M (connected to whether an individual is or is not included in the sample.)S 82 568 :M f1_12 sf S 87 568 :M ( )S 100 568 :M f3_12 sf (The causal structure may involve feedback loops.)S 82 584 :M f1_12 sf S 87 584 :M ( )S 100 584 :M f3_12 sf (The functional relations between causes and their effects may be non-linear.)S 82 600 :M f1_12 sf S 87 600 :M 7 .7( )J 100 600 :M f3_12 sf 2.302 .23(Actual distributions may not be closely approximated by any well known)J 100 616 :M (probability distributions.)S endp %%Page: 13 13 %%BeginPageSetup initializepage (peter; page: 13 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (13)S gR gS 0 0 552 730 rC 81 54 :M f3_12 sf 1.316 .132(In the last fifteen years a movement in computer science and statistics has made)J 59 70 :M .18 .018(theoretical progress on a number of these issues,)J 294 65 :M f3_7 sf (8)S 297 70 :M f3_12 sf .195 .02( progress that has led to computer based)J 59 86 :M 1.298 .13(methods to aid in model specification. These results and the methods that implement)J 59 102 :M .808 .081(them appear to be little known and rarely used in the social science communities. This)J 59 118 :M .53 .053(paper is an introductory description of some of the more important theoretical ideas and)J 59 134 :M .052 .005(of some of the computational procedures that have arisen out of these discoveries, offered)J 59 150 :M .176 .018(in the hope that social and behavioral scientists will make more use of these methods and)J 59 166 :M .907 .091(help to improve them. \(Mathematical details and related results can be found in Pearl,)J 59 182 :M (1988; Spirtes, et al., 1993; Scheines, et al., 1994\).)S 59 210 :M f4_12 sf (1)S 65 210 :M (.)S 68 210 :M (5)S 74 210 :M ( )S 81 210 :M (Search Procedures)S 81 232 :M f3_12 sf 1.741 .174(Two approaches to RSEM specification have been pursued in the statistics and)J 59 248 :M .395 .039(computer science literature. The first focuses on searching for the RSEM or RSEMs that)J 59 264 :M .587 .059(maximize some score.)J 168 259 :M f3_7 sf (9)S 171 264 :M f3_12 sf .741 .074( The second approach focuses on searching for the RSEMs that)J 59 280 :M .853 .085(satisfy a set of constraints judged to hold in the population \(e.g., Spirtes, et al., 1993\).)J 59 296 :M 1.222 .122(\(See Richardson \(1996\) for a correct algorithm for searching for non-recursive SEMs)J 59 312 :M .201 .02(without latent variables.\) Searches based on maximizing a score have been developed for)J 59 328 :M .207 .021(RSEMs with no latent variables \(e.g., Geiger & Heckerman, 1994; Cooper & Herskovits,)J 59 344 :M 1.055 .105(1992\); typically they are either stepwise forward \(they add edges\), stepwise backward)J 59 360 :M .82 .082(\(they take away edges\), or some combination of stepwise forward and backward. Most)J 59 376 :M 1.602 .16(regression searches are of this type, although they are restricted to searching a very)J 59 392 :M 1.282 .128(restricted class of RSEMs. The \322modification index\323 searches based on the Lagrange)J 59 408 :M 1.855 .186(Multiplier statistic \(Bentler, 1986; Kaplan, 1989, 1990; J)J 356 408 :M 2.244 .224(\232reskog & S)J 423 408 :M 1.603 .16(\232rbom, 1993;)J 59 424 :M (S)S 66 424 :M 2.221 .222(\232rbom, 1989\) in LISREL and EQS are restricted versions of this strategy. They)J 59 440 :M .81 .081(typically begin with a given SEM M and perform a stepwise forward search \(EQS can)J 59 456 :M .493 .049(also perform a stepwise backward search\). One difficulty with searches that maximize a)J 59 472 :M .689 .069(score is that no proofs of correctness are yet available. A more difficult problem is that)J 59 488 :M 1.417 .142(there are as of now no feasible score-maximization searches that include SEMs with)J 59 504 :M .885 .089(latent variables. The modification index searches cannot suggest adding or removing a)J 59 520 :M 1.124 .112(latent variable, for example. Also, these searches output a single SEM, rather than an)J 59 536 :M .155 .015(equivalence class of SEMs. Another search strategy based upon maximizing a score is to)J 59 552 :M 1.121 .112(search not RSEMs themselves, but covariance equivalence classes of RSEMs \(Spirtes)J 59 568 :M (and Meek, 1995\).)S 59 590 :M ( )S 59 587.48 -.48 .48 203.48 587 .48 59 587 @a 81 599 :M f3_6 sf (8)S 84 603 :M f3_10 sf .46 .046( Some of this literature is published in the annual proceedings of the conferences on Uncertainty in)J 59 615 :M .968 .097(Artificial Intelligence, Knowledge Discovery in Data Bases, and the bi-annual conference on Artificial)J 59 627 :M .366 .037(Intelligence and Statistics. Examples of important papers in this tradition include: \(Buntine 1991; Cooper)J 59 639 :M .83 .083(& Herskovits, 1992; Geiger, 1990; Geiger & Heckerman, 1991, 1994; Geiger, Verma, and Pearl, 1990,)J 59 651 :M .045 .004(Hand, 1993; Lauritzen, et al., 1990; Lauritzen & Wermuth, 1984; Pearl, 1988; Pearl & Dechter, 1989; Pearl)J 59 663 :M (and Verma, 1991; Robins, 1986; Spiegelhalter, 1986\).)S 81 671 :M f3_6 sf (9)S 84 675 :M f3_10 sf 1.382 .138( For example, the Bayesian Information Criterion \(Raftery, 1993\), or the posterior probability,)J 59 687 :M (\(Geiger & Heckerman, 1994\).)S endp %%Page: 14 14 %%BeginPageSetup initializepage (peter; page: 14 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (14)S gR gS 0 0 552 730 rC 81 54 :M f3_12 sf 1.241 .124(In contrast to a score maximization search, a constraint search uses some testing)J 59 70 :M .708 .071(procedure for conditional independence, vanishing partial correlations, vanishing tetrad)J 59 86 :M .312 .031(differences, or other constraints on the covariance matrix. One advantage of this kind of)J 59 102 :M .479 .048(search is that there are provably correct search algorithms for certain classes of RSEMs.)J 59 118 :M 1.138 .114(For example, we will later discuss correct algorithms for multivariate normal RSEMs)J 59 134 :M (even when the population RSEM may contain latent variables \(Spirtes, et al. 1993\).)S 81 150 :M .782 .078(In order to understand model specification search procedures based on constraints,)J 59 166 :M .203 .02(one must first understand how SEMs entail constraints on the covariance matrix. Various)J 59 182 :M .037 .004(equivalence relations between SEMs also need to be explained. We turn to those topics in)J 59 198 :M (the next sections.)S 182 226 :M f4_14 sf (2)S 189 226 :M (.)S 192 226 :M 4 0 rm ( )S 199 226 :M (Constraints Entailed by SEMs)S 81 254 :M f3_12 sf .546 .055(We use two kinds of correlation constraints in our searches: zero partial correlation)J 59 270 :M (constraints, and vanishing tetrad constraints.)S 59 298 :M f4_12 sf (2)S 65 298 :M (.)S 68 298 :M (1)S 74 298 :M ( )S 81 298 :M (Zero Partial Correlation Constraints)S 81 320 :M f3_12 sf (In a SEM some partial correlations may be equal to zero for )S 372 320 :M f4_12 sf (all)S 385 320 :M f3_12 sf ( values of the model\325s)S 59 336 :M .42 .042(free parameters \(for which the partial correlation is defined\). \(See Blalock 1962; Kiiveri)J 59 352 :M .402 .04(& Speed, 1982\). In this case we will say that the SEM )J 329 352 :M f0_12 sf .069(entails)A f3_12 sf .27 .027( that the partial correlation)J 59 368 :M (is zero.)S 95 363 :M f3_7 sf (1)S 98 363 :M (0)S 101 368 :M f3_12 sf .025 .002( For example, in SEM S)J f3_7 sf 0 3 rm (1 )S 0 -3 rm f3_12 sf <28>S 226 368 :M .02 .002(Figure 1\), where all of the error terms are uncorrelated,)J 59 384 :M f1_12 sf (r)S 66 387 :M f3_7 sf (X1,X3.X2 )S 97 384 :M f3_12 sf (= 0 for all values of the free parameters of S)S 309 387 :M f3_7 sf (1)S 313 384 :M f3_12 sf (.)S 81 400 :M .881 .088(Judea Pearl \(1988\) discovered a fast procedure that can be used to decide, for any)J 59 416 :M 1.161 .116(partial correlation )J 153 416 :M f1_12 sf (r)S 160 419 :M f3_7 sf .217(A,B.)A f0_7 sf .331(C)A f3_12 sf 0 -3 rm 1.323 .132( and any RSEM with uncorrelated errors, whether the RSEM)J 0 3 rm 59 432 :M 1.457 .146(entails that )J f1_12 sf (r)S 126 435 :M f3_7 sf .347(A,B.)A f0_7 sf .53(C)A f3_12 sf 0 -3 rm 1.621 .162( is zero. Pearl defined a relation called )J 0 3 rm 353 432 :M f0_12 sf 1.223 .122(d-separation )J 424 432 :M f3_12 sf 1.682 .168(that can hold)J 59 448 :M .358 .036(between three disjoint sets of vertices in a directed acyclic graph. A simple consequence)J 59 464 :M 2.206 .221(of theorems proved by Pearl, Geiger, and Verma shows that in an RSEM R with)J 59 480 :M .168 .017(uncorrelated errors a partial correlation )J f1_12 sf (r)S 258 483 :M f3_7 sf .051(A,B.)A f0_7 sf .077(C)A f3_12 sf 0 -3 rm .209 .021( is entailed to be zero if and only if {A} and)J 0 3 rm 59 496 :M 1.348 .135({B} are d-separated by )J f0_12 sf .793 .079(C )J 193 496 :M f3_12 sf 1.147 .115(in the directed graph associated with R \(Pearl 1988\). More)J 59 512 :M 1.12 .112(details about their discovery, which is considerably more general than the description)J 59 528 :M 2.224 .222(given here, are given in section )J 233 528 :M 1.895 .189(8.1. Spirtes \(1995\) showed that these connections)J 59 544 :M 1.7 .17(between graphical structure and vanishing partial correlations hold as well for non-)J 59 560 :M 1.519 .152(recursive SEMs, i.e. in a SEM with uncorrelated errors a partial correlation )J f1_12 sf (r)S 459 563 :M f3_7 sf .287(A,B.)A f0_7 sf .439(C)A f3_12 sf 0 -3 rm .796 .08( is)J 0 3 rm 59 576 :M .147 .015(entailed to be zero if and only if {A} and {B} are d-separated given )J f0_12 sf (C)S 399 576 :M f3_12 sf .161 .016(. \(The if part of the)J 59 592 :M (theorem was shown independently in Koster \(forthcoming\)\).)S 59 638 :M ( )S 59 635.48 -.48 .48 203.48 635 .48 59 635 @a 81 647 :M f3_6 sf (1)S 84 647 :M (0)S 87 651 :M f3_10 sf 1.224 .122( Correlations and partial correlations are zero exactly when the corresponding covariances and)J 59 663 :M 1.671 .167(partial covariances are zero. While there may be important different statistical properties of partial)J 59 675 :M .246 .025(correlations and partial covariances, they are not germane to the discussion of the constraints entailed by a)J 59 687 :M (SEM.)S endp %%Page: 15 15 %%BeginPageSetup initializepage (peter; page: 15 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (15)S gR gS 0 0 552 730 rC 81 54 :M f3_12 sf .538 .054(There is also a way to decide which partial correlations are entailed to be zero by a)J 59 70 :M .022 .002(SEM with correlated errors, such as S)J 242 73 :M f3_7 sf (2)S 246 70 :M f3_12 sf .024 .002( \(Figure 2\). This is done by first creating a directed)J 59 86 :M .513 .051(graph G with latent variables and then applying d-separation to G to determine if a zero)J 59 102 :M 1.167 .117(partial correlation is entailed. The directed graph G \(with latent variables but without)J 59 118 :M .423 .042(correlated errors\) that we associate with a SEM S with correlated errors is created in the)J 59 134 :M .024 .002(following way. Start with the usual graphical representation of S, that contains undirected)J 59 150 :M .567 .057(lines connecting correlated errors \(e.g. SEM S)J f3_7 sf 0 3 rm (2)S 0 -3 rm 291 150 :M f3_12 sf .672 .067( in Figure 2\). For each pair of correlated)J 59 166 :M .502 .05(error terms )J 117 166 :M f1_12 sf (e)S f3_7 sf 0 3 rm (i)S 0 -3 rm 124 166 :M f3_12 sf .475 .048( and )J f1_12 sf .268(e)A f3_7 sf 0 3 rm (j)S 0 -3 rm 155 166 :M f3_12 sf .496 .05(, introduce a new latent variable T)J 324 169 :M f3_7 sf (ij)S 328 166 :M f3_12 sf .514 .051(, and edges from T)J f3_7 sf 0 3 rm .128(ij)A 0 -3 rm 425 166 :M f3_12 sf .576 .058( to X)J f3_7 sf 0 3 rm (i)S 0 -3 rm 452 166 :M f3_12 sf .622 .062( and X)J 486 169 :M f3_7 sf (j)S 488 166 :M f3_12 sf (.)S 59 182 :M .381 .038(Finally replace )J f1_12 sf .097(e)A f3_7 sf 0 3 rm (i)S 0 -3 rm 142 182 :M f3_12 sf .627 .063( and )J 167 182 :M f1_12 sf (e)S f3_7 sf 0 3 rm (j)S 0 -3 rm 174 182 :M f3_12 sf .43 .043( with uncorrelated errors )J f1_12 sf .123(e)A f3_7 sf 0 3 rm (i)S 0 -3 rm 305 182 :M f3_12 sf .458 .046(\325 and )J f1_12 sf .229(e)A f3_7 sf 0 3 rm (j)S 0 -3 rm 340 182 :M f3_12 sf .47 .047(\325. When this process is applied)J 59 198 :M (to SEM S)S 106 201 :M f3_7 sf (2)S 110 198 :M f3_12 sf (, the result is shown in )S 221 198 :M (Figure 8.)S 172 217 206 83 rC 173.5 217.5 27 26 rS 179 222 13 15 rC 179 234 :M f8_12 sf (X)S gR gS 188 228 9 11 rC 188 237 :M f8_10 sf (1)S gR gS 172 217 206 83 rC 264.5 217.5 27 26 rS 271 222 13 15 rC 271 234 :M f8_12 sf (X)S gR gS 280 228 9 11 rC 280 237 :M f8_10 sf (2)S gR gS 172 217 206 83 rC 350.5 217.5 27 26 rS 357 222 13 15 rC 357 234 :M f8_12 sf (X)S gR gS 366 228 9 11 rC 366 237 :M f8_10 sf (3)S gR gS 172 217 206 83 rC np 257 230 :M 245 235 :L 245 235 :L 245 234 :L 245 234 :L 245 234 :L 245 234 :L 245 234 :L 245 233 :L 245 233 :L 244 233 :L 244 233 :L 244 232 :L 244 232 :L 244 232 :L 244 232 :L 244 232 :L 244 231 :L 244 231 :L 244 231 :L 244 231 :L 244 230 :L 244 230 :L 244 230 :L 244 230 :L 244 230 :L 244 229 :L 244 229 :L 244 229 :L 244 229 :L 244 228 :L 244 228 :L 244 228 :L 244 228 :L 244 228 :L 244 227 :L 244 227 :L 244 227 :L 244 227 :L 244 226 :L 244 226 :L 245 226 :L 245 226 :L 245 226 :L 245 225 :L 245 225 :L 245 225 :L 245 225 :L 245 225 :L 257 230 :L 257 230 :L eofill 205 231 -1 1 253 230 1 205 230 @a np 346 229 :M 334 234 :L 334 234 :L 334 233 :L 334 233 :L 334 233 :L 333 233 :L 333 233 :L 333 232 :L 333 232 :L 333 232 :L 333 232 :L 333 232 :L 333 231 :L 333 231 :L 333 231 :L 333 231 :L 333 230 :L 333 230 :L 333 230 :L 333 230 :L 333 229 :L 333 229 :L 333 229 :L 333 229 :L 333 229 :L 333 228 :L 333 228 :L 333 228 :L 333 228 :L 333 228 :L 333 227 :L 333 227 :L 333 227 :L 333 227 :L 333 226 :L 333 226 :L 333 226 :L 333 226 :L 333 226 :L 333 225 :L 333 225 :L 333 225 :L 333 225 :L 333 224 :L 334 224 :L 334 224 :L 334 224 :L 334 224 :L 346 229 :L 346 229 :L eofill 297 230 -1 1 341 229 1 297 229 @a 318 276 31 15 rC 318 288 :M f8_12 sf (T)S 326 290 :M f8_8 sf (2)S 330 290 :M (3)S gR gS 172 217 206 83 rC 25 26 326 285.5 @f np 289 245 :M 301 250 :L 301 250 :L 301 251 :L 301 251 :L 300 251 :L 300 251 :L 300 252 :L 300 252 :L 300 252 :L 300 252 :L 300 252 :L 300 252 :L 300 253 :L 299 253 :L 299 253 :L 299 253 :L 299 253 :L 299 254 :L 299 254 :L 299 254 :L 298 254 :L 298 254 :L 298 254 :L 298 255 :L 298 255 :L 298 255 :L 298 255 :L 297 255 :L 297 255 :L 297 255 :L 297 256 :L 297 256 :L 296 256 :L 296 256 :L 296 256 :L 296 256 :L 296 256 :L 296 257 :L 295 257 :L 295 257 :L 295 257 :L 295 257 :L 295 257 :L 294 257 :L 294 257 :L 294 257 :L 294 257 :L 293 257 :L 289 245 :L 289 245 :L eofill 293 251 -1 1 317 274 1 293 250 @a np 352 245 :M 348 258 :L 348 258 :L 347 258 :L 347 258 :L 347 257 :L 347 257 :L 347 257 :L 346 257 :L 346 257 :L 346 257 :L 346 257 :L 346 257 :L 345 257 :L 345 257 :L 345 257 :L 345 256 :L 345 256 :L 344 256 :L 344 256 :L 344 256 :L 344 256 :L 344 256 :L 344 255 :L 343 255 :L 343 255 :L 343 255 :L 343 255 :L 343 255 :L 342 255 :L 342 254 :L 342 254 :L 342 254 :L 342 254 :L 342 254 :L 342 254 :L 341 253 :L 341 253 :L 341 253 :L 341 253 :L 341 253 :L 341 252 :L 341 252 :L 341 252 :L 340 252 :L 340 252 :L 340 252 :L 340 251 :L 340 251 :L 352 245 :L 352 245 :L eofill -1 -1 330 274 1 1 348 251 @b gR gS 0 0 552 730 rC 70 325 :M f0_12 sf (Figure )S 107 325 :M (8. SEM S)S 155 328 :M f0_7 sf (2)S 159 325 :M f0_12 sf S 170 325 :M (Correlated Errors )S 267 325 :M (in S)S 287 328 :M f0_7 sf (2)S 291 325 :M f0_12 sf ( Replaced b)S 351 325 :M (y Latent Common Cause)S 81 353 :M f3_12 sf .581 .058(In a SEM like S)J 161 356 :M f3_7 sf (2)S 165 353 :M f3_12 sf .473 .047(, with correlated errors, one can decide whether )J f1_12 sf (r)S 409 356 :M f3_7 sf .439 .044(X1,X3.X2 )J 441 353 :M f3_12 sf .392 .039(is entailed)J 59 369 :M (to be zero by determining whether {X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 245 369 :M f3_12 sf (} and {X)S 289 372 :M f3_7 sf (3)S 293 369 :M f3_12 sf (} are d-separated given {X)S 422 372 :M f3_7 sf (2)S 426 369 :M f3_12 sf (} in the graph)S 59 385 :M .078 .008(in Figure 8. In this way the problem of determining whether a SEM with correlated errors)J 59 401 :M .432 .043(entails a zero partial correlation is reduced to the already solved problem of determining)J 59 417 :M .149 .015(whether a SEM without correlated errors entails a zero partial correlation. \(In general if S)J 59 433 :M 1.286 .129(is a SEM with correlated errors, and G is the latent variable graph with uncorrelated)J 59 449 :M .687 .069(erorrs associated with S, it is )J 207 449 :M f4_12 sf .17(not)A f3_12 sf .581 .058( the case that for every linear parameterization )J f1_12 sf .208(q)A f2_7 sf 0 3 rm (1)S 0 -3 rm 466 449 :M f1_12 sf .251 .025( )J f3_12 sf 1.39 .139(of S)J 59 465 :M .657 .066(there is a linear parameterization )J f1_12 sf .218(q)A f2_7 sf 0 3 rm (2)S 0 -3 rm 235 465 :M f2_12 sf .219 .022( )J f3_12 sf 1.052 .105(of G such that )J 315 465 :M f2_12 sf (S)S f0_7 sf 0 3 rm (S)S 0 -3 rm 326 465 :M f3_12 sf <28>S 330 465 :M f2_12 sf (q)S f2_7 sf 0 3 rm (1)S 0 -3 rm 340 465 :M f3_12 sf .497 .05(\) = )J f2_12 sf .505(S)A f0_7 sf 0 3 rm .387(G)A 0 -3 rm f3_12 sf <28>S 375 465 :M f2_12 sf (q)S f2_7 sf 0 3 rm (2)S 0 -3 rm 385 465 :M f3_12 sf .803 .08(\). We are making the)J 59 481 :M 2.031 .203(weaker claim that d-separation applied to G correctly describes which zero partial)J 59 497 :M (correlations are entailed by S. For the proof, see Spirtes, et. al, 1996.)S 180 516 189 103 rC 185.5 591.5 27 26 rS 192 607 :M f8_12 sf (X)S 201 611 :M f8_10 sf (1)S 244.5 590.5 27 26 rS 251 606 :M f8_12 sf (X)S 260 610 :M f8_10 sf (2)S 294.5 590.5 27 26 rS 301 606 :M f8_12 sf (X)S 310 610 :M f8_10 sf (3)S 340.5 591.5 27 26 rS 347 607 :M f8_12 sf (X)S 356 611 :M f8_10 sf (4)S 248 536 :M f8_12 sf -.24(Intelligence)A 70 24 275.5 532.5 @f 13 -65 -17 210 587 @k -1 -1 215 584 1 1 259 544 @b 13 -102 -54 261 587 @k -1 -1 263 582 1 1 270 546 @b 13 225 273 306 588 @k 289 544 -1 1 305 582 1 289 543 @a 13 203 251 348 588 @k 304 541 -1 1 345 584 1 304 540 @a gR gS 0 0 552 730 rC 178 644 :M f0_12 sf (Figure )S 215 644 :M (9. Factor)S 261 644 :M ( Model)S 297 644 :M ( of Intelligence)S endp %%Page: 16 16 %%BeginPageSetup initializepage (peter; page: 16 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (16)S gR gS 0 0 552 730 rC 59 54 :M f4_12 sf (2)S 65 54 :M (.)S 68 54 :M (2)S 74 54 :M ( )S 81 54 :M (Vanishing Tetrad Constraints)S 81 76 :M f3_12 sf .742 .074(In SEMs containing latent variables, zero partial correlation constraints among the)J 59 92 :M 1.47 .147(measured covariances )J 174 92 :M f2_12 sf .611(S)A f1_12 sf .234 .023( )J f3_12 sf 1.859 .186(are often uninformative. For example, consider )J 433 92 :M 1.96 .196(Figure 9 in)J 59 108 :M .735 .073(which Intelligence is a latent variable. The only correlations entailed to be zero by this)J 59 124 :M 3.157 .316(SEM are those that are partialed on at least Intelligence. Since Intelligence is)J 59 140 :M .11 .011(unmeasured, however, our data will only include partial correlations among the measured)J 59 156 :M (variables )S f0_12 sf (X)S 114 156 :M f3_12 sf ( = {X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 145 156 :M f3_12 sf (, X)S 160 159 :M f3_7 sf (2)S 164 156 :M f3_12 sf (, X)S 179 159 :M f3_7 sf (3)S 183 156 :M f3_12 sf (, X)S 198 159 :M f3_7 sf (4)S 202 156 :M f3_12 sf .013 .001(}, and there is no partial correlation involving only variables)J 59 172 :M (in )S f0_12 sf (X)S 80 172 :M f3_12 sf ( that is entailed to be zero by this SEM.)S 81 188 :M .676 .068(The vanishing tetrad difference \(Spearman, 1904, Glymour, et al., 1987\), however,)J 59 204 :M .747 .075(can provide extra information about the specification of this model. A tetrad difference)J 59 220 :M .361 .036(involves two products of correlations, each of which involve the same four variables but)J 59 236 :M .039 .004(in different permutations. In the SEM of )J 257 236 :M .037 .004(Figure 9 there are three tetrad differences among)J 59 252 :M .41 .041(the measured correlations that are entailed to vanish for all values of the free parameters)J 59 268 :M (\(for which the correlations are defined\):)S 212 300 :M f1_12 sf (r)S 219 303 :M f3_7 sf (X1,X2)S 238 300 :M .739 0 rm f7_12 sf ( )S f1_12 sf (r)S 249 303 :M f3_7 sf (X3,X4)S 268 300 :M f7_12 sf <20D020>S f1_12 sf (r)S 288 303 :M f3_7 sf (X1,X3)S 307 300 :M .739 0 rm f7_12 sf ( )S f1_12 sf (r)S 318 303 :M f3_7 sf (X2,X4)S 213 316 :M f1_12 sf (r)S 220 319 :M f3_7 sf (X1,X2)S 239 316 :M .739 0 rm f7_12 sf ( )S f1_12 sf (r)S 250 319 :M f3_7 sf (X3,X4)S 269 316 :M f7_12 sf <20D020>S f1_12 sf (r)S 289 319 :M f3_7 sf (X1,X4)S 308 319 :M .432 0 rm f7_7 sf ( )S f1_12 sf 0 -3 rm (r)S 0 3 rm 317 319 :M f3_7 sf (X2,X3)S 213 332 :M f1_12 sf (r)S 220 335 :M f3_7 sf (X1,X3)S 239 332 :M .739 0 rm f7_12 sf ( )S f1_12 sf (r)S 250 335 :M f3_7 sf (X2,X4)S 269 332 :M f7_12 sf <20D020>S f1_12 sf (r)S 289 335 :M f3_7 sf (X1,X4)S 308 335 :M .432 0 rm f7_7 sf ( )S f1_12 sf 0 -3 rm (r)S 0 3 rm 317 335 :M f3_7 sf (X2,X3)S 81 364 :M f3_12 sf 1.288 .129(If a SEM S entails that )J f1_12 sf (r)S 211 367 :M f3_7 sf (X1,X2)S 230 364 :M f7_12 sf 2.648 .265( )J 236 364 :M f1_12 sf (r)S 243 367 :M f7_7 sf .019(X3,X4)A 264 364 :M f7_12 sf 1.716 .172<20D020>J 281 364 :M f1_12 sf (r)S 288 367 :M f7_7 sf .019(X1,X3)A 309 364 :M f7_12 sf 2.648 .265( )J 315 364 :M f1_12 sf (r)S 322 367 :M f7_7 sf .019(X2,X4)A 343 364 :M f3_12 sf 1.389 .139( = 0 for all values of its free)J 59 380 :M .873 .087(parameters we say that S )J 188 380 :M f0_12 sf .147(entails)A f3_12 sf .642 .064( the vanishing tetrad difference. The tetrad differences)J 59 396 :M 2.017 .202(that are entailed to vanish by a SE)J 244 396 :M (M)S 255 396 :M 1.816 .182( without correlated errors )J 392 396 :M 1.577 .158(are also completely)J 59 412 :M 4.639 .464(determined by the directed graph associated with the SEM. The graphical)J 59 428 :M .108 .011(characterization is given by the Tetrad Representation Theorem \(Spirtes, 1989; Spirtes, et)J 59 444 :M 1.547 .155(al. 1993; Shafer et al., 1993\), which leads to a general procedure for computing the)J 59 460 :M .829 .083(vanishing tetrad differences entailed by a SEM, implemented in the Tetrads module of)J 59 476 :M .243 .024(the TETRAD II program \(Scheines, Spirtes, Glymour and Meek, 1994\))J 406 476 :M .271 .027(. Bollen and Ting)J 59 492 :M .233 .023(\(1993\) discuss the advantages of using vanishing tetrad differences in SEM analysis, e.g.)J 59 508 :M (they can be used to compare underidentified SEMs.)S 115 536 :M f4_14 sf (3)S 122 536 :M (.)S 125 536 :M 4 0 rm ( )S 132 536 :M (Assumptions Relating )S 256 536 :M (Probability to Causal Relations)S 59 576 :M f4_12 sf (3)S 65 576 :M (.)S 68 576 :M (1)S 74 576 :M ( )S 81 576 :M (The Causal Independence Assumption)S 81 598 :M f3_12 sf 1.267 .127(The most fundamental assumption relating causality and probability that we will)J 59 614 :M (make is the following:)S 95 646 :M f0_12 sf .58 .058(Causal Independence Assumption:)J 276 646 :M f3_12 sf 1.061 .106( If A does not cause B, and B does)J 95 662 :M 1.111 .111(not cause A, and there is no third variable which causes both A and B,)J 95 678 :M (then A and B are uncorrelated.)S endp %%Page: 17 17 %%BeginPageSetup initializepage (peter; page: 17 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (17)S gR gS 0 0 552 730 rC 81 70 :M f3_12 sf .154 .015(This assumption provides a bridge between statistical facts and causal features of the)J 59 86 :M 1.389 .139(process that underlies the data. In certain cases the assumption allows us to draw a)J 59 102 :M f4_12 sf .465(causal)A f3_12 sf 1.631 .163( conclusion from )J 185 102 :M f4_12 sf (statistical)S 232 102 :M f3_12 sf 2.317 .232( data and lies at the foundation of the theory of)J 59 118 :M .812 .081(randomized experiments. If the value of A is randomized, the experimenter knows that)J 59 134 :M .449 .045(the randomizing device is the sole cause of A. Hence the experimenter knows B did not)J 59 150 :M .292 .029(cause A, and that there is no other variable which causes both A and B. This leaves only)J 59 166 :M 1.758 .176(two alternatives: either A causes B or it does not. If A and B are correlated in the)J 59 182 :M .879 .088(experimental population, the experimenter concludes that A does cause B, which is an)J 59 198 :M (application of the Causal Independence assumption.)S 81 214 :M .723 .072(The Causal Independence assumption entails that if two error terms are correlated,)J 59 230 :M .095 .009(such as )J 98 230 :M f1_12 sf (e)S f3_7 sf 0 3 rm (2 )S 0 -3 rm f3_12 sf .094 .009(and )J f1_12 sf (e)S f3_7 sf 0 3 rm (3)S 0 -3 rm 137 230 :M f3_12 sf .088 .009( in S)J f3_7 sf 0 3 rm (2)S 0 -3 rm 163 230 :M f3_12 sf .091 .009( \(see Figure )J 223 230 :M .084 .008(2\), then there is at least one latent common cause of the)J 59 246 :M (explicitly modeled variables associated with these errors, i.e., X)S 366 249 :M f3_7 sf (2 )S f3_12 sf 0 -3 rm (and X)S 0 3 rm 400 249 :M f3_7 sf (3)S 404 246 :M f3_12 sf (.)S 177 265 195 128 rC 294 282 :M f8_12 sf -.328(Tax)A 294 297 :M (Rate)S 202 346 :M (Economy)S 322 373 :M -.328(Tax)A 322 388 :M -.093(Revenues)A 287.5 268.5 35 34 rS 195.5 329.5 58 22 rS 315.5 357.5 55 34 rS 13 -57 -9 249 326 @k -1 -1 255 324 1 1 285 302 @b 13 177 225 310 368 @k 258 349 -1 1 305 366 1 258 348 @a 13 219 267 335 353 @k 312 308 -1 1 334 348 1 312 307 @a 325 326 :M f1_12 sf (b)S 331 331 :M (1)S 262 305 :M (b)S 268 310 :M (2)S 275 350 :M (b)S 281 355 :M (3)S gR gS 0 0 552 730 rC 115 418 :M f0_12 sf (Figure )S 152 418 :M (10. Distribution is Unfaithful to SEM when )S f2_12 sf (b)S 382 421 :M f0_7 sf (1)S 386 418 :M f0_12 sf ( = \320\()S 409 418 :M f2_12 sf (b)S 416 421 :M f2_7 sf (2)S 420 418 :M f2_12 sf (b)S 427 421 :M f0_7 sf (3)S 431 418 :M f0_12 sf <29>S 59 458 :M f4_12 sf (3)S 65 458 :M (.)S 68 458 :M (2)S 74 458 :M ( )S 81 458 :M (The Faithfulness Assumption)S 81 480 :M f3_12 sf .803 .08(In addition to the zero partial correlations and vanishing tetrad differences that are)J 59 496 :M 1.867 .187(entailed for )J 123 496 :M f4_12 sf (all)S 136 496 :M f3_12 sf 1.992 .199( values of the free parameters of a SEM, there may be zero partial)J 59 512 :M .043 .004(correlations or vanishing tetrad differences that hold only for )J 356 512 :M f4_12 sf (particular)S 405 512 :M f3_12 sf .05 .005( values of the free)J 59 528 :M .841 .084(parameters of a SEM. For example, suppose )J 284 528 :M .833 .083(Figure )J 320 528 :M .915 .091(10 is the directed graph of a SEM)J 59 544 :M (that describes the relations among the Tax Rate, the Economy, and Tax Revenues.)S 81 560 :M 1.469 .147(In this case there are no vanishing partial correlation constraints entailed for all)J 59 576 :M .53 .053(values of the free parameters. But if )J f1_12 sf (b)S 247 579 :M f3_7 sf (1)S 251 576 :M f3_12 sf .707 .071( = \320\()J 276 576 :M f1_12 sf (b)S 283 579 :M f3_7 sf (2)S 287 576 :M f1_12 sf (b)S 294 579 :M f3_7 sf (3)S 298 576 :M f3_12 sf .56 .056(\), then Tax Rate and Tax Revenues are)J 59 592 :M .877 .088(uncorrelated. The SEM postulates a direct effect of Tax Rate on Revenue \()J 435 592 :M f1_12 sf (b)S 442 595 :M f3_7 sf (1)S 446 592 :M f3_12 sf <29>S 450 592 :M 1.276 .128(, )J 458 592 :M .936 .094(and an)J 59 608 :M .263 .026(indirect effect through the Economy \()J f1_12 sf (b)S 249 611 :M f3_7 sf (2)S 253 608 :M f1_12 sf (b)S 260 611 :M f3_7 sf (3)S 264 608 :M f3_12 sf .266 .027(\). The parameter constraint indicates that these)J 59 624 :M .065 .007(effects )J f4_12 sf .019(exactly)A 128 624 :M f3_12 sf .15 .015( offset each other, leaving no total effect whatsoever. In such a case we say)J 59 640 :M .415 .041(that the population is )J 166 640 :M f0_12 sf .103(Unfaithful)A f3_12 sf .232 .023( to the SE)J 268 640 :M (M)S 279 640 :M .334 .033( that generated it. A distribution is )J f0_12 sf .127(Faithful)A 59 656 :M f3_12 sf .546 .055(to SE)J 85 656 :M (M)S 96 656 :M .48 .048( M \(or its corresponding directed graph\) if each partial correlation that is zero in)J endp %%Page: 18 18 %%BeginPageSetup initializepage (peter; page: 18 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (18)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .453 .045(the distribution is entailed to be zero by M, and each tetrad difference that is zero in the)J 59 70 :M (distribution is entailed to be zero by M.)S 95 102 :M f0_12 sf .205 .021(Faithfulness Assumption:)J f3_12 sf .089 .009( If the directed graph associated with a SEM M)J 95 118 :M .312 .031(correctly describes the causal structure in the population, then each partial)J 95 134 :M .402 .04(correlation and each tetrad difference that is zero in )J f2_12 sf .165(S)A f3_7 sf 0 3 rm .145(M)A 0 -3 rm f3_12 sf <28>S 368 134 :M f2_12 sf .151(q)A f0_7 sf 0 3 rm .091(pop)A 0 -3 rm f3_12 sf .368 .037(\) is entailed to)J 95 150 :M (be zero by M.)S 84 166 :M ( )S 81 182 :M .214 .021(The Faithfulness assumption is a kind of simplicity assumption. If a distribution P is)J 59 198 :M .871 .087(faithful to an RSEM R)J f3_7 sf 0 3 rm (1)S 0 -3 rm 177 198 :M f3_12 sf .831 .083( without latent variables or correlated errors, and P also results)J 59 214 :M .047 .005(from a parameterization of another RSEM R)J f3_7 sf 0 3 rm (2)S 0 -3 rm 277 214 :M f3_12 sf .057 .006( to which P is not faithful, then R)J 438 217 :M f3_7 sf (1)S 442 214 :M f3_12 sf .056 .006( has fewer)J 59 230 :M (free parameters than R)S 168 233 :M f3_7 sf (2)S 172 230 :M f3_12 sf (.)S 81 246 :M 2.959 .296(The Faithfulness assumption limits the SEMs considered to those in which)J 59 262 :M 3.067 .307(population constraints are entailed by structure, not by particular values of the)J 59 278 :M (parameters)S 112 278 :M .903 .09(. If one assumes Faithfulness, then if A and B are )J f4_12 sf .31(not)A f3_12 sf 1.15 .115( d-separated given )J 479 278 :M f0_12 sf (C)S 488 278 :M f3_12 sf (,)S 59 294 :M 2.329 .233(then )J 86 294 :M f1_12 sf (r)S 93 297 :M f3_7 sf .498(A,B.)A f0_7 sf .761(C)A f3_7 sf .239 .024( )J f3_12 sf 0 -3 rm cF f1_12 sf .221A sf 2.214 .221( 0, \(because it is not entailed to equal zero for all values of the free)J 0 3 rm 59 310 :M 3.546 .355(parameters.\) Faithfulness should not be assumed when there are deterministic)J 59 326 :M 3.148 .315(relationships among the substantive variables, or equality constraints upon free)J 59 342 :M .48 .048(parameters, since either of these can lead to violations of the assumption. S)J 429 342 :M .522 .052(ome form of)J 59 358 :M .332 .033(the assumption of Faithfulness is used in every science, and amounts to no more that the)J 59 374 :M 1.534 .153(belief that an improbable and unstable cancellation of parameters does not hide real)J 59 390 :M .168 .017(causal influences. When a theory cannot explain an empirical regularity save by invoking)J 59 406 :M 1.17 .117(a special parameterization, most scientists are uneasy with the theory and look for an)J 59 422 :M (alternative)S 110 422 :M (.)S 81 438 :M .076 .008(It is also possible to give a personalist Bayesian argument for assuming Faithfulness.)J 59 454 :M .56 .056(For any SEM with free parameters, the set of parameterizations of the SEM that lead to)J 59 470 :M .681 .068(violations of Faithfulness are Lebesgue measure zero. Hence any Bayesian whose prior)J 59 486 :M .283 .028(over the parameters is absolutely continuous with Lebesgue measure assigns a zero prior)J 59 502 :M .281 .028(probability to violations of Faithfulness. Of course, this argument is not relevant to those)J 59 518 :M .722 .072(Bayesians who place a prior over the parameters that is not absolutely continuous with)J 59 534 :M .363 .036(Lebesgue measure and assign a non-zero probability to violations of Faithfulness. All of)J 59 550 :M .555 .056(the algorithms we have developed assume Faithfulness, and from here on we use it as a)J 59 566 :M (working assumption.)S 81 582 :M .567 .057(The Faithfulness assumption is necessary to guarantee the correctness of the model)J 59 598 :M .999 .1(specification algorithms used in TETRAD II. It does )J 328 598 :M f4_12 sf .269(not)A f3_12 sf .946 .095( guarantee that on samples of)J 59 614 :M (finite size the model specification algorithms are reliable.)S endp %%Page: 19 19 %%BeginPageSetup initializepage (peter; page: 19 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (19)S gR gS 0 0 552 730 rC 224 54 :M f4_12 sf (4)S 230 54 :M (.)S 233 54 :M 4 0 rm ( )S 240 54 :M (SEM Equivalence)S 81 82 :M f3_12 sf .106 .011(Two SEMs S)J f3_7 sf 0 3 rm (1)S 0 -3 rm 149 82 :M f3_12 sf .165 .017( and S)J f3_7 sf 0 3 rm .068 .007(2 )J 0 -3 rm 185 82 :M f3_12 sf .093 .009(with the same substantive variables \(or their respective directed)J 59 98 :M .284 .028(graphs\) are )J 117 98 :M f0_12 sf .373 .037(covariance equivalent)J f3_12 sf .186 .019( if for every parameterization )J 375 98 :M f2_12 sf (q)S f0_7 sf 0 3 rm (i)S 0 -3 rm 383 98 :M f3_12 sf .367 .037( of S)J 407 101 :M f3_7 sf (1)S 411 98 :M f3_12 sf .252 .025( with covariance)J 59 114 :M .357 .036(matrix )J 94 114 :M f2_12 sf (S)S f3_7 sf 0 3 rm (S1)S 0 -3 rm f1_12 sf <28>S 112 114 :M f2_12 sf (q)S f0_7 sf 0 3 rm (i)S 0 -3 rm 120 114 :M f1_12 sf <29>S 124 114 :M f3_12 sf .403 .04( there is a parameterization )J f2_12 sf .15(q)A f0_7 sf 0 3 rm .089 .009(j )J 0 -3 rm 270 114 :M f3_12 sf .439 .044(of S)J 291 117 :M f3_7 sf (2)S 295 114 :M f3_12 sf .363 .036( with covariance matrix )J 414 114 :M f2_12 sf (S)S f3_7 sf 0 3 rm (S2)S 0 -3 rm f1_12 sf <28>S 432 114 :M f2_12 sf .122(q)A f0_7 sf 0 3 rm (j)S 0 -3 rm f1_12 sf .114 .011(\) )J f3_12 sf .467 .047(such that)J 59 130 :M f2_12 sf (S)S f3_7 sf 0 3 rm (S1)S 0 -3 rm f1_12 sf <28>S 77 130 :M f2_12 sf (q)S f0_7 sf 0 3 rm (i)S 0 -3 rm 85 130 :M f1_12 sf .551 .055(\) = )J 103 130 :M f2_12 sf (S)S f3_7 sf 0 3 rm (S2)S 0 -3 rm f1_12 sf <28>S 121 130 :M f2_12 sf (q)S f0_7 sf 0 3 rm (j)S 0 -3 rm f1_12 sf <29>S 133 130 :M f3_12 sf .398 .04(, and vice versa. Two SEMs with the same substantive variables \(or their)J 59 146 :M .295 .029(respective directed graphs\) are )J 211 146 :M f0_12 sf .214 .021(partial correlation equivalent)J 363 146 :M f3_12 sf .361 .036( if they entail the same set)J 59 162 :M (of zero partial correlations among the substantive variables.)S 81 178 :M .394 .039(If two SEMs contain latent variables, and the same set of measured variables )J f0_12 sf (V)S 470 178 :M f3_12 sf .494 .049(, we)J 59 194 :M .299 .03(may be interested if they are equivalent on the measured variables. Two SEMs S)J f3_7 sf 0 3 rm (1)S 0 -3 rm 456 194 :M f3_12 sf .387 .039( and S)J 487 197 :M f3_7 sf (2)S 59 210 :M f3_12 sf .476 .048(\(or their respective directed graphs\) are )J 256 210 :M f0_12 sf .267 .027(covariance equivalent)J 369 210 :M f3_12 sf .801 .08( )J 373 210 :M f0_12 sf .525 .053(over a set of measured)J 59 226 :M .127 .013(variables V)J f3_12 sf .082 .008( if for every parameterization )J 262 226 :M f2_12 sf (q)S f0_7 sf 0 3 rm (i)S 0 -3 rm 270 226 :M f3_12 sf .128 .013( of S)J 293 229 :M f3_7 sf (1)S 297 226 :M f3_12 sf .098 .01( with covariance matrix )J 415 226 :M f1_12 sf (S)S f3_7 sf 0 3 rm (S1)S 0 -3 rm f1_12 sf <28>S 433 226 :M f2_12 sf (q)S f0_7 sf 0 3 rm (i)S 0 -3 rm 441 226 :M f1_12 sf <29>S 445 226 :M f3_12 sf .115 .012( there is a)J 59 242 :M .591 .059(parameterization )J 145 242 :M f2_12 sf .641(q)A f0_7 sf 0 3 rm .349 .035(j )J 0 -3 rm f3_12 sf 1.552 .155(of S)J 178 245 :M f3_7 sf (2)S 182 242 :M f3_12 sf .976 .098( with covariance matrix )J 305 242 :M f1_12 sf (S)S f3_7 sf 0 3 rm (S2)S 0 -3 rm f1_12 sf <28>S 323 242 :M f2_12 sf .329(q)A f0_7 sf 0 3 rm .122(j)A 0 -3 rm f1_12 sf .306 .031(\) )J f3_12 sf 1.255 .125(such that)J 384 242 :M f1_12 sf .237 .024( )J f3_12 sf 1.37 .137(the margin of )J 461 242 :M f2_12 sf (S)S f3_7 sf 0 3 rm (S1)S 0 -3 rm f1_12 sf <28>S 479 242 :M f2_12 sf (q)S f0_7 sf 0 3 rm (i)S 0 -3 rm 487 242 :M f1_12 sf <29>S 59 258 :M f3_12 sf .129 .013(over )J f0_12 sf .084 .008(V )J 95 258 :M f1_12 sf .13 .013(= )J 105 258 :M f3_12 sf .1 .01(the margin of )J 174 258 :M f2_12 sf (S)S f3_7 sf 0 3 rm (S2)S 0 -3 rm f1_12 sf <28>S 192 258 :M f2_12 sf (q)S f0_7 sf 0 3 rm (j)S 0 -3 rm f1_12 sf (\) )S f3_12 sf .096 .01(over )J f0_12 sf (V)S 240 258 :M f3_12 sf .1 .01(, and vice versa. Two SEMs are )J 397 258 :M f0_12 sf .051 .005(partial correlation)J 59 274 :M .063(equivalent)A f3_12 sf ( )S 116 274 :M f0_12 sf .431 .043(over a set of measured vertices V)J 289 274 :M f3_12 sf .447 .045( if they entail the same set of zero partial)J 59 290 :M (correlations among variables in )S 213 290 :M f0_12 sf (V)S 222 290 :M f3_12 sf (.)S .75 lw 69 309 411 118 rC 275.5 373.5 25 23 rS 277 375 22 20 rC 277 384 :M ( )S 280 384 :M (X)S 289 387 :M f3_7 sf (3)S gR .75 lw gS 69 309 411 118 rC 243.5 373.5 25 23 rS 245 375 22 20 rC 245 384 :M f3_12 sf ( )S 248 384 :M (X)S 257 387 :M f3_7 sf (2)S gR gS 69 309 411 118 rC 212.5 373.5 25 23 rS 214 375 22 20 rC 214 384 :M f3_12 sf ( )S 217 384 :M (X)S 226 387 :M f3_7 sf (1)S gR gS 69 309 411 118 rC 307.5 373.5 25 23 rS 309 375 22 20 rC 309 384 :M f3_12 sf ( )S 312 384 :M (X)S 321 387 :M f3_7 sf (4)S gR gS 69 309 411 118 rC 132.5 372.5 25 23 rS 134 374 22 20 rC 134 383 :M f3_12 sf ( )S 137 383 :M (X)S 146 386 :M f3_7 sf (3)S gR gS 69 309 411 118 rC 100.5 372.5 25 23 rS 102 374 22 20 rC 102 383 :M f3_12 sf ( )S 105 383 :M (X)S 114 386 :M f3_7 sf (2)S gR gS 69 309 411 118 rC 69.5 372.5 25 23 rS 71 374 22 20 rC 71 383 :M f3_12 sf ( )S 74 383 :M (X)S 83 386 :M f3_7 sf (1)S gR gS 69 309 411 118 rC 164.5 372.5 25 23 rS 166 374 22 20 rC 166 383 :M f3_12 sf ( )S 169 383 :M (X)S 178 386 :M f3_7 sf (4)S gR gS 69 309 411 118 rC 421.5 373.5 25 23 rS 423 375 22 20 rC 423 384 :M f3_12 sf ( )S 426 384 :M (X)S 435 387 :M f3_7 sf (3)S gR gS 69 309 411 118 rC 389.5 373.5 25 23 rS 391 375 22 20 rC 391 384 :M f3_12 sf ( )S 394 384 :M (X)S 403 387 :M f3_7 sf (2)S gR gS 69 309 411 118 rC 358.5 373.5 25 23 rS 360 375 22 20 rC 360 384 :M f3_12 sf ( )S 363 384 :M (X)S 372 387 :M f3_7 sf (1)S gR gS 69 309 411 118 rC 453.5 373.5 25 23 rS 455 375 22 20 rC 455 384 :M f3_12 sf ( )S 458 384 :M (X)S 467 387 :M f3_7 sf (4)S gR gS 113 319 19 18 rC 113 328 :M f3_12 sf (T)S 120 331 :M f3_7 sf (1)S gR gS 69 309 411 118 rC 23 24 120 325.5 @f 248 316 16 21 rC 248 325 :M f3_12 sf (T)S 255 328 :M f3_7 sf (1)S gR gS 69 309 411 118 rC 20 27 254.5 325 @f 415 408 24 17 rC 415 417 :M f0_12 sf <28>S 420 417 :M (i)S 424 417 :M (i)S 428 417 :M (i)S 432 417 :M <29>S gR gS 273 407 29 16 rC 273 416 :M f0_12 sf <28>S 278 416 :M (i)S 282 416 :M (i)S 286 416 :M <29>S gR gS 121 406 30 20 rC 121 415 :M f0_12 sf <28>S 126 415 :M (i)S 130 415 :M <29>S gR gS 312 316 16 21 rC 312 325 :M f3_12 sf (T)S 319 328 :M f3_7 sf (2)S gR gS 69 309 411 118 rC 20 27 318.5 325 @f 391 314 16 21 rC 391 323 :M f3_12 sf (T)S 398 326 :M f3_7 sf (1)S gR gS 69 309 411 118 rC 20 27 397.5 323 @f 448 315 16 21 rC 448 324 :M f3_12 sf (T)S 455 327 :M f3_7 sf (2)S gR gS 69 309 411 118 rC 20 27 454.5 324 @f -.75 -.75 87.75 368.75 .75 .75 110 334 @b np 91 370 :M 85 366 :L 86 371 :L 91 370 :L eofill 85 366.75 -.75 .75 91.75 370 .75 85 366 @a 85 366.75 -.75 .75 86.75 371 .75 85 366 @a -.75 -.75 86.75 371.75 .75 .75 91 370 @b -.75 -.75 114.75 368.75 .75 .75 117 339 @b np 118 367 :M 111 367 :L 114 371 :L 118 367 :L eofill 111 367.75 -.75 .75 118.75 367 .75 111 367 @a 111 367.75 -.75 .75 114.75 371 .75 111 367 @a -.75 -.75 114.75 371.75 .75 .75 118 367 @b 129 336.75 -.75 .75 143.75 368 .75 129 336 @a np 146 366 :M 139 369 :L 144 371 :L 146 366 :L eofill -.75 -.75 139.75 369.75 .75 .75 146 366 @b 139 369.75 -.75 .75 144.75 371 .75 139 369 @a -.75 -.75 144.75 371.75 .75 .75 146 366 @b 135 332.75 -.75 .75 169.75 367 .75 135 332 @a np 171 364 :M 166 369 :L 171 369 :L 171 364 :L eofill -.75 -.75 166.75 369.75 .75 .75 171 364 @b 166 369.75 -.75 .75 171.75 369 .75 166 369 @a -.75 -.75 171.75 369.75 .75 .75 171 364 @b -.75 -.75 226.75 368.75 .75 .75 246 335 @b np 230 369 :M 224 365 :L 225 370 :L 230 369 :L eofill 224 365.75 -.75 .75 230.75 369 .75 224 365 @a 224 365.75 -.75 .75 225.75 370 .75 224 365 @a -.75 -.75 225.75 370.75 .75 .75 230 369 @b 254 338.75 -.75 .75 256.75 369 .75 254 338 @a np 259 368 :M 252 368 :L 256 371 :L 259 368 :L eofill 252 368.75 -.75 .75 259.75 368 .75 252 368 @a 252 368.75 -.75 .75 256.75 371 .75 252 368 @a -.75 -.75 256.75 371.75 .75 .75 259 368 @b 265 333.75 -.75 .75 282.75 369 .75 265 333 @a np 285 367 :M 279 370 :L 284 371 :L 285 367 :L eofill -.75 -.75 279.75 370.75 .75 .75 285 367 @b 279 370.75 -.75 .75 284.75 371 .75 279 370 @a -.75 -.75 284.75 371.75 .75 .75 285 367 @b 264 324.75 -.75 .75 302.75 324 .75 264 324 @a np 302 320 :M 302 328 :L 305 324 :L 302 320 :L eofill -.75 -.75 302.75 328.75 .75 .75 302 320 @b -.75 -.75 302.75 328.75 .75 .75 305 324 @b 302 320.75 -.75 .75 305.75 324 .75 302 320 @a -.75 -.75 317.75 368.75 .75 .75 318 340 @b np 321 368 :M 314 367 :L 317 371 :L 321 368 :L eofill 314 367.75 -.75 .75 321.75 368 .75 314 367 @a 314 367.75 -.75 .75 317.75 371 .75 314 367 @a -.75 -.75 317.75 371.75 .75 .75 321 368 @b -.75 -.75 374.75 368.75 .75 .75 389 334 @b np 378 368 :M 371 365 :L 373 370 :L 378 368 :L eofill 371 365.75 -.75 .75 378.75 368 .75 371 365 @a 371 365.75 -.75 .75 373.75 370 .75 371 365 @a -.75 -.75 373.75 370.75 .75 .75 378 368 @b 399 338.75 -.75 .75 403.75 370 .75 399 338 @a np 406 368 :M 399 369 :L 403 372 :L 406 368 :L eofill -.75 -.75 399.75 369.75 .75 .75 406 368 @b 399 369.75 -.75 .75 403.75 372 .75 399 369 @a -.75 -.75 403.75 372.75 .75 .75 406 368 @b 410 322.75 -.75 .75 438.75 322 .75 410 322 @a np 438 318 :M 438 326 :L 441 322 :L 438 318 :L eofill -.75 -.75 438.75 326.75 .75 .75 438 318 @b -.75 -.75 438.75 326.75 .75 .75 441 322 @b 438 318.75 -.75 .75 441.75 322 .75 438 318 @a -.75 -.75 437.75 369.75 .75 .75 446 336 @b np 441 369 :M 434 367 :L 436 371 :L 441 369 :L eofill 434 367.75 -.75 .75 441.75 369 .75 434 367 @a 434 367.75 -.75 .75 436.75 371 .75 434 367 @a -.75 -.75 436.75 371.75 .75 .75 441 369 @b 458 337.75 -.75 .75 466.75 369 .75 458 337 @a np 469 367 :M 463 369 :L 467 371 :L 469 367 :L eofill -.75 -.75 463.75 369.75 .75 .75 469 367 @b 463 369.75 -.75 .75 467.75 371 .75 463 369 @a -.75 -.75 467.75 371.75 .75 .75 469 367 @b gR gS 0 0 552 730 rC 215 452 :M f0_12 sf (Figure )S 252 452 :M (11. Three SEMs)S 81 480 :M f3_12 sf .497 .05(We illustrate the difference between equivalence and equivalence over a set )J f0_12 sf (V)S 466 480 :M f3_12 sf .615 .061( with)J 59 496 :M 1.988 .199(the models in )J 136 496 :M 1.89 .189(Figure 11. Models i and ii do not share the same set of substantive)J 59 512 :M .659 .066(variables, so they are not covariance or partial correlation equivalent. Models ii and iii)J 59 528 :M 2.399 .24(share the same substantive variables, but are not covariance equivalent or partial)J 59 544 :M .424 .042(correlation equivalent because, for example, model iii entails )J f1_12 sf (r)S 368 547 :M f3_7 sf .115(X2,X3.T2)A f3_12 sf 0 -3 rm .435 .044( = 0 while model ii)J 0 3 rm 59 560 :M .857 .086(does not. For )J 129 560 :M f0_12 sf (V)S 138 560 :M f3_12 sf .871 .087( = {X)J f3_7 sf 0 3 rm (1)S 0 -3 rm 171 560 :M f3_12 sf 1.024 .102(, X)J 188 563 :M f3_7 sf (2)S 192 560 :M f3_12 sf 1.024 .102(, X)J 208 563 :M f3_7 sf (3)S 212 560 :M f3_12 sf 1.024 .102(, X)J 228 563 :M f3_7 sf (4)S 232 560 :M f3_12 sf .734 .073(}, however, the situation is quite different. All three)J 59 576 :M 1.39 .139(models are partial correlation equivalent over )J 295 576 :M f0_12 sf (V)S 304 576 :M f3_12 sf 1.637 .164(, and models i and ii are covariance)J 59 592 :M .712 .071(equivalent over )J f0_12 sf (V)S 148 592 :M f3_12 sf 1.029 .103(. Models ii and iii are not covariance equivalent over)J f0_12 sf .39 .039( V )J 432 592 :M f3_12 sf .679 .068(because, for)J 59 608 :M .652 .065(example, model ii entails that )J 210 608 :M f1_12 sf (r)S 217 611 :M f3_7 sf (X1,X2)S 236 608 :M f7_12 sf .536 .054( )J f1_12 sf (r)S 248 611 :M f3_7 sf (X3,X4)S 267 608 :M f7_12 sf .945 .095( = )J f1_12 sf (r)S 291 611 :M f3_7 sf (X1,X3)S 310 608 :M f7_12 sf .536 .054( )J f1_12 sf (r)S 322 611 :M f3_7 sf (X2,X4)S 341 608 :M f3_12 sf .703 .07( while model iii does not. The)J 59 624 :M .04 .004(next four subsections will outline what is known about these various kinds of equivalence)J 59 640 :M (in both recursive and non-recursive SEMs.)S endp %%Page: 20 20 %%BeginPageSetup initializepage (peter; page: 20 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (20)S gR gS 0 0 552 730 rC 59 54 :M f4_12 sf (4)S 65 54 :M (.)S 68 54 :M (1)S 74 54 :M ( )S 81 54 :M (Covariance and Partial Correlation Equivalence in Recursive SEMs)S 81 76 :M f3_12 sf .642 .064(In this section we consider equivalence over RSEMs with no correlated errors. For)J 59 92 :M 1.19 .119(two such RSEMs, covariance equivalence holds if and only if zero partial correlation)J 59 108 :M .304 .03(equivalence holds \(Spirtes et al. 1993\). In RSEMs, only two concepts need to be defined)J 59 124 :M .177 .018(to graphically characterize covariance \(or partial correlation\) equivalence: )J f0_12 sf .044(adjacency)A f3_12 sf .121 .012( and)J 59 140 :M f0_12 sf .441 .044(unshielded collider)J 158 140 :M f3_12 sf .843 .084(. Two variables X and Y are adjacent in a directed graph G just in)J 59 156 :M (case X )S f1_12 sf S 106 156 :M f3_12 sf ( Y is in G, or Y)S f1_12 sf S 192 156 :M f3_12 sf ( X is in G.)S 145 175 259 106 rC 153 219 :M f8_12 sf (X)S 145.5 201.5 23 21 rS 194 276 :M (Z)S 186.5 259.5 23 21 rS 233 217 :M (Y)S 225.5 199.5 22 21 rS 306 216 :M (X)S 299.5 198.5 22 21 rS 347 273 :M (Z)S 339.5 256.5 23 21 rS 386 214 :M (Y)S 378.5 196.5 23 21 rS 145 187 :M -.11(Unshielded Collider)A 145 175 109 18 rC 145 187 94 1 rF gR gS 145 175 259 106 rC 301 188 :M f8_12 sf -.208(Shielded Collider)A 301 176 102 19 rC 301 188 82 1 rF gR gS 145 175 259 106 rC 169 227 -1 1 187 250 1 169 226 @a np 187 246 :M 182 250 :L 189 255 :L 187 246 :L eofill -1 -1 183 251 1 1 187 246 @b 182 251 -1 1 190 255 1 182 250 @a 187 247 -1 1 190 255 1 187 246 @a -1 -1 210 250 1 1 228 223 @b np 212 250 :M 207 246 :L 205 254 :L 212 250 :L eofill 207 247 -1 1 213 250 1 207 246 @a -1 -1 206 255 1 1 207 246 @b -1 -1 206 255 1 1 212 250 @b 326 225 -1 1 343 246 1 326 224 @a np 343 242 :M 338 246 :L 345 251 :L 343 242 :L eofill -1 -1 339 247 1 1 343 242 @b 338 247 -1 1 346 251 1 338 246 @a 343 243 -1 1 346 251 1 343 242 @a -1 -1 366 247 1 1 387 221 @b np 369 246 :M 364 242 :L 362 250 :L 369 246 :L eofill 364 243 -1 1 370 246 1 364 242 @a -1 -1 363 251 1 1 364 242 @b -1 -1 363 251 1 1 369 246 @b 324 209 -1 1 369 208 1 324 208 @a np 365 205 :M 365 212 :L 374 208 :L 365 205 :L eofill -1 -1 366 213 1 1 365 205 @b -1 -1 366 213 1 1 374 208 @b 365 206 -1 1 375 208 1 365 205 @a gR gS 0 0 552 730 rC 249 306 :M f0_12 sf (Figure )S 286 306 :M (12.)S 59 334 :M f3_12 sf .101 .01(A triple of variables is a )J f0_12 sf .035(collider)A 265 334 :M f3_12 sf .128 .013( in G just in case X )J f1_12 sf S 373 334 :M f3_12 sf .171 .017( Z )J 387 334 :M f1_12 sf S 399 334 :M f3_12 sf .148 .015( Y is in G, and Z is)J 59 350 :M .094 .009(an )J f0_12 sf .296 .03(unshielded collider)J 171 350 :M f3_12 sf .337 .034( between X and Y just in case is a collider and X and Y)J 59 366 :M .236 .024(are not adjacent \()J 144 366 :M .233 .023(Figure 12\). The first theorem stated below is a simple consequence of a)J 59 382 :M (theorem proved in Verma and Pearl \(1990\), and in Frydenberg \(1990\).)S 95 414 :M f0_12 sf 1.101 .11(RSEM Partial Correlation Equivalence Theorem)J 355 414 :M f3_12 sf 1.523 .152(: Two RSEMs with)J 95 430 :M 3.086 .309(the same variables and no correlated errors are partial correlation)J 95 446 :M 1.078 .108(equivalent if and only if their respective directed graphs have the same)J 95 462 :M (adjacencies and the same unshielded colliders.)S 95 494 :M f0_12 sf .663 .066(RSEM Covariance Equivalence Theorem)J 310 494 :M f3_12 sf .955 .096(: Two RSEMs with the same)J 95 510 :M .514 .051(variables and no correlated errors are covariance equivalent if and only if)J 95 526 :M .891 .089(their respective directed graphs have the same adjacencies and the same)J 95 542 :M (unshielded colliders.)S 81 574 :M .461 .046(By the first theorem, if two RSEMs with the same variables and no corelated errors)J 59 590 :M 1.57 .157(have the same adjacencies and unshielded colliders, then they are partial correlation)J 59 606 :M .392 .039(equivalent. The latter theorem can be proved by induction on the number of edges taken)J 59 622 :M 2.293 .229(out of two exactly identified SEMs. Take an exactly identified RSEM R with no)J 59 638 :M 2.339 .234(correlated errors and no latent variables. Then R can be parameterized to fit any)J 59 654 :M .424 .042(covariance matrix. Take an edge out of R to form R\325. R\325 now entails a single vanishing)J 59 670 :M .202 .02(partial correlation )J 149 670 :M f1_12 sf (r)S 156 673 :M f3_7 sf (XY)S f3_12 sf 0 -3 rm (.)S 0 3 rm f0_7 sf (Z)S 174 670 :M f3_12 sf .256 .026(, so it can be parameterized to fit any covariance matrix in which)J 59 686 :M f1_12 sf (r)S 66 689 :M f3_7 sf (XY)S f3_12 sf 0 -3 rm (.)S 0 3 rm f0_7 sf (Z)S 84 686 :M f3_12 sf .146 .015(, = 0. If M is partial correlation equivalent to R, and also has no correlated errors or)J endp %%Page: 21 21 %%BeginPageSetup initializepage (peter; page: 21 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (21)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .763 .076(latent variables, then it too is exactly identified and thus can also can parameterize any)J 59 70 :M .142 .014(covariance matrix. If we form M\325 by taking an edge away from M that corresponds to the)J 59 86 :M .644 .064(same adjacency we took out of R to form R\325, and M\325 and R\325 are still partial correlation)J 59 102 :M (equivalent, then M\325 can parameterize any covariance matrix in which )S f1_12 sf (r)S 402 105 :M f3_7 sf (XY.)S 414 105 :M f0_7 sf (Z)S 419 102 :M f3_12 sf (, = 0.)S 59 130 :M f4_12 sf (4)S 65 130 :M (.)S 68 130 :M (2)S 74 130 :M ( )S 81 130 :M (Covariance and Partial Correlation Equivalence Over the Measured Variables in)S 59 146 :M (RSEMs)S 81 168 :M f3_12 sf 1.376 .138(We now consider the case where there may be latent variables and/or correlated)J 59 184 :M 2.041 .204(errors, and the question is whether two SEMs are covariance equivalent or partial)J 59 200 :M (correlation equivalent over a set of measured variables )S f0_12 sf (V)S 333 200 :M f3_12 sf (. Since an RSEM with correlated)S 59 216 :M 1.248 .125(errors is partial correlation equivalent to another RSEM with a latent variable but no)J 59 232 :M 2.241 .224(correlated errors, the problem of deciding partial correlation equivalence over the)J 59 248 :M .671 .067(measured variables when there are correlated errors reduces to the problem of deciding)J 59 264 :M .444 .044(partial correlation equivalence over the measured variables when there are no correlated)J 59 280 :M (errors.)S 81 296 :M 2.237 .224(Covariance equivalence over the measured variables entails partial correlation)J 59 312 :M .874 .087(equivalence over the measured variables, but the converse does not hold. Consider the)J 59 328 :M (directed graphs i and ii in Figure 13, where the set of measured variables )S 411 328 :M f0_12 sf (V)S 420 328 :M f3_12 sf ( = {X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 451 328 :M f3_12 sf (, X)S 466 331 :M f3_7 sf (2)S 470 328 :M f3_12 sf (, X)S 485 331 :M f3_7 sf (3)S 489 328 :M f3_12 sf (,)S 59 344 :M (X)S 68 347 :M f3_7 sf (4)S 72 344 :M f3_12 sf 2.251 .225(} and the errors are uncorrelated. Although these graphs are partial correlation)J 59 360 :M .095 .009(equivalent over )J 137 360 :M f0_12 sf (V)S 146 360 :M f3_12 sf .092 .009( \(neither entails any partial correlations among the measured variables\),)J 59 376 :M (they are not covariance equivalent over )S 251 376 :M f0_12 sf (V)S 260 376 :M f3_12 sf (, since model i but not model ii entails that)S 172 398 :M f1_12 sf (r)S 179 401 :M .075 0 rm f7_7 sf (X1,X2)S 200 398 :M .739 0 rm f7_12 sf ( )S f1_12 sf (r)S 211 401 :M .075 0 rm f7_7 sf (X3,X4)S 232 398 :M 1.122 0 rm f7_12 sf ( = )S f1_12 sf (r)S 254 401 :M .075 0 rm f7_7 sf (X1,X3)S 275 398 :M .739 0 rm f7_12 sf ( )S f1_12 sf (r)S 286 401 :M .507 0 rm f7_7 sf (X2,X4 )S 309 398 :M .383 0 rm f7_12 sf (= )S f1_12 sf (r)S 327 401 :M .075 0 rm f7_7 sf (X1,X4)S 348 401 :M .432 0 rm ( )S f1_12 sf 0 -3 rm (r)S 0 3 rm 357 401 :M .075 0 rm f7_7 sf (X2,X3)S .75 lw 103 439 343 128 rC 232.5 514.5 24 20 rS 234 516 21 17 rC 234 525 :M f3_12 sf ( )S 237 525 :M (X)S 246 528 :M f3_7 sf (4)S gR .75 lw gS 103 439 343 128 rC 189.5 515.5 24 20 rS 191 517 21 17 rC 191 526 :M f3_12 sf ( )S 194 526 :M (X)S 203 529 :M f3_7 sf (3)S gR gS 103 439 343 128 rC 147.5 515.5 24 20 rS 149 517 21 17 rC 149 526 :M f3_12 sf ( )S 152 526 :M (X)S 161 529 :M f3_7 sf (2)S gR gS 103 439 343 128 rC 103.5 515.5 24 20 rS 105 517 21 17 rC 105 526 :M f3_12 sf ( )S 108 526 :M (X)S 117 529 :M f3_7 sf (1)S gR gS 103 439 343 128 rC 389.5 450.5 24 20 rS 391 452 21 17 rC 391 461 :M f3_12 sf ( )S 394 461 :M (X)S 403 464 :M f3_7 sf (4)S gR gS 103 439 343 128 rC 420.5 514.5 24 20 rS 422 516 21 17 rC 422 525 :M f3_12 sf ( )S 425 525 :M (X)S 434 528 :M f3_7 sf (3)S gR gS 103 439 343 128 rC 378.5 514.5 24 20 rS 380 516 21 17 rC 380 525 :M f3_12 sf ( )S 383 525 :M (X)S 392 528 :M f3_7 sf (2)S gR gS 103 439 343 128 rC 334.5 514.5 24 20 rS 336 516 21 17 rC 336 525 :M f3_12 sf ( )S 339 525 :M (X)S 348 528 :M f3_7 sf (1)S gR gS 175 446 32 17 rC 175 455 :M f3_12 sf ( )S 178 455 :M ( )S 181 455 :M (T)S gR gS 103 439 343 128 rC 30 26 185.5 452.5 @f -.75 -.75 126.75 507.75 .75 .75 174 463 @b np 129 509 :M 124 504 :L 124 509 :L 129 509 :L eofill 124 504.75 -.75 .75 129.75 509 .75 124 504 @a -.75 -.75 124.75 509.75 .75 .75 124 504 @b 124 509.75 -.75 .75 129.75 509 .75 124 509 @a -.75 -.75 165.75 508.75 .75 .75 178 467 @b np 168 508 :M 162 506 :L 164 510 :L 168 508 :L eofill 162 506.75 -.75 .75 168.75 508 .75 162 506 @a 162 506.75 -.75 .75 164.75 510 .75 162 506 @a -.75 -.75 164.75 510.75 .75 .75 168 508 @b 192 463.75 -.75 .75 202.75 508 .75 192 463 @a np 205 507 :M 198 508 :L 202 511 :L 205 507 :L eofill -.75 -.75 198.75 508.75 .75 .75 205 507 @b 198 508.75 -.75 .75 202.75 511 .75 198 508 @a -.75 -.75 202.75 511.75 .75 .75 205 507 @b 200 462.75 -.75 .75 237.75 507 .75 200 462 @a np 239 504 :M 234 509 :L 239 509 :L 239 504 :L eofill -.75 -.75 234.75 509.75 .75 .75 239 504 @b 234 509.75 -.75 .75 239.75 509 .75 234 509 @a -.75 -.75 239.75 509.75 .75 .75 239 504 @b 361 520.75 -.75 .75 373.75 520 .75 361 520 @a np 373 516 :M 373 524 :L 376 520 :L 373 516 :L eofill -.75 -.75 373.75 524.75 .75 .75 373 516 @b -.75 -.75 373.75 524.75 .75 .75 376 520 @b 373 516.75 -.75 .75 376.75 520 .75 373 516 @a 403 521.75 -.75 .75 416.75 521 .75 403 521 @a np 416 517 :M 416 525 :L 419 521 :L 416 517 :L eofill -.75 -.75 416.75 525.75 .75 .75 416 517 @b -.75 -.75 416.75 525.75 .75 .75 419 521 @b 416 517.75 -.75 .75 419.75 521 .75 416 517 @a -.75 -.75 354.75 508.75 .75 .75 387 468 @b np 357 510 :M 352 505 :L 352 510 :L 357 510 :L eofill 352 505.75 -.75 .75 357.75 510 .75 352 505 @a -.75 -.75 352.75 510.75 .75 .75 352 505 @b 352 510.75 -.75 .75 357.75 510 .75 352 510 @a -.75 -.75 394.75 505.75 .75 .75 398 473 @b np 398 505 :M 391 504 :L 394 508 :L 398 505 :L eofill 391 504.75 -.75 .75 398.75 505 .75 391 504 @a 391 504.75 -.75 .75 394.75 508 .75 391 504 @a -.75 -.75 394.75 508.75 .75 .75 398 505 @b 411 473.75 -.75 .75 432.75 507 .75 411 473 @a np 434 504 :M 428 508 :L 433 509 :L 434 504 :L eofill -.75 -.75 428.75 508.75 .75 .75 434 504 @b 428 508.75 -.75 .75 433.75 509 .75 428 508 @a -.75 -.75 433.75 509.75 .75 .75 434 504 @b 391 543 31 19 rC 391 552 :M f3_12 sf ( )S 394 552 :M f0_12 sf <28>S 399 552 :M (i)S 403 552 :M (i)S 407 552 :M <29>S gR gS 172 545 34 21 rC 172 554 :M f3_12 sf ( )S 175 554 :M f0_12 sf <28>S 180 554 :M (i)S 184 554 :M <29>S gR gS 0 0 552 730 rC 62 592 :M f0_12 sf (Figure )S 99 592 :M (13: Two graphs that are partial correlation equivalent over {X)S 418 595 :M f0_7 sf (1)S 422 592 :M f0_12 sf (, X)S 437 595 :M f0_7 sf (2)S 441 592 :M f0_12 sf (, X)S 456 595 :M f0_7 sf (3)S 460 592 :M f0_12 sf (, X)S 475 595 :M f0_7 sf (4)S 479 592 :M f0_12 sf (},)S 143 608 :M (but not covariance equivalent over {X)S 337 611 :M f0_7 sf (1)S 341 608 :M f0_12 sf (, X)S 356 611 :M f0_7 sf (2)S 360 608 :M f0_12 sf (, X)S 375 611 :M f0_7 sf (3)S 379 608 :M f0_12 sf (, X)S 394 611 :M f0_7 sf (4)S 398 608 :M f0_12 sf (}.)S 81 636 :M f3_12 sf .941 .094(Spirtes, Meek, and Richardson \(1995\) have given a polynomial \(in the number of)J 59 652 :M 2.195 .219(variables in the two RSEMs\) time algorithm for deciding when two RSEMs with)J 59 668 :M 1.08 .108(uncorrelated errors are partial correlation equivalent over the measured variables. The)J 59 684 :M 1.866 .187(algorithm is too complex to present here, but some examples of partial correlation)J endp %%Page: 22 22 %%BeginPageSetup initializepage (peter; page: 22 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (22)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf 2.156 .216(equivalence are given in section )J 234 54 :M 1.964 .196(5.1. A feasible algorithm for deciding covariance)J 59 70 :M (equivalence over a set of measured variables is not known.)S 59 98 :M f4_12 sf (4)S 65 98 :M (.)S 68 98 :M (3)S 74 98 :M ( )S 81 98 :M (Covariance and Partial Correlation Equivalence in Non-recursive SEMs)S 81 120 :M f3_12 sf .662 .066(Assuming uncorrelated errors, Richardson \(1994, 1995\) has given an algorithm for)J 59 136 :M 2.687 .269(deciding when two non-recursive SEMs are partial correlation equivalent that is)J 59 152 :M .449 .045(polynomial in the number of variables in the two SEMs. It is not known whether partial)J 59 168 :M (correlation equivalence entails covariance equivalence in this case.)S 115 187 319 92 rC 121.5 193.5 27 26 rS 128 209 :M f8_12 sf (X)S 137 213 :M f8_10 sf (1)S 121.5 250.5 27 26 rS 128 266 :M f8_12 sf (X)S 137 270 :M f8_10 sf (2)S 214.5 192.5 27 26 rS 221 208 :M f8_12 sf (X)S 230 212 :M f8_10 sf (3)S 13 156 204 206 206 @k 153 207 -1 1 201 206 1 153 206 @a 13 66 114 221 218 @k -1 -1 222 249 1 1 221 224 @b 13 156 204 210 263 @k 152 264 -1 1 205 263 1 152 263 @a 217.5 251.5 27 26 rS 224 267 :M f8_12 sf (X)S 233 271 :M f8_10 sf (4)S 13 -114 -66 234 249 @k -1 -1 235 244 1 1 234 219 @b 309.5 193.5 27 26 rS 316 209 :M f8_12 sf (X)S 325 213 :M f8_10 sf (1)S 309.5 250.5 27 26 rS 316 266 :M f8_12 sf (X)S 325 270 :M f8_10 sf (2)S 402.5 192.5 27 26 rS 409 208 :M f8_12 sf (X)S 418 212 :M f8_10 sf (3)S 13 190 238 401 256 @k 341 216 -1 1 397 253 1 341 215 @a 13 66 114 409 218 @k -1 -1 410 249 1 1 409 224 @b 13 121 169 400 213 @k -1 -1 341 257 1 1 395 216 @b 405.5 251.5 27 26 rS 412 267 :M f8_12 sf (X)S 421 271 :M f8_10 sf (4)S 13 -114 -66 422 249 @k -1 -1 423 244 1 1 422 219 @b gR gS 0 0 552 730 rC 134 308 :M f0_12 sf (Figure )S 171 308 :M (14. Partial Correlation Equivalent Cyclic )S 384 308 :M (SEMs)S 81 340 :M f3_12 sf 1.289 .129(One noteworthy corollary of Richardson\325s theorem is that for every SE)J 444 340 :M 1.604 .16(M with a)J 59 356 :M .503 .05(directed cycle, there is another partial correlation equivalent SE)J 372 356 :M .588 .059(M with a cycle reversed)J 59 372 :M .91 .091(in direction. And while partial correlation equivalent RSEMs without correlated errors)J 59 388 :M .114 .011(always have the same adjacencies, partial correlation equivalent SEMs without correlated)J 59 404 :M .798 .08(errors can have directed cyclic graphs with different adjacencies. For example, the two)J 59 420 :M 3.355 .335(SEMs in )J 113 420 :M 2.831 .283(Figure )J 152 420 :M 2.813 .281(14 are partial correlation equivalent but do not have the same)J 59 436 :M (adjacencies.)S 59 464 :M f4_12 sf (4)S 65 464 :M (.)S 68 464 :M (4)S 74 464 :M ( )S 81 464 :M (Covariance and Partial Correlation Equivalence over the Measured Variables in)S 59 480 :M (Non-recursive SEMs)S 81 502 :M f3_12 sf 1.05 .105(No feasible general algorithm for deciding either partial correlation or covariance)J 59 518 :M .287 .029(equivalence over a set of measured variables is known for non-recursive SEMs when the)J 59 534 :M (measured variables are a proper subset of the substantive variables in the SEM.)S 175 578 :M f4_14 sf (5)S 182 578 :M (.)S 185 578 :M 4 0 rm ( )S 192 578 :M (Search )S 233 578 :M (Algo)S 258 578 :M (rithms in TETRAD II)S 81 606 :M f3_12 sf 1.274 .127(In this section we describe some of the constraint based, provably correct search)J 59 622 :M 2.415 .241(procedures that we have implemented in TETRAD II. Our approach is to design)J 59 638 :M .62 .062(algorithms that search for all RSEMs consistent with background knowledge that )J 464 638 :M (entail)S 59 654 :M .177 .018(constraints on the covariance matrix that are judged to hold in the population. Depending)J 59 670 :M .498 .05(on the type of background knowledge, and what kind of RSEM is sought, we use either)J 59 686 :M .327 .033(vanishing partial correlation constraints or vanishing tetrad constraints. Because in many)J endp %%Page: 23 23 %%BeginPageSetup initializepage (peter; page: 23 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (23)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .037 .004(cases the number of possible constraints is too large to examine exhaustively, some of the)J 59 70 :M .576 .058(algorithms we describe make sequential decisions about constraints and thus test only a)J 59 86 :M .263 .026(subset of the possible constraints during the search process. These sequential procedures)J 59 102 :M 1.165 .116(are still correct in the sense we defined in section 1.3.2, but might not be optimal on)J 59 118 :M 1.501 .15(realistic samples because mistakes about constraints made early in the sequence can)J 59 134 :M (ramify into mistakes made later.)S 59 162 :M f4_12 sf (5)S 65 162 :M (.)S 68 162 :M (1)S 74 162 :M ( )S 81 162 :M (The Build Algorithm)S 81 184 :M f3_12 sf (The Build module)S 168 179 :M f3_7 sf (1)S 171 179 :M (1)S 174 184 :M f3_12 sf ( of TETRAD II takes as input:)S 104 200 :M (1)S 110 200 :M (.)S 113 200 :M ( )S 122 200 :M (sample data \(either raw, or as a covariance matrix\))S 365 200 :M ( and)S 104 216 :M (2)S 110 216 :M (.)S 113 216 :M ( )S 122 216 :M (background knowledge that constrains )S 310 216 :M (RSEM)S 343 216 :M ( specification,)S 81 232 :M (and gives as output:)S 104 248 :M (1)S 110 248 :M (.)S 113 248 :M 6 .6( )J 122 248 :M .027 .003(a representation of the partial correlation equivalence class of RSEMs that is)J 122 264 :M (consistent with the background knowledge, and)S 104 280 :M (2)S 110 280 :M (.)S 113 280 :M ( )S 122 280 :M (a set of features that this class of RSEMs has in common.)S 81 312 :M (Build)S 108 312 :M .33 .033( performs statistical tests of hypotheses that specific partial correlations vanish)J 59 328 :M .444 .044(in the population, and if it cannot reject the null hypothesis at a significance level set by)J 59 344 :M .302 .03(the user, then the procedure )J 198 344 :M .283 .028(accepts the null hypothesis \(see the appendix in Scheines, et)J 59 360 :M .125 .012(al., 1994\))J 105 360 :M .215 .022(. )J 112 360 :M .134 .013(Because Build uses only information about which partial correlations are zero,)J 59 376 :M 1.314 .131(it cannot distinguish between any members of a partial correlation equivalence class;)J 59 392 :M .744 .074(hence its output is a representation of a partial correlation equivalence class of RSEMs)J 59 408 :M 1.675 .168(consistent with background knowledge. In order to achieve enough efficiency to be)J 59 424 :M 1.466 .147(practical for large numbers of variables \(up to 100\), the algorithms in Build use the)J 59 440 :M .891 .089(results of test)J 126 440 :M .86 .086(s of lower order partial correlation \(i.e., correlations conditional on small)J 59 456 :M .445 .044(sets of variables\) to restrict the tests it needs to perform on partial correlations of higher)J 59 472 :M (order.)S 81 488 :M 1.347 .135(The algorithms are correct in the sense of section 1.3.2. That is, the background)J 59 504 :M (knowledge a user enters may include assumptions about:)S 77 520 :M (1)S 83 520 :M (.)S 86 520 :M 3 .3( )J 95 520 :M 2.103 .21(whether the population RSEM contains correlated errors, or latent common)J 95 536 :M (causes;)S 77 552 :M (2)S 83 552 :M (.)S 86 552 :M ( )S 95 552 :M (time order among the variables;)S 77 568 :M (3)S 83 568 :M (.)S 86 568 :M ( )S 95 568 :M (known causal relationships among the variables;)S 77 584 :M (4)S 83 584 :M (.)S 86 584 :M ( )S 95 584 :M (causal relationships among the variables known not to hold.)S 59 662 :M ( )S 59 659.48 -.48 .48 203.48 659 .48 59 659 @a 81 671 :M f3_6 sf (1)S 84 671 :M (1)S 87 675 :M f3_10 sf .053 .005( The Build module is documented in \(Scheines, et al., 1994\), and its algorithms described in detail in)J 59 687 :M (\(Spirtes, et al, 1993\).)S endp %%Page: 24 24 %%BeginPageSetup initializepage (peter; page: 24 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (24)S gR gS 0 0 552 730 rC 59 54 :M f4_12 sf (5)S 65 54 :M (.)S 68 54 :M (1)S 74 54 :M (.)S 77 54 :M (1)S 83 54 :M ( )S 90 54 :M (Build for RSEMs Without Correlated Errors or Latent Common Causes)S 81 73 :M f3_12 sf 1.065 .107(If you assume that the generating RSEM contains no latent common causes, then)J 59 89 :M 1.017 .102(Build runs the PC algorithm, which is documented and traced in the appendix, and in)J 59 105 :M .123 .012(\(Spirtes, et. al, 1993; Scheines, et. al, 1994\). The output of the PC algorithm is a )J 450 105 :M f0_12 sf (pattern)S 488 105 :M f3_12 sf (,)S 59 121 :M .607 .061(\(Verma and Pearl, 1990\) which is a compact representation of a partial correlation \(and)J 59 137 :M 1.508 .151(covariance\) equivalence class of RSEMs without correlated errors or latent common)J 59 153 :M 1.956 .196(causes. A pattern contains a mixture of directed and undirected edges. If a pattern)J 59 169 :M 1.316 .132(contains an edge A )J 162 169 :M f1_12 sf S 174 169 :M f3_12 sf 1.119 .112( B, then the directed graph of )J f4_12 sf .512(every)A 356 169 :M f3_12 sf 1.212 .121( RSEM represented by the)J 59 185 :M .362 .036(pattern contains the edge A )J f1_12 sf S 208 185 :M f3_12 sf .446 .045( B. If a pattern contains an edge A )J f1_12 sf .403A f3_12 sf .41 .041( B then A and B are)J 59 201 :M .503 .05(adjacent in the directed graph of )J 222 201 :M f4_12 sf (every)S 248 201 :M f3_12 sf .491 .049( RSEM represented by the pattern, but the graphs)J 59 217 :M .169 .017(of some RSEMs represented by the pattern may contain the edge A )J f1_12 sf S 400 217 :M f3_12 sf .189 .019( B, and others may)J 59 233 :M .749 .075(contain the edge A )J 157 233 :M f1_12 sf S 169 233 :M f3_12 sf .749 .075( B. If a pattern contains no adjacency between A and B, then in)J 59 249 :M (every RSEM represented by the pattern A and B are not adjacent.)S 81 265 :M .325 .033(Suppose we measure only two variables A and B and find that they are significantly)J 59 281 :M .361 .036(correlated. There are two RSEMs without correlated errors or latent variables containing)J 59 297 :M (just A and B that are compatible with A and B being correlated in the population: A )S 465 297 :M f1_12 sf S 477 297 :M f3_12 sf ( B,)S 59 313 :M .489 .049(and A )J 93 313 :M f1_12 sf S 105 313 :M f3_12 sf .434 .043( B. The output of Build in this case is the pattern A )J 361 313 :M f1_12 sf .23A f3_12 sf .352 .035( B, which represents the)J 59 329 :M 1.784 .178(two RSEMs in this equivalence class. This illustrates the slogan \322correlation is not)J 59 345 :M .261 .026(causation\323, because the statistical information is not sufficient to predict the results of an)J 59 361 :M (ideal intervention on A or B.)S 81 377 :M .863 .086(In this example the output of Build is not useful for predicting the effects of ideal)J 59 393 :M 1.821 .182(interventions. The next example shows how the output of Build can in some cases)J 59 409 :M .58 .058(provide more useful causal knowledge. Suppose that for four measured variables, A, B,)J 59 425 :M .547 .055(C, and D, from sample data we conclude that in the population, )J f1_12 sf (r)S 383 428 :M f3_7 sf .228(A,B)A f3_12 sf 0 -3 rm .41 .041( = 0, )J 0 3 rm f1_12 sf 0 -3 rm (r)S 0 3 rm 428 428 :M f3_7 sf .156(A,D.C)A f3_12 sf 0 -3 rm .387 .039( = 0,)J 0 3 rm 470 425 :M f1_12 sf .851 .085( )J 474 425 :M f3_12 sf (and)S 59 441 :M f1_12 sf (r)S 66 444 :M f3_7 sf (B,D.C)S 84 441 :M f3_12 sf .378 .038( = 0, but that no other partial correlations \(other than those entailed by those listed\))J 59 457 :M .777 .078(vanish. In that case the output of Build is the pattern in Figure 15, which represents an)J 59 473 :M (equivalence class of RSEMs with only one member, also shown in Figure 15.)S 431 468 :M f3_7 sf (1)S 434 468 :M (2)S 1 G 108 492 333 141 rC 149 496 24 23 rF 0 G 148.5 495.5 25 24 rS 157 500 12 13 rC 157 509 :M f3_12 sf (A)S gR 1 G gS 108 492 333 141 rC 208 496 24 23 rF 0 G 207.5 495.5 25 24 rS 216 500 12 13 rC 216 509 :M f3_12 sf (B)S gR gS 108 492 333 141 rC 181 553 24 22 rF 0 G 180.5 552.5 25 23 rS 189 557 12 13 rC 189 566 :M f3_12 sf (C)S gR gS 108 492 333 141 rC 182 609 24 23 rF 0 G 181.5 608.5 25 24 rS 190 614 12 13 rC 190 623 :M f3_12 sf (D)S gR 0 G gS 108 492 333 141 rC np 189 552 :M 179 544 :L 181 542 :L 183 540 :L 189 552 :L 189 552 :L eofill 161 519 -1 1 183 543 1 161 518 @a np 196 552 :M 200 540 :L 202 542 :L 205 543 :L 196 552 :L 196 552 :L eofill -1 -1 204 544 1 1 222 518 @b np 192 609 :M 189 597 :L 192 597 :L 195 597 :L 192 609 :L 192 609 :L eofill -1 -1 194 598 1 1 193 575 @b 1 G 275 494 24 23 rF 0 G 274.5 493.5 25 24 rS 284 498 12 13 rC 284 507 :M f3_12 sf (A)S gR 1 G gS 108 492 333 141 rC 334 494 24 23 rF 0 G 333.5 493.5 25 24 rS 343 498 12 13 rC 343 507 :M f3_12 sf (B)S gR gS 108 492 333 141 rC 307 551 24 22 rF 0 G 306.5 550.5 25 23 rS 316 555 12 13 rC 316 564 :M f3_12 sf (C)S gR gS 108 492 333 141 rC 308 607 24 23 rF 0 G 307.5 606.5 25 24 rS 317 612 12 13 rC 317 621 :M f3_12 sf (D)S gR 0 G gS 108 492 333 141 rC np 316 550 :M 306 542 :L 308 540 :L 310 538 :L 316 550 :L 316 550 :L eofill 287 517 -1 1 309 541 1 287 516 @a np 323 550 :M 327 538 :L 329 540 :L 332 541 :L 323 550 :L 323 550 :L eofill -1 -1 330 542 1 1 348 516 @b np 319 607 :M 316 595 :L 319 595 :L 322 595 :L 319 607 :L 319 607 :L eofill -1 -1 320 596 1 1 319 573 @b 353 561 87 54 rC 353 570 :M f0_12 sf (E)S 361 570 :M (q)S 368 570 :M (u)S 375 570 :M (i)S 379 570 :M (v)S 386 570 :M (a)S 392 570 :M (l)S 396 570 :M (e)S 402 570 :M (n)S 409 570 :M (c)S 415 570 :M (e)S 353 582 :M (C)S 362 582 :M (l)S 366 582 :M (a)S 372 582 :M (s)S 378 582 :M (s)S 353 594 :M (R)S 362 594 :M (e)S 368 594 :M (p)S 375 594 :M (r)S 380 594 :M (e)S 386 594 :M (s)S 392 594 :M (e)S 398 594 :M (n)S 405 594 :M (t)S 409 594 :M (e)S 415 594 :M (d)S 353 606 :M (b)S 360 606 :M (y)S 367 606 :M 1 .1( )J 371 606 :M (t)S 375 606 :M (h)S 382 606 :M (e)S 388 606 :M 1 .1( )J 392 606 :M (P)S 400 606 :M (a)S 406 606 :M (t)S 410 606 :M (t)S 414 606 :M (e)S 420 606 :M (r)S 425 606 :M (n)S gR gS 109 580 47 17 rC 109 589 :M f0_12 sf (P)S 117 589 :M (a)S 123 589 :M (t)S 127 589 :M (t)S 131 589 :M (e)S 137 589 :M (r)S 142 589 :M (n)S gR gS 108 492 333 141 rC -2 -2 253 629 2 2 251 498 @b gR gS 0 0 552 730 rC 250 658 :M f0_12 sf (Figure )S 287 658 :M (15)S 59 674 :M f3_12 sf ( )S 59 671.48 -.48 .48 203.48 671 .48 59 671 @a 81 683 :M f3_6 sf (1)S 84 683 :M (2)S 87 687 :M f3_10 sf ( See the appendix for an example in which the equivalence class is larger.)S endp %%Page: 25 25 %%BeginPageSetup initializepage (peter; page: 25 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (25)S gR gS 0 0 552 730 rC 81 54 :M f3_12 sf 2.252 .225(In this case, the output of Build is sufficient to predict the results of ideally)J 59 70 :M .437 .044(intervening on A, B, C, or D. Of course, the assumption of no correlated errors or latent)J 59 86 :M .412 .041(variables is a very strong one, and in the next section we consider what happens when it)J 59 102 :M (is abandoned.)S 59 130 :M f4_12 sf (5)S 65 130 :M (.)S 68 130 :M (1)S 74 130 :M (.)S 77 130 :M (2)S 83 130 :M ( )S 90 130 :M (Build for RSEMs with Correlated Errors)S 81 149 :M f3_12 sf .625 .063(If you allow that the RSEM that generated the data might have correlated errors or)J 59 165 :M 1.881 .188(latent common causes, then Build runs the FCI algorithm, which is documented in)J 59 181 :M .612 .061(\(Spirtes, et. al., 1993, chapter 6\). The output of the FCI algorithm is a )J 409 181 :M f0_12 sf .373 .037(partial ancestor)J 59 197 :M (graph)S 90 197 :M f3_12 sf .633 .063( \(PAG\).)J 129 192 :M f3_7 sf (1)S 132 192 :M (3)S 135 197 :M f3_12 sf .388 .039( A is an )J f0_12 sf .258(ancestor)A f3_12 sf .703 .07( of B in a directed graph when there is a directed path)J 59 213 :M 2.544 .254(from A to B. Just as patterns represent features common to a partial correlation)J 59 229 :M .668 .067(equivalence class of RSEMs without latent variables, PAGs represent features common)J 59 245 :M .297 .03(to a set of RSEMs that are partial correlation equivalent over the measured variables. \(In)J 59 261 :M .722 .072(this section, for the sake of brevity, we will refer to the PAG simply as an equivalence)J 59 277 :M (class.\) We will illustrate with the two examples from the previous section.)S 81 293 :M 1.903 .19(Again, suppose we measure two variables A and B, and find that they have a)J 59 309 :M .198 .02(significant correlation and conclude that they are correlated in the population. The output)J 59 325 :M .65 .065(of Build in this case is the partial ancestor graph shown in )J 352 325 :M .601 .06(Figure 16. Because we have)J 59 341 :M 2.643 .264(placed no limit on the number of distinct latent variables, the equivalence class)J 59 357 :M .304 .03(represented by the output is actually infinite, and we have shown only a few members of)J 59 373 :M .841 .084(the equivalence class in )J 181 373 :M .747 .075(Figure 16)J 229 373 :M .966 .097(. The presence of a \322o\323 at both ends of an edge in a)J 59 389 :M .966 .097(PAG makes no claim about the ancestor relationship common to every member of the)J 59 405 :M .362 .036(equivalence class. Note that in some of the RSEMs represented by the PAG \(e.g. \(i\) and)J 59 421 :M .129 .013(\(iv\) of )J 93 421 :M .118 .012(Figure 16\), A is an ancestor of B, and in others \(e.g. \(ii\) and \(iii\) of Figure 16)J 466 421 :M .134 .013(\) it is)J 59 437 :M .322 .032(not. Similarly, in some of the members of the equivalence class \(e.g. \(ii\) of )J 429 437 :M .278 .028(Figure 16)J 476 437 :M .416 .042(\) B)J 59 453 :M .555 .056(is an ancestor of A, and in others \(e.g. \(i\), \(iii\) and \(iv\) of Figure 16\) it is not. Thus this)J 59 469 :M .063 .006(PAG shows us that we cannot predict the results of ideally intervening to change either A)J 59 485 :M (or B from this data without further background knowledge.)S 59 650 :M ( )S 59 647.48 -.48 .48 203.48 647 .48 59 647 @a 81 659 :M f3_6 sf (1)S 84 659 :M (3)S 87 663 :M f3_10 sf .383 .038( In fact the output is described in \(Spirtes, et. al., 1993, and Scheines, et. al., 1994\) as a POIPG, or)J 59 675 :M .479 .048(partially oriented inducing path graph. POIPGs can, without loss of generality, be interpreted much more)J 59 687 :M (naturally as PAGs.)S endp %%Page: 26 26 %%BeginPageSetup initializepage (peter; page: 26 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (26)S gR 1 G gS 88 41 373 194 rC 19 21 273.5 94.5 @j 0 G .9 lw 20 22 273.5 94.5 @f 1 G 93 88 18 19 rF 0 G 92.5 87.5 19 20 rS 1 G 149 88 19 19 rF 0 G 148.5 87.5 20 20 rS 98 92 76 19 rC 98 100 :M f3_11 sf (A)S 106 100 :M .25 .025( )J 109 100 :M .25 .025( )J 112 100 :M .25 .025( )J 115 100 :M .25 .025( )J 118 100 :M .25 .025( )J 121 100 :M .25 .025( )J 124 100 :M .25 .025( )J 127 100 :M .25 .025( )J 130 100 :M .25 .025( )J 133 100 :M .25 .025( )J 136 100 :M .25 .025( )J 139 100 :M .25 .025( )J 142 100 :M .25 .025( )J 145 100 :M .25 .025( )J 148 100 :M .25 .025( )J 151 100 :M .25 .025( )J 154 100 :M .25 .025( )J 157 100 :M .25 .025( )J 160 100 :M (B)S gR gS 88 41 373 194 rC 235 42 18 19 rF 0 G .9 lw 234.5 41.5 19 20 rS 1 G 291 42 19 19 rF 0 G 290.5 41.5 20 20 rS 240 46 81 20 rC 240 54 :M f3_11 sf (A)S 248 54 :M .25 .025( )J 251 54 :M .25 .025( )J 254 54 :M .25 .025( )J 257 54 :M .25 .025( )J 260 54 :M .25 .025( )J 263 54 :M .25 .025( )J 266 54 :M .25 .025( )J 269 54 :M .25 .025( )J 272 54 :M .25 .025( )J 275 54 :M .25 .025( )J 278 54 :M .25 .025( )J 281 54 :M .25 .025( )J 284 54 :M .25 .025( )J 287 54 :M .25 .025( )J 290 54 :M .25 .025( )J 293 54 :M .25 .025( )J 296 54 :M .25 .025( )J 299 54 :M .25 .025( )J 302 54 :M (B)S gR gS 88 41 373 194 rC 376 42 18 19 rF 0 G .9 lw 375.5 41.5 19 20 rS 1 G 432 42 22 22 rF 0 G 431.5 41.5 23 23 rS 381 46 80 22 rC 381 54 :M f3_11 sf (A)S 389 54 :M .25 .025( )J 392 54 :M .25 .025( )J 395 54 :M .25 .025( )J 398 54 :M .25 .025( )J 401 54 :M .25 .025( )J 404 54 :M .25 .025( )J 407 54 :M .25 .025( )J 410 54 :M .25 .025( )J 413 54 :M .25 .025( )J 416 54 :M .25 .025( )J 419 54 :M .25 .025( )J 422 54 :M .25 .025( )J 425 54 :M .25 .025( )J 428 54 :M .25 .025( )J 431 54 :M .25 .025( )J 434 54 :M .25 .025( )J 437 54 :M .25 .025( )J 440 54 :M .25 .025( )J 443 54 :M (B)S gR gS 88 41 373 194 rC 235 122 18 19 rF 0 G .9 lw 234.5 121.5 19 20 rS 1 G 291 122 19 19 rF 0 G 290.5 121.5 20 20 rS 240 126 77 16 rC 240 134 :M f3_11 sf (A)S 248 134 :M .25 .025( )J 251 134 :M .25 .025( )J 254 134 :M .25 .025( )J 257 134 :M .25 .025( )J 260 134 :M .25 .025( )J 263 134 :M .25 .025( )J 266 134 :M .25 .025( )J 269 134 :M .25 .025( )J 272 134 :M .25 .025( )J 275 134 :M .25 .025( )J 278 134 :M .25 .025( )J 281 134 :M .25 .025( )J 284 134 :M .25 .025( )J 287 134 :M .25 .025( )J 290 134 :M .25 .025( )J 293 134 :M .25 .025( )J 296 134 :M .25 .025( )J 299 134 :M .25 .025( )J 302 134 :M (B)S gR gS 110 92 10 12 rC 110 100 :M 0 G f3_11 sf (o)S gR gS 143 92 10 12 rC 143 100 :M 0 G f3_11 sf (o)S gR 0 G gS 88 41 373 194 rC 115 97.9 -.9 .9 143.9 97 .9 115 97 @a np 289 51 :M 278 54 :L 278 51 :L 278 48 :L 289 51 :L 289 51 :L eofill 253 51.9 -.9 .9 279.9 51 .9 253 51 @a np 392 51 :M 403 48 :L 403 51 :L 403 54 :L 392 51 :L 392 51 :L eofill 403 51.9 -.9 .9 432.9 51 .9 403 51 @a 271 90 10 12 rC 271 98 :M f3_11 sf (T)S gR gS 88 41 373 194 rC np 242 121 :M 250 113 :L 252 116 :L 252 118 :L 242 121 :L 242 121 :L eofill -.9 -.9 252.9 116.9 .9 .9 273 104 @b np 301 121 :M 290 118 :L 291 115 :L 293 113 :L 301 121 :L 301 121 :L eofill 273 105.9 -.9 .9 291.9 115 .9 273 105 @a 1 G 19 21 407.5 94.5 @j 0 G .9 lw 20 22 407.5 94.5 @f 1 G 369 122 18 19 rF 0 G 368.5 121.5 19 20 rS 1 G 424 122 20 19 rF 0 G 423.5 121.5 21 20 rS 373 126 77 15 rC 373 134 :M f3_11 sf (A)S 381 134 :M .25 .025( )J 384 134 :M .25 .025( )J 387 134 :M .25 .025( )J 390 134 :M .25 .025( )J 393 134 :M .25 .025( )J 396 134 :M .25 .025( )J 399 134 :M .25 .025( )J 402 134 :M .25 .025( )J 405 134 :M .25 .025( )J 408 134 :M .25 .025( )J 411 134 :M .25 .025( )J 414 134 :M .25 .025( )J 417 134 :M .25 .025( )J 420 134 :M .25 .025( )J 423 134 :M .25 .025( )J 426 134 :M .25 .025( )J 429 134 :M .25 .025( )J 432 134 :M .25 .025( )J 435 134 :M (B)S gR gS 405 90 10 12 rC 405 98 :M f3_11 sf (T)S gR gS 88 41 373 194 rC np 376 121 :M 384 113 :L 385 116 :L 386 118 :L 376 121 :L 376 121 :L eofill -.9 -.9 386.9 116.9 .9 .9 407 104 @b np 435 121 :M 424 118 :L 425 115 :L 427 113 :L 435 121 :L 435 121 :L eofill 407 105.9 -.9 .9 425.9 115 .9 407 105 @a np 423 130 :M 412 133 :L 412 130 :L 412 128 :L 423 130 :L 423 130 :L eofill 387 131.9 -.9 .9 412.9 131 .9 387 131 @a 265 67 173 12 rC 265 75 :M f3_11 sf <28>S 269 75 :M (i)S 272 75 :M <29>S 276 75 :M .25 .025( )J 279 75 :M .25 .025( )J 282 75 :M .25 .025( )J 285 75 :M .25 .025( )J 288 75 :M .25 .025( )J 291 75 :M .25 .025( )J 294 75 :M .25 .025( )J 297 75 :M .25 .025( )J 300 75 :M .25 .025( )J 303 75 :M .25 .025( )J 306 75 :M .25 .025( )J 309 75 :M .25 .025( )J 312 75 :M .25 .025( )J 315 75 :M .25 .025( )J 318 75 :M .25 .025( )J 321 75 :M .25 .025( )J 324 75 :M .25 .025( )J 327 75 :M .25 .025( )J 330 75 :M .25 .025( )J 333 75 :M .25 .025( )J 336 75 :M .25 .025( )J 339 75 :M .25 .025( )J 342 75 :M .25 .025( )J 345 75 :M .25 .025( )J 348 75 :M .25 .025( )J 351 75 :M .25 .025( )J 354 75 :M .25 .025( )J 357 75 :M .25 .025( )J 360 75 :M .25 .025( )J 363 75 :M .25 .025( )J 366 75 :M .25 .025( )J 369 75 :M .25 .025( )J 372 75 :M .25 .025( )J 375 75 :M .25 .025( )J 378 75 :M .25 .025( )J 381 75 :M .25 .025( )J 384 75 :M .25 .025( )J 387 75 :M .25 .025( )J 390 75 :M .25 .025( )J 393 75 :M .25 .025( )J 396 75 :M .25 .025( )J 399 75 :M .25 .025( )J 402 75 :M .25 .025( )J 405 75 :M .25 .025( )J 408 75 :M .25 .025( )J 411 75 :M .25 .025( )J 414 75 :M <28>S 418 75 :M (i)S 421 75 :M (i)S 424 75 :M <29>S gR gS 261 148 170 20 rC 261 156 :M f3_11 sf <28>S 265 156 :M (i)S 268 156 :M (i)S 271 156 :M (i)S 274 156 :M <29>S 278 156 :M .25 .025( )J 281 156 :M .25 .025( )J 284 156 :M .25 .025( )J 287 156 :M .25 .025( )J 290 156 :M .25 .025( )J 293 156 :M .25 .025( )J 296 156 :M .25 .025( )J 299 156 :M .25 .025( )J 302 156 :M .25 .025( )J 305 156 :M .25 .025( )J 308 156 :M .25 .025( )J 311 156 :M .25 .025( )J 314 156 :M .25 .025( )J 317 156 :M .25 .025( )J 320 156 :M .25 .025( )J 323 156 :M .25 .025( )J 326 156 :M .25 .025( )J 329 156 :M .25 .025( )J 332 156 :M .25 .025( )J 335 156 :M .25 .025( )J 338 156 :M .25 .025( )J 341 156 :M .25 .025( )J 344 156 :M .25 .025( )J 347 156 :M .25 .025( )J 350 156 :M .25 .025( )J 353 156 :M .25 .025( )J 356 156 :M .25 .025( )J 359 156 :M .25 .025( )J 362 156 :M .25 .025( )J 365 156 :M .25 .025( )J 368 156 :M .25 .025( )J 371 156 :M .25 .025( )J 374 156 :M .25 .025( )J 377 156 :M .25 .025( )J 380 156 :M .25 .025( )J 383 156 :M .25 .025( )J 386 156 :M .25 .025( )J 389 156 :M .25 .025( )J 392 156 :M .25 .025( )J 395 156 :M .25 .025( )J 398 156 :M .25 .025( )J 401 156 :M .25 .025( )J 404 156 :M .25 .025( )J 407 156 :M <28>S 411 156 :M (i)S 414 156 :M (v)S 419 156 :M <29>S gR gS 89 143 84 43 rC 112 152 :M f0_12 sf (P)S 120 152 :M (a)S 126 152 :M (r)S 131 152 :M (t)S 135 152 :M (i)S 139 152 :M (a)S 145 152 :M (l)S 104 164 :M (A)S 113 164 :M (n)S 120 164 :M (c)S 126 164 :M (e)S 132 164 :M (s)S 138 164 :M (t)S 142 164 :M (r)S 147 164 :M (a)S 153 164 :M (l)S 113 176 :M (G)S 123 176 :M (r)S 128 176 :M (a)S 134 176 :M (p)S 141 176 :M (h)S gR gS 272 182 177 52 rC 283 191 :M f0_12 sf (E)S 291 191 :M (x)S 298 191 :M (a)S 304 191 :M (m)S 314 191 :M (p)S 321 191 :M (l)S 325 191 :M (e)S 331 191 :M (s)S 337 191 :M 1 .1( )J 341 191 :M (o)S 348 191 :M (f)S 353 191 :M 1 .1( )J 357 191 :M (R)S 366 191 :M (S)S 374 191 :M (E)S 382 191 :M (M)S 393 191 :M (S)S 401 191 :M 1 .1( )J 405 191 :M (i)S 409 191 :M (n)S 416 191 :M 1 .1( )J 420 191 :M (t)S 424 191 :M (h)S 431 191 :M (e)S 272 203 :M (E)S 280 203 :M (q)S 287 203 :M (u)S 294 203 :M (i)S 298 203 :M (v)S 305 203 :M (a)S 311 203 :M (l)S 315 203 :M (e)S 321 203 :M (n)S 328 203 :M (c)S 334 203 :M (e)S 340 203 :M 1 .1( )J 344 203 :M (C)S 353 203 :M (l)S 357 203 :M (a)S 363 203 :M (s)S 369 203 :M (s)S 375 203 :M 1 .1( )J 379 203 :M (R)S 388 203 :M (e)S 394 203 :M (p)S 401 203 :M (r)S 406 203 :M (e)S 412 203 :M (s)S 418 203 :M (e)S 424 203 :M (n)S 431 203 :M (t)S 435 203 :M (e)S 441 203 :M (d)S 340 215 :M (b)S 347 215 :M (y)S 354 215 :M 1 .1( )J 358 215 :M 1 .1( )J 362 215 :M (t)S 366 215 :M (h)S 373 215 :M (e)S 291 227 :M (P)S 299 227 :M (a)S 305 227 :M (r)S 310 227 :M (t)S 314 227 :M (i)S 318 227 :M (a)S 324 227 :M (l)S 328 227 :M 1 .1( )J 332 227 :M (A)S 341 227 :M (n)S 348 227 :M (c)S 354 227 :M (e)S 360 227 :M (t)S 364 227 :M (s)S 370 227 :M (t)S 374 227 :M (r)S 379 227 :M (a)S 385 227 :M (l)S 389 227 :M 1 .1( )J 393 227 :M (G)S 403 227 :M (r)S 408 227 :M (a)S 414 227 :M (p)S 421 227 :M (h)S gR gS 88 41 373 194 rC -2 -2 210 199 2 2 208 51 @b gR gS 0 0 552 730 rC 250 260 :M f0_12 sf (Figure )S 287 260 :M (16)S 81 288 :M f3_12 sf .182 .018(Whereas the pattern A )J 194 288 :M f1_12 sf .183A f3_12 sf .195 .019( B informed us that either A is a cause of B or B is a cause)J 59 304 :M .15 .015(of A, the PAG A o)J 150 304 :M f1_12 sf .127A f3_12 sf .138 .014(o B informs us that either A is a cause of B, B is a cause of A, there)J 59 320 :M 1.509 .151(is a latent common cause, or there is some combination of these causal connections)J 59 336 :M .401 .04(responsible for the correlation. The next example shows how PAGs output by Build can)J 59 352 :M 1.114 .111(be used to predict the effects of some ideal interventions. Consider the example from)J 59 368 :M 1.937 .194(Figure 15 again, where there are four measured variables A, B, C, and D, and we)J 59 384 :M (conclude from the data that in the population )S 278 384 :M f1_12 sf (r)S 285 387 :M f3_7 sf (A,B)S f3_12 sf 0 -3 rm ( = 0, )S 0 3 rm 321 384 :M f1_12 sf (r)S 328 387 :M f3_7 sf (A,D.C)S f3_12 sf 0 -3 rm ( = 0, and )S 0 3 rm f1_12 sf 0 -3 rm (r)S 0 3 rm 398 387 :M f3_7 sf (B,D.C)S 416 384 :M f3_12 sf ( = 0, but that no)S 59 400 :M 1.779 .178(other partial correlations vanish. Assuming that correlated errors might exist in the)J 59 416 :M (generating RSEM, the output of Build is the PAG in the left hand side of )S 412 416 :M (Figure 17.)S 81 432 :M .466 .047(A and C are adjacent in the PAG because the correlation of A and C conditional on)J 59 448 :M .564 .056(every subset of the measured variables does not vanish \(i.e. )J f1_12 sf (r)S 363 451 :M f3_7 sf .204(A,C)A f3_12 sf 0 -3 rm .29 .029(, )J 0 3 rm 381 448 :M f1_12 sf (r)S 388 451 :M f3_7 sf (A,C.B)S 406 448 :M f3_12 sf .844 .084(, )J 413 448 :M f1_12 sf (r)S 420 451 :M f3_7 sf .14(A,C.D)A f3_12 sf 0 -3 rm .209 .021(, )J 0 3 rm 445 448 :M f1_12 sf (r)S 452 451 :M f3_7 sf (A,C.BD)S 475 448 :M f3_12 sf .774 .077( do)J 59 464 :M .075 .007(not vanish.\) The \322o\323 at the A end of the edge between A and C entails neither that A is an)J 59 480 :M .142 .014(ancestor of B in every member of the equivalence class nor that A is not an ancestor of B)J 59 496 :M .681 .068(in every member of the equivalence class. The \322>\323 at the C end of the edge between A)J 59 512 :M 1.421 .142(and C in the PAG means that C is not an ancestor of A in any RSEM in the partial)J 59 528 :M 1.654 .165(correlation equivalence class. Similarly, C is not an ancestor of B, and D is not an)J 59 544 :M .259 .026(ancestor of C in any RSEM in the partial correlation equivalence class. Finally, a \322\321\323 at)J 59 560 :M .23 .023(the C end of the edge between C and D means that C is an ancestor of D in every RSEM)J 59 576 :M (in the equivalence class.)S 81 592 :M 3.468 .347(From this PAG we can make predictions about the effects of some ideal)J 59 608 :M 1.396 .14(interventions, but not others. For example, it is not possible to determine if an ideal)J 59 624 :M .222 .022(intervention on A will affect C, because in some members of the equivalence class A is a)J 59 640 :M .591 .059(cause of C, and in others it is not. On the other hand, it is possible to determine that an)J 59 656 :M .84 .084(ideal intervention on C will affect D, because C is a cause of D in every RSEM in the)J 59 672 :M 2.339 .234(equivalence class. \(And given the distributional assumption, it is also possible to)J endp %%Page: 27 27 %%BeginPageSetup initializepage (peter; page: 27 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (27)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf 1.571 .157(determine the size of the effect that an ideal intervention on C will have on D. See)J 59 70 :M (Spirtes, et al. 1993, chapter 7\).)S 271 142 26 21 rC 271 154 :M f0_12 sf <28>S 276 154 :M f9_12 sf (i)S 280 154 :M (i)S 284 154 :M <29>S gR gS 146 277 26 21 rC 146 289 :M f0_12 sf <28>S 151 289 :M f9_12 sf (i)S 155 289 :M (i)S 159 289 :M (i)S 163 289 :M <29>S gR gS 278 275 23 17 rC 278 287 :M f0_12 sf <28>S 283 287 :M f9_12 sf (i)S 287 287 :M (v)S 292 287 :M <29>S gR gS 148 145 16 15 rC 148 157 :M f0_12 sf <28>S 153 157 :M f9_12 sf (i)S 157 157 :M <29>S gR gS 69 299 32 14 rC 69 308 :M f3_12 sf ( )S 72 308 :M f0_12 sf (P)S 80 308 :M (A)S 89 308 :M (G)S gR gS 362 141 127 55 rC 370 150 :M f3_12 sf ( )S 373 150 :M f0_12 sf (E)S 381 150 :M (x)S 388 150 :M (a)S 394 150 :M (m)S 404 150 :M (p)S 411 150 :M (l)S 415 150 :M (e)S 421 150 :M (s)S 427 150 :M 1 .1( )J 431 150 :M (f)S 436 150 :M (r)S 441 150 :M (o)S 448 150 :M (m)S 458 150 :M 1 .1( )J 462 150 :M (t)S 466 150 :M (h)S 473 150 :M (e)S 373 162 :M (E)S 381 162 :M (q)S 388 162 :M (u)S 395 162 :M (i)S 399 162 :M (v)S 406 162 :M (a)S 412 162 :M (l)S 416 162 :M (e)S 422 162 :M (n)S 429 162 :M (c)S 435 162 :M (e)S 441 162 :M 1 .1( )J 445 162 :M (C)S 454 162 :M (l)S 458 162 :M (a)S 464 162 :M (s)S 470 162 :M (s)S 370 174 :M (R)S 379 174 :M (e)S 385 174 :M (p)S 392 174 :M (r)S 397 174 :M (e)S 403 174 :M (s)S 409 174 :M (e)S 415 174 :M (n)S 422 174 :M (t)S 426 174 :M (e)S 432 174 :M (d)S 439 174 :M 1 .1( )J 443 174 :M (b)S 450 174 :M (y)S 457 174 :M 1 .1( )J 461 174 :M (t)S 465 174 :M (h)S 472 174 :M (e)S 411 186 :M (P)S 419 186 :M (A)S 428 186 :M (G)S gR .75 lw gS 60 89 429 247 rC 82.5 233.5 17 17 rS 84 235 14 14 rC 84 244 :M f3_12 sf ( )S 87 244 :M (D)S gR gS 60 89 429 247 rC 84.5 187.5 17 17 rS 86 189 14 14 rC 86 198 :M f3_12 sf ( )S 89 198 :M (C)S gR gS 60 89 429 247 rC 103.5 146.5 17 17 rS 105 148 14 14 rC 105 157 :M f3_12 sf ( )S 108 157 :M (B)S gR gS 60 89 429 247 rC 61.5 145.5 16 17 rS 63 147 13 14 rC 63 156 :M f3_12 sf ( )S 66 156 :M (A)S gR gS 60 89 429 247 rC 3 5 76 166 @f 3 5 111 167 @f 77 168.75 -.75 .75 86.75 184 .75 77 168 @a np 89 181 :M 83 185 :L 88 186 :L 89 181 :L eofill -.75 -.75 83.75 185.75 .75 .75 89 181 @b 83 185.75 -.75 .75 88.75 186 .75 83 185 @a -.75 -.75 88.75 186.75 .75 .75 89 181 @b -.75 -.75 100.75 184.75 .75 .75 109 170 @b np 103 185 :M 97 181 :L 98 186 :L 103 185 :L eofill 97 181.75 -.75 .75 103.75 185 .75 97 181 @a 97 181.75 -.75 .75 98.75 186 .75 97 181 @a -.75 -.75 98.75 186.75 .75 .75 103 185 @b -.75 -.75 92.75 228.75 .75 .75 92 204 @b np 95 227 :M 88 227 :L 92 231 :L 95 227 :L eofill 88 227.75 -.75 .75 95.75 227 .75 88 227 @a 88 227.75 -.75 .75 92.75 231 .75 88 227 @a -.75 -.75 92.75 231.75 .75 .75 95 227 @b 190.5 180.5 17 17 rS 192 182 14 14 rC 192 191 :M f3_12 sf ( )S 195 191 :M (D)S gR gS 60 89 429 247 rC 192.5 135.5 17 16 rS 194 137 14 13 rC 194 146 :M f3_12 sf ( )S 197 146 :M (C)S gR gS 60 89 429 247 rC 211.5 93.5 17 17 rS 213 95 14 14 rC 213 104 :M f3_12 sf ( )S 216 104 :M (B)S gR gS 60 89 429 247 rC 169.5 92.5 16 17 rS 171 94 13 14 rC 171 103 :M f3_12 sf ( )S 174 103 :M (A)S gR gS 60 89 429 247 rC 184 111.75 -.75 .75 195.75 131 .75 184 111 @a np 197 128 :M 191 132 :L 196 133 :L 197 128 :L eofill -.75 -.75 191.75 132.75 .75 .75 197 128 @b 191 132.75 -.75 .75 196.75 133 .75 191 132 @a -.75 -.75 196.75 133.75 .75 .75 197 128 @b -.75 -.75 207.75 131.75 .75 .75 218 111 @b np 211 132 :M 205 128 :L 206 133 :L 211 132 :L eofill 205 128.75 -.75 .75 211.75 132 .75 205 128 @a 205 128.75 -.75 .75 206.75 133 .75 205 128 @a -.75 -.75 206.75 133.75 .75 .75 211 132 @b -.75 -.75 200.75 175.75 .75 .75 200 151 @b np 203 174 :M 196 174 :L 200 178 :L 203 174 :L eofill 196 174.75 -.75 .75 203.75 174 .75 196 174 @a 196 174.75 -.75 .75 200.75 178 .75 196 174 @a -.75 -.75 200.75 178.75 .75 .75 203 174 @b 322.5 190.5 17 17 rS 324 192 14 14 rC 324 201 :M f3_12 sf ( )S 327 201 :M (D)S gR gS 60 89 429 247 rC 324.5 144.5 17 17 rS 326 146 14 14 rC 326 155 :M f3_12 sf ( )S 329 155 :M (C)S gR gS 60 89 429 247 rC 343.5 103.5 17 16 rS 345 105 14 13 rC 345 114 :M f3_12 sf ( )S 348 114 :M (B)S gR gS 60 89 429 247 rC 313.5 103.5 17 16 rS 315 105 14 13 rC 315 114 :M f3_12 sf ( )S 318 114 :M (A)S gR gS 60 89 429 247 rC -.75 -.75 339.75 140.75 .75 .75 350 121 @b np 343 141 :M 337 138 :L 338 143 :L 343 141 :L eofill 337 138.75 -.75 .75 343.75 141 .75 337 138 @a 337 138.75 -.75 .75 338.75 143 .75 337 138 @a -.75 -.75 338.75 143.75 .75 .75 343 141 @b -.75 -.75 332.75 185.75 .75 .75 332 161 @b np 335 184 :M 328 184 :L 332 187 :L 335 184 :L eofill 328 184.75 -.75 .75 335.75 184 .75 328 184 @a 328 184.75 -.75 .75 332.75 187 .75 328 184 @a -.75 -.75 332.75 187.75 .75 .75 335 184 @b 277 95 15 16 rC 277 104 :M f3_12 sf (T)S 284 107 :M f3_7 sf (1)S gR gS 60 89 429 247 rC 281.5 100.5 10 @e 292 105.75 -.75 .75 309.75 111 .75 292 105 @a np 310 108 :M 307 114 :L 312 112 :L 310 108 :L eofill -.75 -.75 307.75 114.75 .75 .75 310 108 @b -.75 -.75 307.75 114.75 .75 .75 312 112 @b 310 108.75 -.75 .75 312.75 112 .75 310 108 @a 289 111.75 -.75 .75 322.75 141 .75 289 111 @a np 324 138 :M 319 143 :L 324 143 :L 324 138 :L eofill -.75 -.75 319.75 143.75 .75 .75 324 138 @b 319 143.75 -.75 .75 324.75 143 .75 319 143 @a -.75 -.75 324.75 143.75 .75 .75 324 138 @b 179.5 318.5 17 17 rS 181 320 14 14 rC 181 329 :M f3_12 sf ( )S 184 329 :M (D)S gR gS 60 89 429 247 rC 181.5 272.5 16 17 rS 183 274 13 14 rC 183 283 :M f3_12 sf ( )S 186 283 :M (C)S gR gS 60 89 429 247 rC 189.5 230.5 16 17 rS 191 232 13 14 rC 191 241 :M f3_12 sf ( )S 194 241 :M (B)S gR gS 60 89 429 247 rC 161.5 230.5 17 17 rS 163 232 14 14 rC 163 241 :M f3_12 sf ( )S 166 241 :M (A)S gR gS 60 89 429 247 rC 176 247.75 -.75 .75 185.75 270 .75 176 247 @a np 187 268 :M 181 270 :L 185 272 :L 187 268 :L eofill -.75 -.75 181.75 270.75 .75 .75 187 268 @b 181 270.75 -.75 .75 185.75 272 .75 181 270 @a -.75 -.75 185.75 272.75 .75 .75 187 268 @b -.75 -.75 189.75 313.75 .75 .75 189 289 @b np 192 312 :M 185 312 :L 189 315 :L 192 312 :L eofill 185 312.75 -.75 .75 192.75 312 .75 185 312 @a 185 312.75 -.75 .75 189.75 315 .75 185 312 @a -.75 -.75 189.75 315.75 .75 .75 192 312 @b 227 217 15 19 rC 227 226 :M f3_12 sf (T)S 234 229 :M f3_7 sf (1)S gR gS 60 89 429 247 rC 233.5 223.5 10 @e -.75 -.75 210.75 236.75 .75 .75 223 228 @b np 213 238 :M 209 232 :L 208 237 :L 213 238 :L eofill 209 232.75 -.75 .75 213.75 238 .75 209 232 @a -.75 -.75 208.75 237.75 .75 .75 209 232 @b 208 237.75 -.75 .75 213.75 238 .75 208 237 @a -.75 -.75 202.75 269.75 .75 .75 229 234 @b np 206 271 :M 200 267 :L 201 271 :L 206 271 :L eofill 200 267.75 -.75 .75 206.75 271 .75 200 267 @a 200 267.75 -.75 .75 201.75 271 .75 200 267 @a 201 271.75 -.75 .75 206.75 271 .75 201 271 @a 322.5 319.5 17 16 rS 324 321 14 13 rC 324 330 :M f3_12 sf ( )S 327 330 :M (D)S gR gS 60 89 429 247 rC 324.5 273.5 17 17 rS 326 275 14 14 rC 326 284 :M f3_12 sf ( )S 329 284 :M (C)S gR gS 60 89 429 247 rC 337.5 231.5 17 17 rS 339 233 14 14 rC 339 242 :M f3_12 sf ( )S 342 242 :M (B)S gR gS 60 89 429 247 rC 313.5 231.5 17 17 rS 315 233 14 14 rC 315 242 :M f3_12 sf ( )S 318 242 :M (A)S gR gS 60 89 429 247 rC -.75 -.75 332.75 314.75 .75 .75 332 290 @b np 335 313 :M 328 313 :L 332 316 :L 335 313 :L eofill 328 313.75 -.75 .75 335.75 313 .75 328 313 @a 328 313.75 -.75 .75 332.75 316 .75 328 313 @a -.75 -.75 332.75 316.75 .75 .75 335 313 @b 277 224 17 16 rC 277 233 :M f3_12 sf (T)S 284 236 :M f3_7 sf (1)S gR gS 60 89 429 247 rC 281.5 229.5 10 @e 292 234.75 -.75 .75 309.75 240 .75 292 234 @a np 310 237 :M 307 243 :L 312 241 :L 310 237 :L eofill -.75 -.75 307.75 243.75 .75 .75 310 237 @b -.75 -.75 307.75 243.75 .75 .75 312 241 @b 310 237.75 -.75 .75 312.75 241 .75 310 237 @a 289 239.75 -.75 .75 322.75 270 .75 289 239 @a np 324 266 :M 319 272 :L 324 271 :L 324 266 :L eofill -.75 -.75 319.75 272.75 .75 .75 324 266 @b -.75 -.75 319.75 272.75 .75 .75 324 271 @b -.75 -.75 324.75 271.75 .75 .75 324 266 @b 379 224 19 19 rC 379 233 :M f3_12 sf (T)S 386 236 :M f3_7 sf (2)S gR gS 60 89 429 247 rC 385.5 229.5 10 @e -.75 -.75 359.75 240.75 .75 .75 375 235 @b np 361 243 :M 359 237 :L 357 241 :L 361 243 :L eofill 359 237.75 -.75 .75 361.75 243 .75 359 237 @a -.75 -.75 357.75 241.75 .75 .75 359 237 @b 357 241.75 -.75 .75 361.75 243 .75 357 241 @a -.75 -.75 345.75 270.75 .75 .75 381 239 @b np 348 272 :M 343 266 :L 343 271 :L 348 272 :L eofill 343 266.75 -.75 .75 348.75 272 .75 343 266 @a -.75 -.75 343.75 271.75 .75 .75 343 266 @b 343 271.75 -.75 .75 348.75 272 .75 343 271 @a -2 -2 131 316 2 2 129 130 @b -1 -1 254 307 1 1 253 122 @b 157 211 -1 1 362 210 1 157 210 @a gR gS 0 0 552 730 rC 250 361 :M f0_12 sf (Figure )S 287 361 :M (17)S 81 389 :M f3_12 sf .628 .063(The partial correlation equivalence class in )J 297 389 :M .641 .064(Figure 17, which includes RSEMs with)J 59 405 :M 2.565 .257(latent variables, is much larger \(in fact it is infinite\) than the partial correlation)J 59 421 :M .241 .024(equivalence class in Figure 15, which does not include models with latent variables. This)J 59 437 :M .057 .006(in turn means that the conclusions that we can draw are weaker than if we assume that the)J 59 453 :M .685 .068(generating RSEM has no correlated errors or latent common causes. For example, with)J 59 469 :M .623 .062(this assumption we can conclude that A is a cause of C; without it we cannot. With the)J 59 485 :M .046 .005(assumption we can estimate the size of the effect that an ideal intervention on A will have)J 59 501 :M .91 .091(on C; without it we cannot. While the conclusions that can be drawn even without the)J 59 517 :M .272 .027(assumption of no latent variables are weaker than when the assumption is made, they are)J 59 533 :M .574 .057(not trivial. Asymptotically, we can reliably conclude that C is a cause of D, and we can)J 59 549 :M (estimate the size of the effect an ideal intervention on C will have on D.)S 81 565 :M 1.078 .108(It should also be noted that even though in general not all of the members of the)J 59 581 :M 1.032 .103(partial correlation equivalence class are covariance equivalent, this does not affect the)J 59 597 :M .115 .011(reliability of the conclusions. It simply means that there may be stronger conclusions that)J 59 613 :M 1.626 .163(could be drawn if we used more information than simply which partial correlations)J 59 629 :M (vanish.)S 81 645 :M .737 .074(Finally, we note that there are examples in which there is no RSEM without latent)J 59 661 :M .826 .083(variables that is compatible with a correlation matrix, but there are RSEMs with latent)J 59 677 :M .567 .057(variables that are. Suppose that we measure A, B, C, and D and from the data conclude)J endp %%Page: 28 28 %%BeginPageSetup initializepage (peter; page: 28 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (28)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .156 .016(that in the population, )J f1_12 sf (r)S 175 57 :M f3_7 sf .215 .021(A,C )J f3_12 sf 0 -3 rm .206 .021(= 0, )J 0 3 rm 211 54 :M f1_12 sf (r)S 218 57 :M f3_7 sf .206 .021(A,D )J 232 54 :M f3_12 sf .209 .021(= 0, and )J 275 54 :M f1_12 sf (r)S 282 57 :M f3_7 sf .096 .01(B,D )J f3_12 sf 0 -3 rm .177 .018(= 0, but that no other partial correlations)J 0 3 rm 59 70 :M .154 .015(\(other than those entailed by these three\) vanish. In that case the output of Build is A o)J 479 70 :M f1_12 sf S 59 86 :M f3_12 sf .651 .065(B )J 71 86 :M f1_12 sf S 84 86 :M f3_12 sf .176 .018( C )J f1_12 sf .266A f3_12 sf .504 .05( D. The double headed arrow between B and C means that in every member)J 59 102 :M .102 .01(of the equivalence class represented by the PAG B is not an ancestor of C and C is not an)J 59 118 :M .022 .002(ancestor of B. This is only possible in an RSEM with a latent variable causing both B and)J 59 134 :M (C, so every member of the equivalence class contains a latent variable.)S 59 162 :M f4_12 sf (5)S 65 162 :M (.)S 68 162 :M (1)S 74 162 :M (.)S 77 162 :M (3)S 83 162 :M ( )S 90 162 :M (What Can Go Wrong)S 81 181 :M f3_12 sf (In general, the correctness of Build\325s output depends upon several factors:)S 77 213 :M (1)S 83 213 :M (.)S 86 213 :M ( )S 95 213 :M (The correctness of the background knowledge input to the algorithm.)S 77 229 :M (2)S 83 229 :M (.)S 86 229 :M ( )S 95 229 :M (Whether the recursiveness condition holds, i.e., that there are no feedback loops.)S 77 245 :M (3)S 83 245 :M (.)S 86 245 :M ( )S 95 245 :M (Whether the Causal Independence assumption holds.)S 77 261 :M (4)S 83 261 :M (.)S 86 261 :M ( )S 95 261 :M (Whether the Faithfulness assumption holds.)S 77 277 :M (5)S 83 277 :M (.)S 86 277 :M ( )S 95 277 :M (Whether the distributional assumptions made by the statistical tests hold.)S 77 293 :M (6)S 83 293 :M (.)S 86 293 :M ( )S 95 293 :M (The power of the statistical tests against alternatives.)S 77 309 :M (7)S 83 309 :M (.)S 86 309 :M ( )S 95 309 :M (The significance level used in the statistical tests.)S 81 341 :M .387 .039(In the case of Build under the assumption of no latent variables, it is not difficult to)J 59 357 :M 1.155 .115(take the output pattern which represents a partial correlation equivalence class \(and a)J 59 373 :M 2.027 .203(covariance equivalence class\) of RSEMs, and use it to find a single RSEM in the)J 59 389 :M .814 .081(equivalence class. A sketch of this process is described in the TETRAD II manual and)J 59 405 :M 1.166 .117(can be automated \(Meek, 1995\). Once this is done, the user can estimate and test the)J 59 421 :M 1.364 .136(selected RSEM using such programs as EQS, or LISREL. \(All RSEMs in the partial)J 59 437 :M .614 .061(correlation equivalence class parameterized by their respective ML parameter estimates)J 59 453 :M .608 .061(have the same p-value.\) In addition, the user can approximate the sampling distribution)J 59 469 :M .292 .029(using the method described in section )J 246 469 :M .294 .029(1.3.3. The user should keep in mind however, that)J 59 485 :M .578 .058(the sampling distribution of the output may show that even when the RSEMs suggested)J 59 501 :M .539 .054(by TETRAD II fit the data very well, it is possible that there are other RSEMs that will)J 59 517 :M .04 .004(also fit the data well and are equally compatible with background knowledge, particularly)J 59 533 :M .089 .009(when the sample size is small. This suggests that further research on the search is needed,)J 59 549 :M .39 .039(and that Build might be improved by outputting multiple patterns--something which can)J 59 565 :M .346 .035(be done in a limited way in the present implementation by varying the significance level)J 59 581 :M .421 .042(used in the procedure. Also, at large sample sizes, even slight deviations from normality)J 59 597 :M .766 .077(or linearity can lead to the rejection of an otherwise correct RSEM. Finally, if a model)J 59 613 :M .3 .03(produced by search is tested on the data used to find the model specification, the p-value)J 59 629 :M .485 .049(of the test is )J 123 629 :M f4_12 sf .102(not)A f3_12 sf .379 .038( a measure of the error probability of the model specification procedure.)J 59 645 :M 1.194 .119(For a discussion of the meaning of such p-values, see Glymour, et al. \(1987\). Where)J 59 661 :M (possible, models generated from one sample should be cross-validated on others.)S endp %%Page: 29 29 %%BeginPageSetup initializepage (peter; page: 29 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (29)S gR gS 0 0 552 730 rC 81 54 :M f3_12 sf 1.787 .179(In the case of Build under the assumption of latent variables, more research is)J 59 70 :M 1.36 .136(needed to find out how to construct \(efficiently\) from the PAG which represents the)J 59 86 :M .148 .015(entire partial correlation equivalence class a single, representative RSEM. In this case the)J 59 102 :M .419 .042(output partial correlation equivalence class is not a covariance equivalence class, so that)J 59 118 :M 3.728 .373(different RSEMs represented by the output can have different p-values when)J 59 134 :M .749 .075(parameterized by their ML parameter estimates. More research is needed on estimating)J 59 150 :M .945 .095(and testing the output of Build under the assumption of latent variables. Spirtes, et al.)J 59 166 :M .08 .008(1993 describes some algorithms that can be used for predicting the effects of some policy)J 59 182 :M (interventions from a given PAG.)S 59 210 :M f4_12 sf (5)S 65 210 :M (.)S 68 210 :M (2)S 74 210 :M ( )S 81 210 :M (Specification Search for Latent Variable RSEMs: Purify and MIMbuild)S 81 232 :M f3_12 sf 1.303 .13(In many applications of structural equation modeling, the focus of interest is the)J 59 248 :M .78 .078(causal relationships among latent variables. In many such cases the latent variables are)J 59 264 :M .3 .03(measured with multiple indicators, and the output of Build on data for these indicators is)J 59 280 :M 3.098 .31(correct but uninformative; the correct RSEM entails no zero partial correlation)J 59 296 :M .111 .011(constraints on the indicators alone and the output of Build on the indicators is completely)J 59 312 :M .202 .02(connected and completely undirected, whether it is a pattern or a PAG. In these cases the)J 59 328 :M .954 .095(Purify and MIMbuild modules of TETRAD II can help in RSEM specification. Purify)J 59 344 :M .499 .05(helps locate unidimensional measurement models \(Anderson, Gerbing, & Hunter, 1987;)J 59 360 :M .401 .04(Anderson & Gerbing, 1988; Scheines, 1993\). The basic idea of unidimensionality is that)J 59 376 :M .308 .031(each indicator measures exactly one latent and all error terms are uncorrelated \(the exact)J 59 392 :M .567 .057(definition is more complicated, and presented in section 8.3 of the appendix\). Finding a)J 59 408 :M 1.465 .146(unidimensional measurement model is one way in which the correlations among the)J 59 424 :M 3.933 .393(latent variables may be estimated consistently. Also, given a unidimensional)J 59 440 :M -.002(measurement model, the MIMbuild module uses vanishing tetrad constraints to search the)A 59 456 :M (space of structural models, i.e., RSEM models containing only the latent variables.)S 59 484 :M f4_12 sf (5)S 65 484 :M (.)S 68 484 :M (2)S 74 484 :M (.)S 77 484 :M (1)S 83 484 :M ( )S 90 484 :M (Purify)S 81 503 :M f3_12 sf .026 .003(We make our explanation of both Purify and MIMbuild concrete by accompanying it)J 59 519 :M 1.318 .132(with an example taken from the user's manual to TETRAD II \(Scheines, et al, 1994,)J 59 535 :M 2.711 .271(chapter 9\). The example shows how Purify can aid in finding a unidimensional)J 59 551 :M .284 .028(measurement model and why it is important to do so. The population RSEM is shown in)J 59 567 :M (Figure )S 94 567 :M .02 .002(18. Our data for the example consist of the correlations among the X variables in a)J 59 583 :M 1.152 .115(pseudo-random multivariate normal sample drawn from a random parameterization of)J 59 599 :M (this RSEM \(N=2,000\).)S endp %%Page: 30 30 %%BeginPageSetup initializepage (peter; page: 30 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (30)S gR gS 84 65 381 134 rC np 218 130 :M 219 120 :L 220 120 :L 220 120 :L 220 120 :L 220 120 :L 220 120 :L 220 120 :L 221 121 :L 221 121 :L 221 121 :L 221 121 :L 221 121 :L 221 121 :L 222 121 :L 222 121 :L 222 121 :L 222 121 :L 222 121 :L 222 121 :L 222 121 :L 223 121 :L 223 121 :L 223 122 :L 223 122 :L 223 122 :L 223 122 :L 223 122 :L 224 122 :L 224 122 :L 224 122 :L 224 122 :L 224 123 :L 224 123 :L 224 123 :L 225 123 :L 225 123 :L 225 123 :L 225 123 :L 225 123 :L 225 123 :L 225 124 :L 225 124 :L 225 124 :L 226 124 :L 226 124 :L 226 124 :L 226 124 :L 226 125 :L 218 130 :L 218 130 :L eofill -1 -1 222 125 1 1 251 80 @b np 240 129 :M 238 119 :L 239 119 :L 239 119 :L 239 119 :L 239 119 :L 239 119 :L 240 119 :L 240 119 :L 240 119 :L 240 119 :L 240 119 :L 240 119 :L 241 119 :L 241 119 :L 241 119 :L 241 119 :L 241 119 :L 241 119 :L 242 119 :L 242 119 :L 242 119 :L 242 119 :L 242 119 :L 242 120 :L 243 120 :L 243 120 :L 243 120 :L 243 120 :L 243 120 :L 243 120 :L 244 120 :L 244 120 :L 244 120 :L 244 120 :L 244 120 :L 244 120 :L 245 120 :L 245 120 :L 245 121 :L 245 121 :L 245 121 :L 245 121 :L 245 121 :L 246 121 :L 246 121 :L 246 121 :L 246 121 :L 246 122 :L 240 129 :L 240 129 :L eofill -1 -1 242 123 1 1 254 83 @b np 283 130 :M 277 123 :L 277 122 :L 277 122 :L 277 122 :L 277 122 :L 278 122 :L 278 122 :L 278 122 :L 278 122 :L 278 122 :L 278 121 :L 278 121 :L 279 121 :L 279 121 :L 279 121 :L 279 121 :L 279 121 :L 279 121 :L 280 121 :L 280 121 :L 280 121 :L 280 121 :L 280 121 :L 280 121 :L 281 121 :L 281 120 :L 281 120 :L 281 120 :L 281 120 :L 281 120 :L 282 120 :L 282 120 :L 282 120 :L 282 120 :L 282 120 :L 282 120 :L 283 120 :L 283 120 :L 283 120 :L 283 120 :L 283 120 :L 283 120 :L 284 120 :L 284 120 :L 284 120 :L 284 120 :L 284 120 :L 285 120 :L 283 130 :L 283 130 :L eofill 268 84 -1 1 282 123 1 268 83 @a np 309 130 :M 301 125 :L 301 125 :L 301 125 :L 301 125 :L 301 125 :L 301 125 :L 301 124 :L 301 124 :L 301 124 :L 301 124 :L 302 124 :L 302 124 :L 302 124 :L 302 123 :L 302 123 :L 302 123 :L 302 123 :L 302 123 :L 302 123 :L 303 123 :L 303 123 :L 303 123 :L 303 122 :L 303 122 :L 303 122 :L 303 122 :L 304 122 :L 304 122 :L 304 122 :L 304 122 :L 304 122 :L 304 121 :L 304 121 :L 305 121 :L 305 121 :L 305 121 :L 305 121 :L 305 121 :L 305 121 :L 305 121 :L 306 121 :L 306 121 :L 306 121 :L 306 121 :L 306 121 :L 306 121 :L 307 120 :L 307 120 :L 309 130 :L 309 130 :L eofill 272 84 -1 1 305 125 1 272 83 @a np 346 129 :M 349 119 :L 349 120 :L 349 120 :L 350 120 :L 350 120 :L 350 120 :L 350 120 :L 350 120 :L 350 120 :L 350 120 :L 351 120 :L 351 120 :L 351 120 :L 351 120 :L 351 120 :L 351 120 :L 352 121 :L 352 121 :L 352 121 :L 352 121 :L 352 121 :L 352 121 :L 352 121 :L 353 121 :L 353 121 :L 353 122 :L 353 122 :L 353 122 :L 353 122 :L 353 122 :L 353 122 :L 354 122 :L 354 122 :L 354 122 :L 354 123 :L 354 123 :L 354 123 :L 354 123 :L 354 123 :L 354 123 :L 355 123 :L 355 124 :L 355 124 :L 355 124 :L 355 124 :L 355 124 :L 355 124 :L 355 125 :L 346 129 :L 346 129 :L eofill -1 -1 352 125 1 1 386 82 @b np 450 130 :M 441 126 :L 441 125 :L 441 125 :L 441 125 :L 441 125 :L 441 125 :L 442 125 :L 442 125 :L 442 124 :L 442 124 :L 442 124 :L 442 124 :L 442 124 :L 442 124 :L 442 124 :L 443 123 :L 443 123 :L 443 123 :L 443 123 :L 443 123 :L 443 123 :L 443 123 :L 443 123 :L 443 123 :L 444 122 :L 444 122 :L 444 122 :L 444 122 :L 444 122 :L 444 122 :L 444 122 :L 445 122 :L 445 122 :L 445 121 :L 445 121 :L 445 121 :L 445 121 :L 446 121 :L 446 121 :L 446 121 :L 446 121 :L 446 121 :L 446 121 :L 446 121 :L 447 121 :L 447 121 :L 447 121 :L 447 120 :L 450 130 :L 450 130 :L eofill 409 83 -1 1 446 125 1 409 82 @a np 374 130 :M 374 120 :L 374 120 :L 374 120 :L 374 120 :L 374 120 :L 375 120 :L 375 120 :L 375 120 :L 375 120 :L 375 120 :L 375 120 :L 376 120 :L 376 120 :L 376 120 :L 376 120 :L 376 120 :L 376 120 :L 377 120 :L 377 120 :L 377 120 :L 377 121 :L 377 121 :L 377 121 :L 378 121 :L 378 121 :L 378 121 :L 378 121 :L 378 121 :L 378 121 :L 379 121 :L 379 121 :L 379 121 :L 379 121 :L 379 121 :L 379 121 :L 379 122 :L 380 122 :L 380 122 :L 380 122 :L 380 122 :L 380 122 :L 380 122 :L 380 122 :L 381 122 :L 381 123 :L 381 123 :L 381 123 :L 381 123 :L 374 130 :L 374 130 :L eofill -1 -1 378 125 1 1 392 84 @b np 421 129 :M 414 122 :L 414 122 :L 415 122 :L 415 121 :L 415 121 :L 415 121 :L 415 121 :L 415 121 :L 415 121 :L 416 121 :L 416 121 :L 416 121 :L 416 120 :L 416 120 :L 416 120 :L 416 120 :L 417 120 :L 417 120 :L 417 120 :L 417 120 :L 417 120 :L 417 120 :L 418 120 :L 418 120 :L 418 120 :L 418 120 :L 418 120 :L 418 119 :L 419 119 :L 419 119 :L 419 119 :L 419 119 :L 419 119 :L 419 119 :L 420 119 :L 420 119 :L 420 119 :L 420 119 :L 420 119 :L 420 119 :L 421 119 :L 421 119 :L 421 119 :L 421 119 :L 421 119 :L 421 119 :L 422 119 :L 422 119 :L 421 129 :L 421 129 :L eofill 405 85 -1 1 419 122 1 405 84 @a 140 72 17 13 rC 140 81 :M f3_12 sf (T)S 147 81 :M (1)S gR gS 84 65 381 134 rC 25 14 145 79.5 @f np 101 130 :M 104 120 :L 104 121 :L 104 121 :L 104 121 :L 104 121 :L 105 121 :L 105 121 :L 105 121 :L 105 121 :L 105 121 :L 105 121 :L 106 121 :L 106 121 :L 106 121 :L 106 121 :L 106 122 :L 106 122 :L 106 122 :L 107 122 :L 107 122 :L 107 122 :L 107 122 :L 107 122 :L 107 122 :L 107 123 :L 108 123 :L 108 123 :L 108 123 :L 108 123 :L 108 123 :L 108 123 :L 108 123 :L 108 123 :L 109 124 :L 109 124 :L 109 124 :L 109 124 :L 109 124 :L 109 124 :L 109 124 :L 109 125 :L 109 125 :L 109 125 :L 110 125 :L 110 125 :L 110 125 :L 110 125 :L 110 126 :L 101 130 :L 101 130 :L eofill -1 -1 106 126 1 1 139 85 @b np 119 131 :M 119 121 :L 120 121 :L 120 121 :L 120 121 :L 120 121 :L 120 121 :L 120 121 :L 121 121 :L 121 121 :L 121 121 :L 121 121 :L 121 121 :L 121 121 :L 122 121 :L 122 122 :L 122 122 :L 122 122 :L 122 122 :L 122 122 :L 123 122 :L 123 122 :L 123 122 :L 123 122 :L 123 122 :L 123 122 :L 124 122 :L 124 122 :L 124 122 :L 124 122 :L 124 123 :L 124 123 :L 124 123 :L 125 123 :L 125 123 :L 125 123 :L 125 123 :L 125 123 :L 125 123 :L 125 124 :L 126 124 :L 126 124 :L 126 124 :L 126 124 :L 126 124 :L 126 124 :L 126 124 :L 126 124 :L 126 125 :L 119 131 :L 119 131 :L eofill -1 -1 123 126 1 1 141 87 @b np 137 132 :M 135 122 :L 135 122 :L 135 122 :L 135 122 :L 135 122 :L 135 122 :L 136 122 :L 136 122 :L 136 122 :L 136 122 :L 136 122 :L 136 122 :L 137 122 :L 137 122 :L 137 122 :L 137 122 :L 137 122 :L 138 122 :L 138 122 :L 138 122 :L 138 122 :L 138 122 :L 138 122 :L 139 122 :L 139 122 :L 139 122 :L 139 122 :L 139 122 :L 139 122 :L 140 122 :L 140 123 :L 140 123 :L 140 123 :L 140 123 :L 140 123 :L 141 123 :L 141 123 :L 141 123 :L 141 123 :L 141 123 :L 141 123 :L 141 123 :L 142 123 :L 142 123 :L 142 123 :L 142 124 :L 142 124 :L 142 124 :L 137 132 :L 137 132 :L eofill -1 -1 139 126 1 1 145 86 @b np 154 130 :M 149 121 :L 149 121 :L 149 121 :L 149 121 :L 150 121 :L 150 121 :L 150 121 :L 150 121 :L 150 121 :L 150 121 :L 151 121 :L 151 121 :L 151 121 :L 151 120 :L 151 120 :L 151 120 :L 152 120 :L 152 120 :L 152 120 :L 152 120 :L 152 120 :L 152 120 :L 153 120 :L 153 120 :L 153 120 :L 153 120 :L 153 120 :L 153 120 :L 154 120 :L 154 120 :L 154 120 :L 154 120 :L 154 120 :L 154 120 :L 155 120 :L 155 120 :L 155 120 :L 155 120 :L 155 120 :L 156 120 :L 156 120 :L 156 120 :L 156 120 :L 156 120 :L 156 120 :L 157 120 :L 157 121 :L 157 121 :L 154 130 :L 154 130 :L eofill 149 88 -1 1 153 123 1 149 87 @a np 246 78 :M 236 82 :L 236 82 :L 236 82 :L 236 82 :L 236 82 :L 236 82 :L 236 81 :L 236 81 :L 236 81 :L 236 81 :L 236 81 :L 236 81 :L 236 80 :L 236 80 :L 236 80 :L 236 80 :L 236 80 :L 236 80 :L 236 79 :L 236 79 :L 236 79 :L 236 79 :L 236 79 :L 236 79 :L 236 78 :L 236 78 :L 236 78 :L 236 78 :L 236 78 :L 236 78 :L 236 77 :L 236 77 :L 236 77 :L 236 77 :L 236 77 :L 236 76 :L 236 76 :L 236 76 :L 236 76 :L 236 76 :L 236 76 :L 236 75 :L 236 75 :L 236 75 :L 236 75 :L 236 75 :L 236 75 :L 236 74 :L 246 78 :L 246 78 :L eofill 162 80 -1 1 238 79 1 162 79 @a 92 133 16 13 rC 92 142 :M f3_12 sf (x)S 98 142 :M (1)S gR gS 84 65 381 134 rC 89.5 131.5 15 14 rS 111 133 16 13 rC 111 142 :M f3_12 sf (x)S 117 142 :M (2)S gR gS 84 65 381 134 rC 108.5 131.5 15 14 rS 130 133 16 13 rC 130 142 :M f3_12 sf (x)S 136 142 :M (3)S gR gS 84 65 381 134 rC 127.5 131.5 15 14 rS 149 133 16 13 rC 149 142 :M f3_12 sf (x)S 155 142 :M (4)S gR gS 84 65 381 134 rC 146.5 131.5 15 14 rS 168 133 16 13 rC 168 142 :M f3_12 sf (x)S 174 142 :M (5)S gR gS 84 65 381 134 rC 165.5 131.5 15 14 rS 187 133 16 13 rC 187 142 :M f3_12 sf (x)S 193 142 :M (6)S gR gS 84 65 381 134 rC 184.5 131.5 15 14 rS 215 132 16 13 rC 215 141 :M f3_12 sf (x)S 221 141 :M (7)S gR gS 84 65 381 134 rC 212.5 130.5 15 14 rS 234 132 16 13 rC 234 141 :M f3_12 sf (x)S 240 141 :M (8)S gR gS 84 65 381 134 rC 231.5 130.5 15 14 rS 253 132 15 13 rC 253 141 :M f3_12 sf (x)S 259 141 :M (9)S gR gS 84 65 381 134 rC 250.5 130.5 15 14 rS 273 132 21 13 rC 273 141 :M f3_12 sf (x)S 279 141 :M (1)S 285 141 :M (0)S gR gS 84 65 381 134 rC 270.5 130.5 23 15 rS 297 132 22 13 rC 297 141 :M f3_12 sf (x)S 303 141 :M (1)S 309 141 :M (1)S gR gS 84 65 381 134 rC 295.5 130.5 23 15 rS 339 132 22 13 rC 339 141 :M f3_12 sf (x)S 345 141 :M (1)S 351 141 :M (2)S gR gS 84 65 381 134 rC 337.5 130.5 23 15 rS 364 132 22 13 rC 364 141 :M f3_12 sf (x)S 370 141 :M (1)S 376 141 :M (3)S gR gS 84 65 381 134 rC 362.5 130.5 23 15 rS 389 132 22 13 rC 389 141 :M f3_12 sf (x)S 395 141 :M (1)S 401 141 :M (4)S gR gS 84 65 381 134 rC 387.5 130.5 23 15 rS 414 132 22 13 rC 414 141 :M f3_12 sf (x)S 420 141 :M (1)S 426 141 :M (5)S gR gS 84 65 381 134 rC 412.5 130.5 23 15 rS 439 132 22 13 rC 439 141 :M f3_12 sf (x)S 445 141 :M (1)S 451 141 :M (6)S gR gS 84 65 381 134 rC 437.5 130.5 22 15 rS np 172 131 :M 165 124 :L 165 124 :L 165 124 :L 165 124 :L 165 124 :L 165 123 :L 165 123 :L 166 123 :L 166 123 :L 166 123 :L 166 123 :L 166 123 :L 166 123 :L 166 123 :L 167 122 :L 167 122 :L 167 122 :L 167 122 :L 167 122 :L 167 122 :L 168 122 :L 168 122 :L 168 122 :L 168 122 :L 168 122 :L 168 122 :L 168 122 :L 169 122 :L 169 122 :L 169 121 :L 169 121 :L 169 121 :L 169 121 :L 170 121 :L 170 121 :L 170 121 :L 170 121 :L 170 121 :L 171 121 :L 171 121 :L 171 121 :L 171 121 :L 171 121 :L 171 121 :L 172 121 :L 172 121 :L 172 121 :L 172 121 :L 172 131 :L 172 131 :L eofill 153 86 -1 1 169 125 1 153 85 @a np 191 129 :M 182 124 :L 182 124 :L 182 124 :L 182 124 :L 182 123 :L 183 123 :L 183 123 :L 183 123 :L 183 123 :L 183 123 :L 183 123 :L 183 122 :L 183 122 :L 183 122 :L 184 122 :L 184 122 :L 184 122 :L 184 122 :L 184 122 :L 184 122 :L 184 121 :L 184 121 :L 185 121 :L 185 121 :L 185 121 :L 185 121 :L 185 121 :L 185 121 :L 185 121 :L 186 120 :L 186 120 :L 186 120 :L 186 120 :L 186 120 :L 186 120 :L 186 120 :L 187 120 :L 187 120 :L 187 120 :L 187 120 :L 187 120 :L 187 120 :L 188 120 :L 188 120 :L 188 119 :L 188 119 :L 188 119 :L 188 119 :L 191 129 :L 191 129 :L eofill 157 86 -1 1 186 124 1 157 85 @a 257 70 16 13 rC 257 79 :M f3_12 sf (T)S 264 79 :M (2)S gR gS 84 65 381 134 rC 25 15 263 77 @f 391 70 17 13 rC 391 79 :M f3_12 sf (T)S 398 79 :M (3)S gR gS 84 65 381 134 rC 25 15 398 77 @f np 380 76 :M 371 80 :L 371 80 :L 371 80 :L 371 80 :L 371 80 :L 371 80 :L 371 79 :L 371 79 :L 371 79 :L 371 79 :L 371 79 :L 371 79 :L 370 78 :L 370 78 :L 370 78 :L 370 78 :L 370 78 :L 370 78 :L 370 77 :L 370 77 :L 370 77 :L 370 77 :L 370 77 :L 370 77 :L 370 76 :L 370 76 :L 370 76 :L 370 76 :L 370 76 :L 370 76 :L 370 75 :L 370 75 :L 370 75 :L 370 75 :L 370 75 :L 370 74 :L 370 74 :L 371 74 :L 371 74 :L 371 74 :L 371 74 :L 371 73 :L 371 73 :L 371 73 :L 371 73 :L 371 73 :L 371 73 :L 371 72 :L 380 76 :L 380 76 :L eofill 288 78 -1 1 373 77 1 288 77 @a np 261 128 :M 256 119 :L 257 119 :L 257 119 :L 257 119 :L 257 119 :L 257 119 :L 257 119 :L 258 119 :L 258 118 :L 258 118 :L 258 118 :L 258 118 :L 258 118 :L 259 118 :L 259 118 :L 259 118 :L 259 118 :L 259 118 :L 259 118 :L 260 118 :L 260 118 :L 260 118 :L 260 118 :L 260 118 :L 261 118 :L 261 118 :L 261 118 :L 261 118 :L 261 118 :L 261 118 :L 262 118 :L 262 118 :L 262 118 :L 262 118 :L 262 118 :L 262 118 :L 263 118 :L 263 118 :L 263 118 :L 263 118 :L 263 118 :L 263 119 :L 264 119 :L 264 119 :L 264 119 :L 264 119 :L 264 119 :L 264 119 :L 261 128 :L 261 128 :L eofill -1 -1 261 122 1 1 260 84 @b np 399 130 :M 395 121 :L 395 121 :L 395 121 :L 396 121 :L 396 121 :L 396 121 :L 396 121 :L 396 121 :L 396 120 :L 397 120 :L 397 120 :L 397 120 :L 397 120 :L 397 120 :L 397 120 :L 398 120 :L 398 120 :L 398 120 :L 398 120 :L 398 120 :L 398 120 :L 399 120 :L 399 120 :L 399 120 :L 399 120 :L 399 120 :L 400 120 :L 400 120 :L 400 120 :L 400 120 :L 400 120 :L 400 120 :L 401 120 :L 401 120 :L 401 120 :L 401 120 :L 401 120 :L 401 120 :L 402 120 :L 402 120 :L 402 120 :L 402 121 :L 402 121 :L 402 121 :L 403 121 :L 403 121 :L 403 121 :L 403 121 :L 399 130 :L 399 130 :L eofill -1 -1 400 124 1 1 399 86 @b 93 174 8 11 rC 93 183 :M f3_9 sf (1)S gR gS 88 168 9 18 rC 88 181 :M f1_12 sf (e)S gR gS 120 175 8 11 rC 120 184 :M f3_9 sf (2)S gR gS 114 168 9 18 rC 114 181 :M f1_12 sf (e)S gR gS 84 65 381 134 rC np 96 147 :M 100 156 :L 100 156 :L 100 156 :L 99 156 :L 99 156 :L 99 156 :L 99 156 :L 99 156 :L 99 156 :L 99 156 :L 98 156 :L 98 157 :L 98 157 :L 98 157 :L 98 157 :L 97 157 :L 97 157 :L 97 157 :L 97 157 :L 97 157 :L 97 157 :L 96 157 :L 96 157 :L 96 157 :L 96 157 :L 96 157 :L 96 157 :L 95 157 :L 95 157 :L 95 157 :L 95 157 :L 95 157 :L 95 157 :L 94 157 :L 94 157 :L 94 157 :L 94 157 :L 94 157 :L 94 156 :L 93 156 :L 93 156 :L 93 156 :L 93 156 :L 93 156 :L 93 156 :L 92 156 :L 92 156 :L 92 156 :L 96 147 :L 96 147 :L eofill -1 -1 96 171 1 1 95 155 @b np 116 147 :M 120 156 :L 120 156 :L 120 156 :L 119 156 :L 119 156 :L 119 156 :L 119 156 :L 119 156 :L 119 156 :L 118 156 :L 118 156 :L 118 157 :L 118 157 :L 118 157 :L 118 157 :L 117 157 :L 117 157 :L 117 157 :L 117 157 :L 117 157 :L 117 157 :L 116 157 :L 116 157 :L 116 157 :L 116 157 :L 116 157 :L 116 157 :L 115 157 :L 115 157 :L 115 157 :L 115 157 :L 115 157 :L 114 157 :L 114 157 :L 114 157 :L 114 157 :L 114 157 :L 114 157 :L 113 156 :L 113 156 :L 113 156 :L 113 156 :L 113 156 :L 113 156 :L 112 156 :L 112 156 :L 112 156 :L 112 156 :L 116 147 :L 116 147 :L eofill -1 -1 116 171 1 1 115 155 @b np 217 128 :M 202 124 :L 202 124 :L 202 124 :L 202 123 :L 202 123 :L 203 123 :L 203 123 :L 203 122 :L 203 122 :L 203 122 :L 203 122 :L 203 121 :L 203 121 :L 203 121 :L 204 121 :L 204 121 :L 204 120 :L 204 120 :L 204 120 :L 204 120 :L 204 119 :L 205 119 :L 205 119 :L 205 119 :L 205 119 :L 205 118 :L 205 118 :L 206 118 :L 206 118 :L 206 118 :L 206 117 :L 206 117 :L 206 117 :L 207 117 :L 207 117 :L 207 117 :L 207 116 :L 207 116 :L 208 116 :L 208 116 :L 208 116 :L 208 116 :L 208 116 :L 209 115 :L 209 115 :L 209 115 :L 209 115 :L 210 115 :L 217 128 :L 217 128 :L eofill 160 84 -2 2 207 119 2 160 82 @a np 345 127 :M 331 130 :L 331 129 :L 331 129 :L 330 129 :L 330 129 :L 330 128 :L 330 128 :L 330 128 :L 330 128 :L 330 127 :L 330 127 :L 330 127 :L 330 127 :L 330 126 :L 330 126 :L 330 126 :L 330 125 :L 330 125 :L 331 125 :L 331 125 :L 331 124 :L 331 124 :L 331 124 :L 331 124 :L 331 123 :L 331 123 :L 331 123 :L 331 123 :L 331 122 :L 331 122 :L 331 122 :L 331 122 :L 331 121 :L 332 121 :L 332 121 :L 332 121 :L 332 120 :L 332 120 :L 332 120 :L 332 120 :L 332 120 :L 333 119 :L 333 119 :L 333 119 :L 333 119 :L 333 118 :L 333 118 :L 333 118 :L 345 127 :L 345 127 :L 2 lw eofill 161 81 -2 2 332 123 2 161 79 @a np 348 144 :M 354 158 :L 353 158 :L 353 158 :L 353 158 :L 353 158 :L 352 158 :L 352 158 :L 352 158 :L 352 158 :L 351 158 :L 351 159 :L 351 159 :L 351 159 :L 350 159 :L 350 159 :L 350 159 :L 350 159 :L 349 159 :L 349 159 :L 349 159 :L 349 159 :L 348 159 :L 348 159 :L 348 159 :L 348 159 :L 347 159 :L 347 159 :L 347 159 :L 347 159 :L 346 159 :L 346 159 :L 346 159 :L 345 159 :L 345 158 :L 345 158 :L 345 158 :L 344 158 :L 344 158 :L 344 158 :L 344 158 :L 343 158 :L 343 158 :L 343 158 :L 343 158 :L 342 158 :L 342 158 :L 342 157 :L 342 157 :L 348 144 :L 348 144 :L eofill -2 -2 347 165 2 2 345 156 @b np 374 144 :M 380 158 :L 380 158 :L 380 158 :L 380 158 :L 379 158 :L 379 158 :L 379 158 :L 379 158 :L 378 158 :L 378 158 :L 378 158 :L 378 158 :L 377 158 :L 377 159 :L 377 159 :L 377 159 :L 376 159 :L 376 159 :L 376 159 :L 376 159 :L 375 159 :L 375 159 :L 375 159 :L 374 159 :L 374 159 :L 374 159 :L 374 159 :L 373 159 :L 373 159 :L 373 159 :L 373 159 :L 372 159 :L 372 159 :L 372 159 :L 372 159 :L 371 159 :L 371 158 :L 371 158 :L 371 158 :L 370 158 :L 370 158 :L 370 158 :L 370 158 :L 369 158 :L 369 158 :L 369 158 :L 369 158 :L 368 158 :L 374 144 :L 374 144 :L eofill -2 -2 374 162 2 2 372 156 @b 0 90 210 58 268 162 @n 90 180 150 90 268 146 @n 0 90 146 32 273 162 @n 90 180 208 64 276 146 @n 0 90 22 20 108 185 @n 90 180 36 18 111 186 @n gR gS 0 0 552 730 rC 200 224 :M f0_12 sf (Figure )S 237 224 :M (18: Population RSEM)S 81 252 :M f3_12 sf .48 .048(Suppose our interest is in the causal relationships between the three latent variables)J 59 268 :M (T)S f3_7 sf 0 3 rm (1)S 0 -3 rm 70 268 :M f3_12 sf 1.213 .121(, T)J f3_7 sf 0 3 rm (2)S 0 -3 rm 89 268 :M f3_12 sf 1.532 .153(, and T)J 127 271 :M f3_7 sf (3)S 131 268 :M f3_12 sf 1.224 .122(. The part of the RSEM specifying the relationships between the latent)J 59 284 :M .384 .038(variables is called the structural model; the rest is called the measurement model. In this)J 59 300 :M 2.858 .286(case the population structural model is shown in Figure 19, and the population)J 59 316 :M (measurement model is shown in )S 217 316 :M (Figure 20.)S 180 340 191 12 rC 181 349 :M (T1 T2 T3)S gR gS 169 335 211 20 rC 31 15 187 345 @f 31 15 274 346 @f 31 15 364 347 @f np 258 345 :M 246 348 :L 246 345 :L 246 342 :L 258 345 :L eofill 201 346 -1 1 247 345 1 201 345 @a np 347 346 :M 335 349 :L 335 346 :L 335 343 :L 347 346 :L eofill 290 347 -1 1 336 346 1 290 346 @a gR gS 0 0 552 730 rC 173 380 :M f0_12 sf (Figure )S 210 380 :M (19: Population Structural Model)S 84 411 381 134 rC np 218 476 :M 219 466 :L 220 466 :L 220 466 :L 220 466 :L 220 466 :L 220 466 :L 220 466 :L 221 467 :L 221 467 :L 221 467 :L 221 467 :L 221 467 :L 221 467 :L 222 467 :L 222 467 :L 222 467 :L 222 467 :L 222 467 :L 222 467 :L 222 467 :L 223 467 :L 223 467 :L 223 468 :L 223 468 :L 223 468 :L 223 468 :L 223 468 :L 224 468 :L 224 468 :L 224 468 :L 224 468 :L 224 469 :L 224 469 :L 224 469 :L 225 469 :L 225 469 :L 225 469 :L 225 469 :L 225 469 :L 225 469 :L 225 470 :L 225 470 :L 225 470 :L 226 470 :L 226 470 :L 226 470 :L 226 470 :L 226 471 :L 218 476 :L 218 476 :L eofill -1 -1 222 471 1 1 251 426 @b np 240 475 :M 238 465 :L 239 465 :L 239 465 :L 239 465 :L 239 465 :L 239 465 :L 240 465 :L 240 465 :L 240 465 :L 240 465 :L 240 465 :L 240 465 :L 241 465 :L 241 465 :L 241 465 :L 241 465 :L 241 465 :L 241 465 :L 242 465 :L 242 465 :L 242 465 :L 242 465 :L 242 465 :L 242 466 :L 243 466 :L 243 466 :L 243 466 :L 243 466 :L 243 466 :L 243 466 :L 244 466 :L 244 466 :L 244 466 :L 244 466 :L 244 466 :L 244 466 :L 245 466 :L 245 466 :L 245 467 :L 245 467 :L 245 467 :L 245 467 :L 245 467 :L 246 467 :L 246 467 :L 246 467 :L 246 467 :L 246 468 :L 240 475 :L 240 475 :L eofill -1 -1 242 469 1 1 254 429 @b np 283 476 :M 277 469 :L 277 468 :L 277 468 :L 277 468 :L 277 468 :L 278 468 :L 278 468 :L 278 468 :L 278 468 :L 278 468 :L 278 467 :L 278 467 :L 279 467 :L 279 467 :L 279 467 :L 279 467 :L 279 467 :L 279 467 :L 280 467 :L 280 467 :L 280 467 :L 280 467 :L 280 467 :L 280 467 :L 281 467 :L 281 466 :L 281 466 :L 281 466 :L 281 466 :L 281 466 :L 282 466 :L 282 466 :L 282 466 :L 282 466 :L 282 466 :L 282 466 :L 283 466 :L 283 466 :L 283 466 :L 283 466 :L 283 466 :L 283 466 :L 284 466 :L 284 466 :L 284 466 :L 284 466 :L 284 466 :L 285 466 :L 283 476 :L 283 476 :L eofill 268 430 -1 1 282 469 1 268 429 @a np 309 476 :M 301 471 :L 301 471 :L 301 471 :L 301 471 :L 301 471 :L 301 471 :L 301 470 :L 301 470 :L 301 470 :L 301 470 :L 302 470 :L 302 470 :L 302 470 :L 302 469 :L 302 469 :L 302 469 :L 302 469 :L 302 469 :L 302 469 :L 303 469 :L 303 469 :L 303 469 :L 303 468 :L 303 468 :L 303 468 :L 303 468 :L 304 468 :L 304 468 :L 304 468 :L 304 468 :L 304 468 :L 304 467 :L 304 467 :L 305 467 :L 305 467 :L 305 467 :L 305 467 :L 305 467 :L 305 467 :L 305 467 :L 306 467 :L 306 467 :L 306 467 :L 306 467 :L 306 467 :L 306 467 :L 307 466 :L 307 466 :L 309 476 :L 309 476 :L eofill 272 430 -1 1 305 471 1 272 429 @a np 346 475 :M 349 465 :L 349 466 :L 349 466 :L 350 466 :L 350 466 :L 350 466 :L 350 466 :L 350 466 :L 350 466 :L 350 466 :L 351 466 :L 351 466 :L 351 466 :L 351 466 :L 351 466 :L 351 466 :L 352 467 :L 352 467 :L 352 467 :L 352 467 :L 352 467 :L 352 467 :L 352 467 :L 353 467 :L 353 467 :L 353 468 :L 353 468 :L 353 468 :L 353 468 :L 353 468 :L 353 468 :L 354 468 :L 354 468 :L 354 468 :L 354 469 :L 354 469 :L 354 469 :L 354 469 :L 354 469 :L 354 469 :L 355 469 :L 355 470 :L 355 470 :L 355 470 :L 355 470 :L 355 470 :L 355 470 :L 355 471 :L 346 475 :L 346 475 :L eofill -1 -1 352 471 1 1 386 428 @b np 450 476 :M 441 472 :L 441 471 :L 441 471 :L 441 471 :L 441 471 :L 441 471 :L 442 471 :L 442 471 :L 442 470 :L 442 470 :L 442 470 :L 442 470 :L 442 470 :L 442 470 :L 442 470 :L 443 469 :L 443 469 :L 443 469 :L 443 469 :L 443 469 :L 443 469 :L 443 469 :L 443 469 :L 443 469 :L 444 468 :L 444 468 :L 444 468 :L 444 468 :L 444 468 :L 444 468 :L 444 468 :L 445 468 :L 445 468 :L 445 467 :L 445 467 :L 445 467 :L 445 467 :L 446 467 :L 446 467 :L 446 467 :L 446 467 :L 446 467 :L 446 467 :L 446 467 :L 447 467 :L 447 467 :L 447 467 :L 447 466 :L 450 476 :L 450 476 :L eofill 409 429 -1 1 446 471 1 409 428 @a np 374 476 :M 374 466 :L 374 466 :L 374 466 :L 374 466 :L 374 466 :L 375 466 :L 375 466 :L 375 466 :L 375 466 :L 375 466 :L 375 466 :L 376 466 :L 376 466 :L 376 466 :L 376 466 :L 376 466 :L 376 466 :L 377 466 :L 377 466 :L 377 466 :L 377 467 :L 377 467 :L 377 467 :L 378 467 :L 378 467 :L 378 467 :L 378 467 :L 378 467 :L 378 467 :L 379 467 :L 379 467 :L 379 467 :L 379 467 :L 379 467 :L 379 467 :L 379 468 :L 380 468 :L 380 468 :L 380 468 :L 380 468 :L 380 468 :L 380 468 :L 380 468 :L 381 468 :L 381 469 :L 381 469 :L 381 469 :L 381 469 :L 374 476 :L 374 476 :L eofill -1 -1 378 471 1 1 392 430 @b np 421 475 :M 414 468 :L 414 468 :L 415 468 :L 415 467 :L 415 467 :L 415 467 :L 415 467 :L 415 467 :L 415 467 :L 416 467 :L 416 467 :L 416 467 :L 416 466 :L 416 466 :L 416 466 :L 416 466 :L 417 466 :L 417 466 :L 417 466 :L 417 466 :L 417 466 :L 417 466 :L 418 466 :L 418 466 :L 418 466 :L 418 466 :L 418 466 :L 418 465 :L 419 465 :L 419 465 :L 419 465 :L 419 465 :L 419 465 :L 419 465 :L 420 465 :L 420 465 :L 420 465 :L 420 465 :L 420 465 :L 420 465 :L 421 465 :L 421 465 :L 421 465 :L 421 465 :L 421 465 :L 421 465 :L 422 465 :L 422 465 :L 421 475 :L 421 475 :L eofill 405 431 -1 1 419 468 1 405 430 @a 140 418 17 13 rC 140 427 :M f3_12 sf (T)S 147 427 :M (1)S gR gS 84 411 381 134 rC 25 14 145 425.5 @f np 101 476 :M 104 466 :L 104 467 :L 104 467 :L 104 467 :L 104 467 :L 105 467 :L 105 467 :L 105 467 :L 105 467 :L 105 467 :L 105 467 :L 106 467 :L 106 467 :L 106 467 :L 106 467 :L 106 468 :L 106 468 :L 106 468 :L 107 468 :L 107 468 :L 107 468 :L 107 468 :L 107 468 :L 107 468 :L 107 469 :L 108 469 :L 108 469 :L 108 469 :L 108 469 :L 108 469 :L 108 469 :L 108 469 :L 108 469 :L 109 470 :L 109 470 :L 109 470 :L 109 470 :L 109 470 :L 109 470 :L 109 470 :L 109 471 :L 109 471 :L 109 471 :L 110 471 :L 110 471 :L 110 471 :L 110 471 :L 110 472 :L 101 476 :L 101 476 :L eofill -1 -1 106 472 1 1 139 431 @b np 119 477 :M 119 467 :L 120 467 :L 120 467 :L 120 467 :L 120 467 :L 120 467 :L 120 467 :L 121 467 :L 121 467 :L 121 467 :L 121 467 :L 121 467 :L 121 467 :L 122 467 :L 122 468 :L 122 468 :L 122 468 :L 122 468 :L 122 468 :L 123 468 :L 123 468 :L 123 468 :L 123 468 :L 123 468 :L 123 468 :L 124 468 :L 124 468 :L 124 468 :L 124 468 :L 124 469 :L 124 469 :L 124 469 :L 125 469 :L 125 469 :L 125 469 :L 125 469 :L 125 469 :L 125 469 :L 125 470 :L 126 470 :L 126 470 :L 126 470 :L 126 470 :L 126 470 :L 126 470 :L 126 470 :L 126 470 :L 126 471 :L 119 477 :L 119 477 :L eofill -1 -1 123 472 1 1 141 433 @b np 137 478 :M 135 468 :L 135 468 :L 135 468 :L 135 468 :L 135 468 :L 135 468 :L 136 468 :L 136 468 :L 136 468 :L 136 468 :L 136 468 :L 136 468 :L 137 468 :L 137 468 :L 137 468 :L 137 468 :L 137 468 :L 138 468 :L 138 468 :L 138 468 :L 138 468 :L 138 468 :L 138 468 :L 139 468 :L 139 468 :L 139 468 :L 139 468 :L 139 468 :L 139 468 :L 140 468 :L 140 469 :L 140 469 :L 140 469 :L 140 469 :L 140 469 :L 141 469 :L 141 469 :L 141 469 :L 141 469 :L 141 469 :L 141 469 :L 141 469 :L 142 469 :L 142 469 :L 142 469 :L 142 470 :L 142 470 :L 142 470 :L 137 478 :L 137 478 :L eofill -1 -1 139 472 1 1 145 432 @b np 154 476 :M 149 467 :L 149 467 :L 149 467 :L 149 467 :L 150 467 :L 150 467 :L 150 467 :L 150 467 :L 150 467 :L 150 467 :L 151 467 :L 151 467 :L 151 467 :L 151 466 :L 151 466 :L 151 466 :L 152 466 :L 152 466 :L 152 466 :L 152 466 :L 152 466 :L 152 466 :L 153 466 :L 153 466 :L 153 466 :L 153 466 :L 153 466 :L 153 466 :L 154 466 :L 154 466 :L 154 466 :L 154 466 :L 154 466 :L 154 466 :L 155 466 :L 155 466 :L 155 466 :L 155 466 :L 155 466 :L 156 466 :L 156 466 :L 156 466 :L 156 466 :L 156 466 :L 156 466 :L 157 466 :L 157 467 :L 157 467 :L 154 476 :L 154 476 :L eofill 149 434 -1 1 153 469 1 149 433 @a 92 479 16 13 rC 92 488 :M f3_12 sf (x)S 98 488 :M (1)S gR gS 84 411 381 134 rC 89.5 477.5 15 14 rS 111 479 16 13 rC 111 488 :M f3_12 sf (x)S 117 488 :M (2)S gR gS 84 411 381 134 rC 108.5 477.5 15 14 rS 130 479 16 13 rC 130 488 :M f3_12 sf (x)S 136 488 :M (3)S gR gS 84 411 381 134 rC 127.5 477.5 15 14 rS 149 479 16 13 rC 149 488 :M f3_12 sf (x)S 155 488 :M (4)S gR gS 84 411 381 134 rC 146.5 477.5 15 14 rS 168 479 16 13 rC 168 488 :M f3_12 sf (x)S 174 488 :M (5)S gR gS 84 411 381 134 rC 165.5 477.5 15 14 rS 187 479 16 13 rC 187 488 :M f3_12 sf (x)S 193 488 :M (6)S gR gS 84 411 381 134 rC 184.5 477.5 15 14 rS 215 478 16 13 rC 215 487 :M f3_12 sf (x)S 221 487 :M (7)S gR gS 84 411 381 134 rC 212.5 476.5 15 14 rS 234 478 16 13 rC 234 487 :M f3_12 sf (x)S 240 487 :M (8)S gR gS 84 411 381 134 rC 231.5 476.5 15 14 rS 253 478 15 13 rC 253 487 :M f3_12 sf (x)S 259 487 :M (9)S gR gS 84 411 381 134 rC 250.5 476.5 15 14 rS 273 478 21 13 rC 273 487 :M f3_12 sf (x)S 279 487 :M (1)S 285 487 :M (0)S gR gS 84 411 381 134 rC 270.5 476.5 23 15 rS 297 478 22 13 rC 297 487 :M f3_12 sf (x)S 303 487 :M (1)S 309 487 :M (1)S gR gS 84 411 381 134 rC 295.5 476.5 23 15 rS 339 478 22 13 rC 339 487 :M f3_12 sf (x)S 345 487 :M (1)S 351 487 :M (2)S gR gS 84 411 381 134 rC 337.5 476.5 23 15 rS 364 478 22 13 rC 364 487 :M f3_12 sf (x)S 370 487 :M (1)S 376 487 :M (3)S gR gS 84 411 381 134 rC 362.5 476.5 23 15 rS 389 478 22 13 rC 389 487 :M f3_12 sf (x)S 395 487 :M (1)S 401 487 :M (4)S gR gS 84 411 381 134 rC 387.5 476.5 23 15 rS 414 478 22 13 rC 414 487 :M f3_12 sf (x)S 420 487 :M (1)S 426 487 :M (5)S gR gS 84 411 381 134 rC 412.5 476.5 23 15 rS 439 478 22 13 rC 439 487 :M f3_12 sf (x)S 445 487 :M (1)S 451 487 :M (6)S gR gS 84 411 381 134 rC 437.5 476.5 22 15 rS np 172 477 :M 165 470 :L 165 470 :L 165 470 :L 165 470 :L 165 470 :L 165 469 :L 165 469 :L 166 469 :L 166 469 :L 166 469 :L 166 469 :L 166 469 :L 166 469 :L 166 469 :L 167 468 :L 167 468 :L 167 468 :L 167 468 :L 167 468 :L 167 468 :L 168 468 :L 168 468 :L 168 468 :L 168 468 :L 168 468 :L 168 468 :L 168 468 :L 169 468 :L 169 468 :L 169 467 :L 169 467 :L 169 467 :L 169 467 :L 170 467 :L 170 467 :L 170 467 :L 170 467 :L 170 467 :L 171 467 :L 171 467 :L 171 467 :L 171 467 :L 171 467 :L 171 467 :L 172 467 :L 172 467 :L 172 467 :L 172 467 :L 172 477 :L 172 477 :L eofill 153 432 -1 1 169 471 1 153 431 @a np 191 475 :M 182 470 :L 182 470 :L 182 470 :L 182 470 :L 182 469 :L 183 469 :L 183 469 :L 183 469 :L 183 469 :L 183 469 :L 183 469 :L 183 468 :L 183 468 :L 183 468 :L 184 468 :L 184 468 :L 184 468 :L 184 468 :L 184 468 :L 184 468 :L 184 467 :L 184 467 :L 185 467 :L 185 467 :L 185 467 :L 185 467 :L 185 467 :L 185 467 :L 185 467 :L 186 466 :L 186 466 :L 186 466 :L 186 466 :L 186 466 :L 186 466 :L 186 466 :L 187 466 :L 187 466 :L 187 466 :L 187 466 :L 187 466 :L 187 466 :L 188 466 :L 188 466 :L 188 465 :L 188 465 :L 188 465 :L 188 465 :L 191 475 :L 191 475 :L eofill 157 432 -1 1 186 470 1 157 431 @a 257 416 16 13 rC 257 425 :M f3_12 sf (T)S 264 425 :M (2)S gR gS 84 411 381 134 rC 25 15 263 423 @f 391 416 17 13 rC 391 425 :M f3_12 sf (T)S 398 425 :M (3)S gR gS 84 411 381 134 rC 25 15 398 423 @f np 261 474 :M 256 465 :L 257 465 :L 257 465 :L 257 465 :L 257 465 :L 257 465 :L 257 465 :L 258 465 :L 258 464 :L 258 464 :L 258 464 :L 258 464 :L 258 464 :L 259 464 :L 259 464 :L 259 464 :L 259 464 :L 259 464 :L 259 464 :L 260 464 :L 260 464 :L 260 464 :L 260 464 :L 260 464 :L 261 464 :L 261 464 :L 261 464 :L 261 464 :L 261 464 :L 261 464 :L 262 464 :L 262 464 :L 262 464 :L 262 464 :L 262 464 :L 262 464 :L 263 464 :L 263 464 :L 263 464 :L 263 464 :L 263 464 :L 263 465 :L 264 465 :L 264 465 :L 264 465 :L 264 465 :L 264 465 :L 264 465 :L 261 474 :L 261 474 :L eofill -1 -1 261 468 1 1 260 430 @b np 399 476 :M 395 467 :L 395 467 :L 395 467 :L 396 467 :L 396 467 :L 396 467 :L 396 467 :L 396 467 :L 396 466 :L 397 466 :L 397 466 :L 397 466 :L 397 466 :L 397 466 :L 397 466 :L 398 466 :L 398 466 :L 398 466 :L 398 466 :L 398 466 :L 398 466 :L 399 466 :L 399 466 :L 399 466 :L 399 466 :L 399 466 :L 400 466 :L 400 466 :L 400 466 :L 400 466 :L 400 466 :L 400 466 :L 401 466 :L 401 466 :L 401 466 :L 401 466 :L 401 466 :L 401 466 :L 402 466 :L 402 466 :L 402 466 :L 402 467 :L 402 467 :L 402 467 :L 403 467 :L 403 467 :L 403 467 :L 403 467 :L 399 476 :L 399 476 :L eofill -1 -1 400 470 1 1 399 432 @b 93 520 8 11 rC 93 529 :M f3_9 sf (1)S gR gS 88 514 9 18 rC 88 527 :M f1_12 sf (e)S gR gS 120 521 8 11 rC 120 530 :M f3_9 sf (2)S gR gS 114 514 9 18 rC 114 527 :M f1_12 sf (e)S gR gS 84 411 381 134 rC np 96 493 :M 100 502 :L 100 502 :L 100 502 :L 99 502 :L 99 502 :L 99 502 :L 99 502 :L 99 502 :L 99 502 :L 99 502 :L 98 502 :L 98 503 :L 98 503 :L 98 503 :L 98 503 :L 97 503 :L 97 503 :L 97 503 :L 97 503 :L 97 503 :L 97 503 :L 96 503 :L 96 503 :L 96 503 :L 96 503 :L 96 503 :L 96 503 :L 95 503 :L 95 503 :L 95 503 :L 95 503 :L 95 503 :L 95 503 :L 94 503 :L 94 503 :L 94 503 :L 94 503 :L 94 503 :L 94 502 :L 93 502 :L 93 502 :L 93 502 :L 93 502 :L 93 502 :L 93 502 :L 92 502 :L 92 502 :L 92 502 :L 96 493 :L 96 493 :L eofill -1 -1 96 517 1 1 95 501 @b np 116 493 :M 120 502 :L 120 502 :L 120 502 :L 119 502 :L 119 502 :L 119 502 :L 119 502 :L 119 502 :L 119 502 :L 118 502 :L 118 502 :L 118 503 :L 118 503 :L 118 503 :L 118 503 :L 117 503 :L 117 503 :L 117 503 :L 117 503 :L 117 503 :L 117 503 :L 116 503 :L 116 503 :L 116 503 :L 116 503 :L 116 503 :L 116 503 :L 115 503 :L 115 503 :L 115 503 :L 115 503 :L 115 503 :L 114 503 :L 114 503 :L 114 503 :L 114 503 :L 114 503 :L 114 503 :L 113 502 :L 113 502 :L 113 502 :L 113 502 :L 113 502 :L 113 502 :L 112 502 :L 112 502 :L 112 502 :L 112 502 :L 116 493 :L 116 493 :L eofill -1 -1 116 517 1 1 115 501 @b np 217 474 :M 202 470 :L 202 470 :L 202 470 :L 202 469 :L 202 469 :L 203 469 :L 203 469 :L 203 468 :L 203 468 :L 203 468 :L 203 468 :L 203 467 :L 203 467 :L 203 467 :L 204 467 :L 204 467 :L 204 466 :L 204 466 :L 204 466 :L 204 466 :L 204 465 :L 205 465 :L 205 465 :L 205 465 :L 205 465 :L 205 464 :L 205 464 :L 206 464 :L 206 464 :L 206 464 :L 206 463 :L 206 463 :L 206 463 :L 207 463 :L 207 463 :L 207 463 :L 207 462 :L 207 462 :L 208 462 :L 208 462 :L 208 462 :L 208 462 :L 208 462 :L 209 461 :L 209 461 :L 209 461 :L 209 461 :L 210 461 :L 217 474 :L 217 474 :L eofill 160 430 -2 2 207 465 2 160 428 @a np 345 473 :M 331 476 :L 331 475 :L 331 475 :L 330 475 :L 330 475 :L 330 474 :L 330 474 :L 330 474 :L 330 474 :L 330 473 :L 330 473 :L 330 473 :L 330 473 :L 330 472 :L 330 472 :L 330 472 :L 330 471 :L 330 471 :L 331 471 :L 331 471 :L 331 470 :L 331 470 :L 331 470 :L 331 470 :L 331 469 :L 331 469 :L 331 469 :L 331 469 :L 331 468 :L 331 468 :L 331 468 :L 331 468 :L 331 467 :L 332 467 :L 332 467 :L 332 467 :L 332 466 :L 332 466 :L 332 466 :L 332 466 :L 332 466 :L 333 465 :L 333 465 :L 333 465 :L 333 465 :L 333 464 :L 333 464 :L 333 464 :L 345 473 :L 345 473 :L 2 lw eofill 161 427 -2 2 332 469 2 161 425 @a np 348 490 :M 354 504 :L 353 504 :L 353 504 :L 353 504 :L 353 504 :L 352 504 :L 352 504 :L 352 504 :L 352 504 :L 351 504 :L 351 505 :L 351 505 :L 351 505 :L 350 505 :L 350 505 :L 350 505 :L 350 505 :L 349 505 :L 349 505 :L 349 505 :L 349 505 :L 348 505 :L 348 505 :L 348 505 :L 348 505 :L 347 505 :L 347 505 :L 347 505 :L 347 505 :L 346 505 :L 346 505 :L 346 505 :L 345 505 :L 345 504 :L 345 504 :L 345 504 :L 344 504 :L 344 504 :L 344 504 :L 344 504 :L 343 504 :L 343 504 :L 343 504 :L 343 504 :L 342 504 :L 342 504 :L 342 503 :L 342 503 :L 348 490 :L 348 490 :L eofill -2 -2 347 511 2 2 345 502 @b np 374 490 :M 380 504 :L 380 504 :L 380 504 :L 380 504 :L 379 504 :L 379 504 :L 379 504 :L 379 504 :L 378 504 :L 378 504 :L 378 504 :L 378 504 :L 377 504 :L 377 505 :L 377 505 :L 377 505 :L 376 505 :L 376 505 :L 376 505 :L 376 505 :L 375 505 :L 375 505 :L 375 505 :L 374 505 :L 374 505 :L 374 505 :L 374 505 :L 373 505 :L 373 505 :L 373 505 :L 373 505 :L 372 505 :L 372 505 :L 372 505 :L 372 505 :L 371 505 :L 371 504 :L 371 504 :L 371 504 :L 370 504 :L 370 504 :L 370 504 :L 370 504 :L 369 504 :L 369 504 :L 369 504 :L 369 504 :L 368 504 :L 374 490 :L 374 490 :L eofill -2 -2 374 508 2 2 372 502 @b 0 90 210 58 268 508 @n 90 180 150 90 268 492 @n 0 90 146 32 273 508 @n 90 180 208 64 276 492 @n 0 90 22 20 108 531 @n 90 180 36 18 111 532 @n gR gS 0 0 552 730 rC 164 570 :M f0_12 sf (Figure )S 201 570 :M (20: Population Measurement Model)S 81 614 :M f3_12 sf 3.205 .32(One approach to this problem is to use background knowledge to build a)J 59 630 :M .49 .049(measurement model for each latent variable, and then perform a specification search for)J 59 646 :M -.002(the structural model constrained by background knowledge and aided by a computer, e.g.,)A 59 662 :M 1.837 .184(the Search module of TETRAD II, or the modification indices of LISREL, or the)J 59 678 :M 1.355 .136(Lagrange Multiplier statistic of EQS. There are several problems with this approach.)J endp %%Page: 31 31 %%BeginPageSetup initializepage (peter; page: 31 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (31)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf 1.767 .177(First, while background knowledge may often be sufficient to construct part of the)J 59 70 :M 1.258 .126(population measurement model \(i.e. we may know which of the latent variables each)J 59 86 :M .496 .05(indicator variable is a measure of\), background knowledge is seldom detailed enough to)J 59 102 :M .945 .094(completely specify the full population measurement model \(e.g. an indicator may be a)J 59 118 :M 1.019 .102(measure of several latent variables, or indicator variables may have correlated errors\).)J 59 134 :M 1.741 .174(This means that the specification search must also seek to correct the hypothesized)J 59 150 :M .595 .059(measurement model, as well as discover the structural model. Because there are often a)J 59 166 :M .394 .039(large number of indicator variables, this search space is astronomically large. Moreover,)J 59 182 :M 1.189 .119(a search that at each step chooses to add the edge \(free the parameter\) that will most)J 59 198 :M .991 .099(improve the fit can easily go wrong for several reasons. First, it may be that freeing a)J 59 214 :M .247 .025(number of different parameters improves the fit to the same degree, so there is no way to)J 59 230 :M .205 .021(choose which parameter to free at that point in the search. In addition, there may be pairs)J 59 246 :M 1.231 .123(of parameters which if freed will greatly improve the overall fit, even though freeing)J 59 262 :M .29 .029(either parameter by itself does not improve the fit much. Also, when the initial RSEM to)J 59 278 :M .575 .058(be modified is far from the population RSEM, the parameter estimates may be far from)J 59 294 :M 1.092 .109(their population values, which can affect the estimates of the Lagrange multipliers, or)J 59 310 :M (may prevent the estimation algorithms from converging at all.)S 84 329 381 98 rC np 218 394 :M 219 384 :L 220 384 :L 220 384 :L 220 384 :L 220 384 :L 220 384 :L 220 384 :L 221 385 :L 221 385 :L 221 385 :L 221 385 :L 221 385 :L 221 385 :L 222 385 :L 222 385 :L 222 385 :L 222 385 :L 222 385 :L 222 385 :L 222 385 :L 223 385 :L 223 385 :L 223 386 :L 223 386 :L 223 386 :L 223 386 :L 223 386 :L 224 386 :L 224 386 :L 224 386 :L 224 386 :L 224 387 :L 224 387 :L 224 387 :L 225 387 :L 225 387 :L 225 387 :L 225 387 :L 225 387 :L 225 387 :L 225 388 :L 225 388 :L 225 388 :L 226 388 :L 226 388 :L 226 388 :L 226 388 :L 226 389 :L 218 394 :L 218 394 :L eofill -1 -1 222 389 1 1 251 344 @b np 240 393 :M 238 383 :L 239 383 :L 239 383 :L 239 383 :L 239 383 :L 239 383 :L 240 383 :L 240 383 :L 240 383 :L 240 383 :L 240 383 :L 240 383 :L 241 383 :L 241 383 :L 241 383 :L 241 383 :L 241 383 :L 241 383 :L 242 383 :L 242 383 :L 242 383 :L 242 383 :L 242 383 :L 242 384 :L 243 384 :L 243 384 :L 243 384 :L 243 384 :L 243 384 :L 243 384 :L 244 384 :L 244 384 :L 244 384 :L 244 384 :L 244 384 :L 244 384 :L 245 384 :L 245 384 :L 245 385 :L 245 385 :L 245 385 :L 245 385 :L 245 385 :L 246 385 :L 246 385 :L 246 385 :L 246 385 :L 246 386 :L 240 393 :L 240 393 :L eofill -1 -1 242 387 1 1 254 347 @b np 283 394 :M 277 387 :L 277 386 :L 277 386 :L 277 386 :L 277 386 :L 278 386 :L 278 386 :L 278 386 :L 278 386 :L 278 386 :L 278 385 :L 278 385 :L 279 385 :L 279 385 :L 279 385 :L 279 385 :L 279 385 :L 279 385 :L 280 385 :L 280 385 :L 280 385 :L 280 385 :L 280 385 :L 280 385 :L 281 385 :L 281 384 :L 281 384 :L 281 384 :L 281 384 :L 281 384 :L 282 384 :L 282 384 :L 282 384 :L 282 384 :L 282 384 :L 282 384 :L 283 384 :L 283 384 :L 283 384 :L 283 384 :L 283 384 :L 283 384 :L 284 384 :L 284 384 :L 284 384 :L 284 384 :L 284 384 :L 285 384 :L 283 394 :L 283 394 :L eofill 268 348 -1 1 282 387 1 268 347 @a np 309 394 :M 301 389 :L 301 389 :L 301 389 :L 301 389 :L 301 389 :L 301 389 :L 301 388 :L 301 388 :L 301 388 :L 301 388 :L 302 388 :L 302 388 :L 302 388 :L 302 387 :L 302 387 :L 302 387 :L 302 387 :L 302 387 :L 302 387 :L 303 387 :L 303 387 :L 303 387 :L 303 386 :L 303 386 :L 303 386 :L 303 386 :L 304 386 :L 304 386 :L 304 386 :L 304 386 :L 304 386 :L 304 385 :L 304 385 :L 305 385 :L 305 385 :L 305 385 :L 305 385 :L 305 385 :L 305 385 :L 305 385 :L 306 385 :L 306 385 :L 306 385 :L 306 385 :L 306 385 :L 306 385 :L 307 384 :L 307 384 :L 309 394 :L 309 394 :L eofill 272 348 -1 1 305 389 1 272 347 @a np 346 393 :M 349 383 :L 349 384 :L 349 384 :L 350 384 :L 350 384 :L 350 384 :L 350 384 :L 350 384 :L 350 384 :L 350 384 :L 351 384 :L 351 384 :L 351 384 :L 351 384 :L 351 384 :L 351 384 :L 352 385 :L 352 385 :L 352 385 :L 352 385 :L 352 385 :L 352 385 :L 352 385 :L 353 385 :L 353 385 :L 353 386 :L 353 386 :L 353 386 :L 353 386 :L 353 386 :L 353 386 :L 354 386 :L 354 386 :L 354 386 :L 354 387 :L 354 387 :L 354 387 :L 354 387 :L 354 387 :L 354 387 :L 355 387 :L 355 388 :L 355 388 :L 355 388 :L 355 388 :L 355 388 :L 355 388 :L 355 389 :L 346 393 :L 346 393 :L eofill -1 -1 352 389 1 1 386 346 @b np 450 394 :M 441 390 :L 441 389 :L 441 389 :L 441 389 :L 441 389 :L 441 389 :L 442 389 :L 442 389 :L 442 388 :L 442 388 :L 442 388 :L 442 388 :L 442 388 :L 442 388 :L 442 388 :L 443 387 :L 443 387 :L 443 387 :L 443 387 :L 443 387 :L 443 387 :L 443 387 :L 443 387 :L 443 387 :L 444 386 :L 444 386 :L 444 386 :L 444 386 :L 444 386 :L 444 386 :L 444 386 :L 445 386 :L 445 386 :L 445 385 :L 445 385 :L 445 385 :L 445 385 :L 446 385 :L 446 385 :L 446 385 :L 446 385 :L 446 385 :L 446 385 :L 446 385 :L 447 385 :L 447 385 :L 447 385 :L 447 384 :L 450 394 :L 450 394 :L eofill 409 347 -1 1 446 389 1 409 346 @a np 374 394 :M 374 384 :L 374 384 :L 374 384 :L 374 384 :L 374 384 :L 375 384 :L 375 384 :L 375 384 :L 375 384 :L 375 384 :L 375 384 :L 376 384 :L 376 384 :L 376 384 :L 376 384 :L 376 384 :L 376 384 :L 377 384 :L 377 384 :L 377 384 :L 377 385 :L 377 385 :L 377 385 :L 378 385 :L 378 385 :L 378 385 :L 378 385 :L 378 385 :L 378 385 :L 379 385 :L 379 385 :L 379 385 :L 379 385 :L 379 385 :L 379 385 :L 379 386 :L 380 386 :L 380 386 :L 380 386 :L 380 386 :L 380 386 :L 380 386 :L 380 386 :L 381 386 :L 381 387 :L 381 387 :L 381 387 :L 381 387 :L 374 394 :L 374 394 :L eofill -1 -1 378 389 1 1 392 348 @b np 421 393 :M 414 386 :L 414 386 :L 415 386 :L 415 385 :L 415 385 :L 415 385 :L 415 385 :L 415 385 :L 415 385 :L 416 385 :L 416 385 :L 416 385 :L 416 384 :L 416 384 :L 416 384 :L 416 384 :L 417 384 :L 417 384 :L 417 384 :L 417 384 :L 417 384 :L 417 384 :L 418 384 :L 418 384 :L 418 384 :L 418 384 :L 418 384 :L 418 383 :L 419 383 :L 419 383 :L 419 383 :L 419 383 :L 419 383 :L 419 383 :L 420 383 :L 420 383 :L 420 383 :L 420 383 :L 420 383 :L 420 383 :L 421 383 :L 421 383 :L 421 383 :L 421 383 :L 421 383 :L 421 383 :L 422 383 :L 422 383 :L 421 393 :L 421 393 :L eofill 405 349 -1 1 419 386 1 405 348 @a 140 336 17 13 rC 140 345 :M (T)S 147 345 :M (1)S gR gS 84 329 381 98 rC 25 14 145 343.5 @f np 101 394 :M 104 384 :L 104 385 :L 104 385 :L 104 385 :L 104 385 :L 105 385 :L 105 385 :L 105 385 :L 105 385 :L 105 385 :L 105 385 :L 106 385 :L 106 385 :L 106 385 :L 106 385 :L 106 386 :L 106 386 :L 106 386 :L 107 386 :L 107 386 :L 107 386 :L 107 386 :L 107 386 :L 107 386 :L 107 387 :L 108 387 :L 108 387 :L 108 387 :L 108 387 :L 108 387 :L 108 387 :L 108 387 :L 108 387 :L 109 388 :L 109 388 :L 109 388 :L 109 388 :L 109 388 :L 109 388 :L 109 388 :L 109 389 :L 109 389 :L 109 389 :L 110 389 :L 110 389 :L 110 389 :L 110 389 :L 110 390 :L 101 394 :L 101 394 :L eofill -1 -1 106 390 1 1 139 349 @b np 119 395 :M 119 385 :L 120 385 :L 120 385 :L 120 385 :L 120 385 :L 120 385 :L 120 385 :L 121 385 :L 121 385 :L 121 385 :L 121 385 :L 121 385 :L 121 385 :L 122 385 :L 122 386 :L 122 386 :L 122 386 :L 122 386 :L 122 386 :L 123 386 :L 123 386 :L 123 386 :L 123 386 :L 123 386 :L 123 386 :L 124 386 :L 124 386 :L 124 386 :L 124 386 :L 124 387 :L 124 387 :L 124 387 :L 125 387 :L 125 387 :L 125 387 :L 125 387 :L 125 387 :L 125 387 :L 125 388 :L 126 388 :L 126 388 :L 126 388 :L 126 388 :L 126 388 :L 126 388 :L 126 388 :L 126 388 :L 126 389 :L 119 395 :L 119 395 :L eofill -1 -1 123 390 1 1 141 351 @b np 137 396 :M 135 386 :L 135 386 :L 135 386 :L 135 386 :L 135 386 :L 135 386 :L 136 386 :L 136 386 :L 136 386 :L 136 386 :L 136 386 :L 136 386 :L 137 386 :L 137 386 :L 137 386 :L 137 386 :L 137 386 :L 138 386 :L 138 386 :L 138 386 :L 138 386 :L 138 386 :L 138 386 :L 139 386 :L 139 386 :L 139 386 :L 139 386 :L 139 386 :L 139 386 :L 140 386 :L 140 387 :L 140 387 :L 140 387 :L 140 387 :L 140 387 :L 141 387 :L 141 387 :L 141 387 :L 141 387 :L 141 387 :L 141 387 :L 141 387 :L 142 387 :L 142 387 :L 142 387 :L 142 388 :L 142 388 :L 142 388 :L 137 396 :L 137 396 :L eofill -1 -1 139 390 1 1 145 350 @b np 154 394 :M 149 385 :L 149 385 :L 149 385 :L 149 385 :L 150 385 :L 150 385 :L 150 385 :L 150 385 :L 150 385 :L 150 385 :L 151 385 :L 151 385 :L 151 385 :L 151 384 :L 151 384 :L 151 384 :L 152 384 :L 152 384 :L 152 384 :L 152 384 :L 152 384 :L 152 384 :L 153 384 :L 153 384 :L 153 384 :L 153 384 :L 153 384 :L 153 384 :L 154 384 :L 154 384 :L 154 384 :L 154 384 :L 154 384 :L 154 384 :L 155 384 :L 155 384 :L 155 384 :L 155 384 :L 155 384 :L 156 384 :L 156 384 :L 156 384 :L 156 384 :L 156 384 :L 156 384 :L 157 384 :L 157 385 :L 157 385 :L 154 394 :L 154 394 :L eofill 149 352 -1 1 153 387 1 149 351 @a 92 397 16 13 rC 92 406 :M f3_12 sf (x)S 98 406 :M (1)S gR gS 84 329 381 98 rC 89.5 395.5 15 14 rS 111 397 16 13 rC 111 406 :M f3_12 sf (x)S 117 406 :M (2)S gR gS 84 329 381 98 rC 108.5 395.5 15 14 rS 130 397 16 13 rC 130 406 :M f3_12 sf (x)S 136 406 :M (3)S gR gS 84 329 381 98 rC 127.5 395.5 15 14 rS 149 397 16 13 rC 149 406 :M f3_12 sf (x)S 155 406 :M (4)S gR gS 84 329 381 98 rC 146.5 395.5 15 14 rS 168 397 16 13 rC 168 406 :M f3_12 sf (x)S 174 406 :M (5)S gR gS 84 329 381 98 rC 165.5 395.5 15 14 rS 187 397 16 13 rC 187 406 :M f3_12 sf (x)S 193 406 :M (6)S gR gS 84 329 381 98 rC 184.5 395.5 15 14 rS 215 396 16 13 rC 215 405 :M f3_12 sf (x)S 221 405 :M (7)S gR gS 84 329 381 98 rC 212.5 394.5 15 14 rS 234 396 16 13 rC 234 405 :M f3_12 sf (x)S 240 405 :M (8)S gR gS 84 329 381 98 rC 231.5 394.5 15 14 rS 253 396 15 13 rC 253 405 :M f3_12 sf (x)S 259 405 :M (9)S gR gS 84 329 381 98 rC 250.5 394.5 15 14 rS 273 396 21 13 rC 273 405 :M f3_12 sf (x)S 279 405 :M (1)S 285 405 :M (0)S gR gS 84 329 381 98 rC 270.5 394.5 23 15 rS 297 396 22 13 rC 297 405 :M f3_12 sf (x)S 303 405 :M (1)S 309 405 :M (1)S gR gS 84 329 381 98 rC 295.5 394.5 23 15 rS 339 396 22 13 rC 339 405 :M f3_12 sf (x)S 345 405 :M (1)S 351 405 :M (2)S gR gS 84 329 381 98 rC 337.5 394.5 23 15 rS 364 396 22 13 rC 364 405 :M f3_12 sf (x)S 370 405 :M (1)S 376 405 :M (3)S gR gS 84 329 381 98 rC 362.5 394.5 23 15 rS 389 396 22 13 rC 389 405 :M f3_12 sf (x)S 395 405 :M (1)S 401 405 :M (4)S gR gS 84 329 381 98 rC 387.5 394.5 23 15 rS 414 396 22 13 rC 414 405 :M f3_12 sf (x)S 420 405 :M (1)S 426 405 :M (5)S gR gS 84 329 381 98 rC 412.5 394.5 23 15 rS 439 396 22 13 rC 439 405 :M f3_12 sf (x)S 445 405 :M (1)S 451 405 :M (6)S gR gS 84 329 381 98 rC 437.5 394.5 22 15 rS np 172 395 :M 165 388 :L 165 388 :L 165 388 :L 165 388 :L 165 388 :L 165 387 :L 165 387 :L 166 387 :L 166 387 :L 166 387 :L 166 387 :L 166 387 :L 166 387 :L 166 387 :L 167 386 :L 167 386 :L 167 386 :L 167 386 :L 167 386 :L 167 386 :L 168 386 :L 168 386 :L 168 386 :L 168 386 :L 168 386 :L 168 386 :L 168 386 :L 169 386 :L 169 386 :L 169 385 :L 169 385 :L 169 385 :L 169 385 :L 170 385 :L 170 385 :L 170 385 :L 170 385 :L 170 385 :L 171 385 :L 171 385 :L 171 385 :L 171 385 :L 171 385 :L 171 385 :L 172 385 :L 172 385 :L 172 385 :L 172 385 :L 172 395 :L 172 395 :L eofill 153 350 -1 1 169 389 1 153 349 @a np 191 393 :M 182 388 :L 182 388 :L 182 388 :L 182 388 :L 182 387 :L 183 387 :L 183 387 :L 183 387 :L 183 387 :L 183 387 :L 183 387 :L 183 386 :L 183 386 :L 183 386 :L 184 386 :L 184 386 :L 184 386 :L 184 386 :L 184 386 :L 184 386 :L 184 385 :L 184 385 :L 185 385 :L 185 385 :L 185 385 :L 185 385 :L 185 385 :L 185 385 :L 185 385 :L 186 384 :L 186 384 :L 186 384 :L 186 384 :L 186 384 :L 186 384 :L 186 384 :L 187 384 :L 187 384 :L 187 384 :L 187 384 :L 187 384 :L 187 384 :L 188 384 :L 188 384 :L 188 383 :L 188 383 :L 188 383 :L 188 383 :L 191 393 :L 191 393 :L eofill 157 350 -1 1 186 388 1 157 349 @a 257 334 16 13 rC 257 343 :M f3_12 sf (T)S 264 343 :M (2)S gR gS 84 329 381 98 rC 25 15 263 341 @f 391 334 17 13 rC 391 343 :M f3_12 sf (T)S 398 343 :M (3)S gR gS 84 329 381 98 rC 25 15 398 341 @f np 261 392 :M 256 383 :L 257 383 :L 257 383 :L 257 383 :L 257 383 :L 257 383 :L 257 383 :L 258 383 :L 258 382 :L 258 382 :L 258 382 :L 258 382 :L 258 382 :L 259 382 :L 259 382 :L 259 382 :L 259 382 :L 259 382 :L 259 382 :L 260 382 :L 260 382 :L 260 382 :L 260 382 :L 260 382 :L 261 382 :L 261 382 :L 261 382 :L 261 382 :L 261 382 :L 261 382 :L 262 382 :L 262 382 :L 262 382 :L 262 382 :L 262 382 :L 262 382 :L 263 382 :L 263 382 :L 263 382 :L 263 382 :L 263 382 :L 263 383 :L 264 383 :L 264 383 :L 264 383 :L 264 383 :L 264 383 :L 264 383 :L 261 392 :L 261 392 :L eofill -1 -1 261 386 1 1 260 348 @b np 399 394 :M 395 385 :L 395 385 :L 395 385 :L 396 385 :L 396 385 :L 396 385 :L 396 385 :L 396 385 :L 396 384 :L 397 384 :L 397 384 :L 397 384 :L 397 384 :L 397 384 :L 397 384 :L 398 384 :L 398 384 :L 398 384 :L 398 384 :L 398 384 :L 398 384 :L 399 384 :L 399 384 :L 399 384 :L 399 384 :L 399 384 :L 400 384 :L 400 384 :L 400 384 :L 400 384 :L 400 384 :L 400 384 :L 401 384 :L 401 384 :L 401 384 :L 401 384 :L 401 384 :L 401 384 :L 402 384 :L 402 384 :L 402 384 :L 402 385 :L 402 385 :L 402 385 :L 403 385 :L 403 385 :L 403 385 :L 403 385 :L 399 394 :L 399 394 :L eofill -1 -1 400 388 1 1 399 350 @b gR gS 0 0 552 730 rC 157 452 :M f0_12 sf (Figure )S 194 452 :M (21: Hypothesized Measurement Model)S 81 480 :M f3_12 sf .416 .042(The Purify module represents a different approach to the problem that is a provably)J 59 496 :M (correct)S 92 491 :M f3_7 sf (1)S 95 491 :M (4)S 98 496 :M f3_12 sf 1.278 .128( algorithm for finding unidimensional measurement models \(Scheines, 1993\).)J 59 512 :M 1.634 .163(Instead of searching for parameters to free, i.e., edges to add, Purify searches for a)J 59 528 :M 1.142 .114(submodel of the originally specified measurement model that contains a subset of the)J 59 544 :M .746 .075(indicators originally specified, but that is correctly specified as unidimensional. Such a)J 59 560 :M .53 .053(submodel can be used to find consistent estimates of the correlations between the latent)J 59 576 :M (variables, and thus aid in the search for structural models.)S 336 571 :M f3_7 sf (1)S 339 571 :M (5)S 81 592 :M f3_12 sf .416 .042(For example, the measurement model in Figure 20 is not unidimensional because of)J 59 608 :M .77 .077(the edges and correlated errors that are in boldface. But note that the population model)J 59 626 :M ( )S 59 623.48 -.48 .48 203.48 623 .48 59 623 @a 81 635 :M f3_6 sf (1)S 84 635 :M (4)S 87 639 :M f3_10 sf .311 .031( Purify is a correct search procedure in the following sense. Given that there are correctly specified)J 59 651 :M .638 .064(unidimensional submodels of the initially specified measurement model with at least three indicators for)J 59 663 :M 1.406 .141(each latent, then as the sample grows without bound the probability that Purify will find one of the)J 59 675 :M (unidimensional submodels converges to one.)S 81 683 :M f3_6 sf (1)S 84 683 :M (5)S 87 687 :M f3_10 sf ( This two stage search process was also suggested by Anderson and Gerbing, \(1988\).)S endp %%Page: 32 32 %%BeginPageSetup initializepage (peter; page: 32 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (32)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf 1.564 .156(does contain a unidimensional submodel, shown in )J 326 54 :M 1.648 .165(Figure 22, which is obtained by)J 59 70 :M (simply removing X)S f3_7 sf 0 3 rm (1)S 0 -3 rm 156 70 :M f3_12 sf (, X)S 171 73 :M f3_7 sf (7)S 175 70 :M f3_12 sf (, X)S 190 73 :M f3_7 sf (12)S f3_12 sf 0 -3 rm (, and X)S 0 3 rm 232 73 :M f3_7 sf (13)S f3_12 sf 0 -3 rm ( from the model.)S 0 3 rm 84 89 381 98 rC np 240 153 :M 238 143 :L 239 143 :L 239 143 :L 239 143 :L 239 143 :L 239 143 :L 240 143 :L 240 143 :L 240 143 :L 240 143 :L 240 143 :L 240 143 :L 241 143 :L 241 143 :L 241 143 :L 241 143 :L 241 143 :L 241 143 :L 242 143 :L 242 143 :L 242 143 :L 242 143 :L 242 143 :L 242 144 :L 243 144 :L 243 144 :L 243 144 :L 243 144 :L 243 144 :L 243 144 :L 244 144 :L 244 144 :L 244 144 :L 244 144 :L 244 144 :L 244 144 :L 245 144 :L 245 144 :L 245 145 :L 245 145 :L 245 145 :L 245 145 :L 245 145 :L 246 145 :L 246 145 :L 246 145 :L 246 145 :L 246 146 :L 240 153 :L 240 153 :L eofill -1 -1 242 147 1 1 254 107 @b np 283 154 :M 277 147 :L 277 146 :L 277 146 :L 277 146 :L 277 146 :L 278 146 :L 278 146 :L 278 146 :L 278 146 :L 278 146 :L 278 145 :L 278 145 :L 279 145 :L 279 145 :L 279 145 :L 279 145 :L 279 145 :L 279 145 :L 280 145 :L 280 145 :L 280 145 :L 280 145 :L 280 145 :L 280 145 :L 281 145 :L 281 144 :L 281 144 :L 281 144 :L 281 144 :L 281 144 :L 282 144 :L 282 144 :L 282 144 :L 282 144 :L 282 144 :L 282 144 :L 283 144 :L 283 144 :L 283 144 :L 283 144 :L 283 144 :L 283 144 :L 284 144 :L 284 144 :L 284 144 :L 284 144 :L 284 144 :L 285 144 :L 283 154 :L 283 154 :L eofill 268 108 -1 1 282 147 1 268 107 @a np 309 154 :M 301 149 :L 301 149 :L 301 149 :L 301 149 :L 301 149 :L 301 149 :L 301 148 :L 301 148 :L 301 148 :L 301 148 :L 302 148 :L 302 148 :L 302 148 :L 302 147 :L 302 147 :L 302 147 :L 302 147 :L 302 147 :L 302 147 :L 303 147 :L 303 147 :L 303 147 :L 303 146 :L 303 146 :L 303 146 :L 303 146 :L 304 146 :L 304 146 :L 304 146 :L 304 146 :L 304 146 :L 304 145 :L 304 145 :L 305 145 :L 305 145 :L 305 145 :L 305 145 :L 305 145 :L 305 145 :L 305 145 :L 306 145 :L 306 145 :L 306 145 :L 306 145 :L 306 145 :L 306 145 :L 307 144 :L 307 144 :L 309 154 :L 309 154 :L eofill 272 108 -1 1 305 149 1 272 107 @a np 450 154 :M 441 150 :L 441 149 :L 441 149 :L 441 149 :L 441 149 :L 441 149 :L 442 149 :L 442 149 :L 442 148 :L 442 148 :L 442 148 :L 442 148 :L 442 148 :L 442 148 :L 442 148 :L 443 147 :L 443 147 :L 443 147 :L 443 147 :L 443 147 :L 443 147 :L 443 147 :L 443 147 :L 443 147 :L 444 146 :L 444 146 :L 444 146 :L 444 146 :L 444 146 :L 444 146 :L 444 146 :L 445 146 :L 445 146 :L 445 145 :L 445 145 :L 445 145 :L 445 145 :L 446 145 :L 446 145 :L 446 145 :L 446 145 :L 446 145 :L 446 145 :L 446 145 :L 447 145 :L 447 145 :L 447 145 :L 447 144 :L 450 154 :L 450 154 :L eofill 409 107 -1 1 446 149 1 409 106 @a np 421 153 :M 414 146 :L 414 146 :L 415 146 :L 415 145 :L 415 145 :L 415 145 :L 415 145 :L 415 145 :L 415 145 :L 416 145 :L 416 145 :L 416 145 :L 416 144 :L 416 144 :L 416 144 :L 416 144 :L 417 144 :L 417 144 :L 417 144 :L 417 144 :L 417 144 :L 417 144 :L 418 144 :L 418 144 :L 418 144 :L 418 144 :L 418 144 :L 418 143 :L 419 143 :L 419 143 :L 419 143 :L 419 143 :L 419 143 :L 419 143 :L 420 143 :L 420 143 :L 420 143 :L 420 143 :L 420 143 :L 420 143 :L 421 143 :L 421 143 :L 421 143 :L 421 143 :L 421 143 :L 421 143 :L 422 143 :L 422 143 :L 421 153 :L 421 153 :L eofill 405 109 -1 1 419 146 1 405 108 @a 140 96 17 13 rC 140 105 :M (T)S 147 105 :M (1)S gR gS 84 89 381 98 rC 25 14 145 103.5 @f np 119 155 :M 119 145 :L 120 145 :L 120 145 :L 120 145 :L 120 145 :L 120 145 :L 120 145 :L 121 145 :L 121 145 :L 121 145 :L 121 145 :L 121 145 :L 121 145 :L 122 145 :L 122 146 :L 122 146 :L 122 146 :L 122 146 :L 122 146 :L 123 146 :L 123 146 :L 123 146 :L 123 146 :L 123 146 :L 123 146 :L 124 146 :L 124 146 :L 124 146 :L 124 146 :L 124 147 :L 124 147 :L 124 147 :L 125 147 :L 125 147 :L 125 147 :L 125 147 :L 125 147 :L 125 147 :L 125 148 :L 126 148 :L 126 148 :L 126 148 :L 126 148 :L 126 148 :L 126 148 :L 126 148 :L 126 148 :L 126 149 :L 119 155 :L 119 155 :L eofill -1 -1 123 150 1 1 141 111 @b np 137 156 :M 135 146 :L 135 146 :L 135 146 :L 135 146 :L 135 146 :L 135 146 :L 136 146 :L 136 146 :L 136 146 :L 136 146 :L 136 146 :L 136 146 :L 137 146 :L 137 146 :L 137 146 :L 137 146 :L 137 146 :L 138 146 :L 138 146 :L 138 146 :L 138 146 :L 138 146 :L 138 146 :L 139 146 :L 139 146 :L 139 146 :L 139 146 :L 139 146 :L 139 146 :L 140 146 :L 140 147 :L 140 147 :L 140 147 :L 140 147 :L 140 147 :L 141 147 :L 141 147 :L 141 147 :L 141 147 :L 141 147 :L 141 147 :L 141 147 :L 142 147 :L 142 147 :L 142 147 :L 142 148 :L 142 148 :L 142 148 :L 137 156 :L 137 156 :L eofill -1 -1 139 150 1 1 145 110 @b np 154 154 :M 149 145 :L 149 145 :L 149 145 :L 149 145 :L 150 145 :L 150 145 :L 150 145 :L 150 145 :L 150 145 :L 150 145 :L 151 145 :L 151 145 :L 151 145 :L 151 144 :L 151 144 :L 151 144 :L 152 144 :L 152 144 :L 152 144 :L 152 144 :L 152 144 :L 152 144 :L 153 144 :L 153 144 :L 153 144 :L 153 144 :L 153 144 :L 153 144 :L 154 144 :L 154 144 :L 154 144 :L 154 144 :L 154 144 :L 154 144 :L 155 144 :L 155 144 :L 155 144 :L 155 144 :L 155 144 :L 156 144 :L 156 144 :L 156 144 :L 156 144 :L 156 144 :L 156 144 :L 157 144 :L 157 145 :L 157 145 :L 154 154 :L 154 154 :L eofill 149 112 -1 1 153 147 1 149 111 @a 111 157 16 13 rC 111 166 :M f3_12 sf (x)S 117 166 :M (2)S gR gS 84 89 381 98 rC 108.5 156.5 15 13 rS 130 157 16 13 rC 130 166 :M f3_12 sf (x)S 136 166 :M (3)S gR gS 84 89 381 98 rC 127.5 156.5 15 13 rS 149 157 16 13 rC 149 166 :M f3_12 sf (x)S 155 166 :M (4)S gR gS 84 89 381 98 rC 146.5 156.5 15 13 rS 168 157 16 13 rC 168 166 :M f3_12 sf (x)S 174 166 :M (5)S gR gS 84 89 381 98 rC 165.5 156.5 15 13 rS 187 157 16 13 rC 187 166 :M f3_12 sf (x)S 193 166 :M (6)S gR gS 84 89 381 98 rC 184.5 156.5 15 13 rS 234 156 16 13 rC 234 165 :M f3_12 sf (x)S 240 165 :M (8)S gR gS 84 89 381 98 rC 231.5 155.5 15 13 rS 253 156 15 13 rC 253 165 :M f3_12 sf (x)S 259 165 :M (9)S gR gS 84 89 381 98 rC 250.5 155.5 15 13 rS 273 156 21 13 rC 273 165 :M f3_12 sf (x)S 279 165 :M (1)S 285 165 :M (0)S gR gS 84 89 381 98 rC 270.5 154.5 23 15 rS 297 156 22 13 rC 297 165 :M f3_12 sf (x)S 303 165 :M (1)S 309 165 :M (1)S gR gS 84 89 381 98 rC 295.5 155.5 23 14 rS 389 156 22 13 rC 389 165 :M f3_12 sf (x)S 395 165 :M (1)S 401 165 :M (4)S gR gS 84 89 381 98 rC 387.5 155.5 23 14 rS 414 156 22 13 rC 414 165 :M f3_12 sf (x)S 420 165 :M (1)S 426 165 :M (5)S gR gS 84 89 381 98 rC 412.5 155.5 23 14 rS 439 156 22 13 rC 439 165 :M f3_12 sf (x)S 445 165 :M (1)S 451 165 :M (6)S gR gS 84 89 381 98 rC 437.5 155.5 22 14 rS np 172 155 :M 165 148 :L 165 148 :L 165 148 :L 165 148 :L 165 148 :L 165 147 :L 165 147 :L 166 147 :L 166 147 :L 166 147 :L 166 147 :L 166 147 :L 166 147 :L 166 147 :L 167 146 :L 167 146 :L 167 146 :L 167 146 :L 167 146 :L 167 146 :L 168 146 :L 168 146 :L 168 146 :L 168 146 :L 168 146 :L 168 146 :L 168 146 :L 169 146 :L 169 146 :L 169 145 :L 169 145 :L 169 145 :L 169 145 :L 170 145 :L 170 145 :L 170 145 :L 170 145 :L 170 145 :L 171 145 :L 171 145 :L 171 145 :L 171 145 :L 171 145 :L 171 145 :L 172 145 :L 172 145 :L 172 145 :L 172 145 :L 172 155 :L 172 155 :L eofill 153 110 -1 1 169 149 1 153 109 @a np 191 153 :M 182 148 :L 182 148 :L 182 148 :L 182 148 :L 182 147 :L 183 147 :L 183 147 :L 183 147 :L 183 147 :L 183 147 :L 183 147 :L 183 146 :L 183 146 :L 183 146 :L 184 146 :L 184 146 :L 184 146 :L 184 146 :L 184 146 :L 184 146 :L 184 145 :L 184 145 :L 185 145 :L 185 145 :L 185 145 :L 185 145 :L 185 145 :L 185 145 :L 185 145 :L 186 144 :L 186 144 :L 186 144 :L 186 144 :L 186 144 :L 186 144 :L 186 144 :L 187 144 :L 187 144 :L 187 144 :L 187 144 :L 187 144 :L 187 144 :L 188 144 :L 188 144 :L 188 143 :L 188 143 :L 188 143 :L 188 143 :L 191 153 :L 191 153 :L eofill 157 110 -1 1 186 148 1 157 109 @a 257 94 16 13 rC 257 103 :M f3_12 sf (T)S 264 103 :M (2)S gR gS 84 89 381 98 rC 25 15 263 101 @f 391 94 17 13 rC 391 103 :M f3_12 sf (T)S 398 103 :M (3)S gR gS 84 89 381 98 rC 25 15 398 101 @f np 261 152 :M 256 143 :L 257 143 :L 257 143 :L 257 143 :L 257 143 :L 257 143 :L 257 143 :L 258 143 :L 258 142 :L 258 142 :L 258 142 :L 258 142 :L 258 142 :L 259 142 :L 259 142 :L 259 142 :L 259 142 :L 259 142 :L 259 142 :L 260 142 :L 260 142 :L 260 142 :L 260 142 :L 260 142 :L 261 142 :L 261 142 :L 261 142 :L 261 142 :L 261 142 :L 261 142 :L 262 142 :L 262 142 :L 262 142 :L 262 142 :L 262 142 :L 262 142 :L 263 142 :L 263 142 :L 263 142 :L 263 142 :L 263 142 :L 263 143 :L 264 143 :L 264 143 :L 264 143 :L 264 143 :L 264 143 :L 264 143 :L 261 152 :L 261 152 :L eofill -1 -1 261 146 1 1 260 108 @b np 399 154 :M 395 145 :L 395 145 :L 395 145 :L 396 145 :L 396 145 :L 396 145 :L 396 145 :L 396 145 :L 396 144 :L 397 144 :L 397 144 :L 397 144 :L 397 144 :L 397 144 :L 397 144 :L 398 144 :L 398 144 :L 398 144 :L 398 144 :L 398 144 :L 398 144 :L 399 144 :L 399 144 :L 399 144 :L 399 144 :L 399 144 :L 400 144 :L 400 144 :L 400 144 :L 400 144 :L 400 144 :L 400 144 :L 401 144 :L 401 144 :L 401 144 :L 401 144 :L 401 144 :L 401 144 :L 402 144 :L 402 144 :L 402 144 :L 402 145 :L 402 145 :L 402 145 :L 403 145 :L 403 145 :L 403 145 :L 403 145 :L 399 154 :L 399 154 :L eofill -1 -1 400 148 1 1 399 110 @b 158 102.75 -.75 .75 242.75 102 .75 158 102 @a np 240 98 :M 240 106 :L 248 102 :L 240 98 :L .75 lw eofill -.75 -.75 240.75 106.75 .75 .75 240 98 @b -.75 -.75 240.75 106.75 .75 .75 248 102 @b 240 98.75 -.75 .75 248.75 102 .75 240 98 @a 277 101.75 -.75 .75 378.75 101 .75 277 101 @a np 376 97 :M 376 105 :L 384 101 :L 376 97 :L eofill -.75 -.75 376.75 105.75 .75 .75 376 97 @b -.75 -.75 376.75 105.75 .75 .75 384 101 @b 376 97.75 -.75 .75 384.75 101 .75 376 97 @a gR gS 0 0 552 730 rC 71 212 :M f0_12 sf (Figure )S 108 212 :M (22. Model with Correctly Specified Unidimensional Measurement Model)S 81 240 :M f3_12 sf 2.273 .227(Purify searches for unidimensional submodels in the following way. First we)J 59 256 :M 2.694 .269(suppose that we are given as input a hypothetical measurement model which is)J 59 272 :M .89 .089(unidimensional, for example, the measurement model shown in Figure )J 415 272 :M .964 .096(21. We assume)J 59 288 :M .605 .06(the input measurement model is a submodel of the population measurement model, that)J 59 304 :M 2.355 .236(is, every edge specified in the input measurement model exists in the population)J 59 320 :M .797 .08(measurement model. However, we do not assume that the input measurement model is)J 59 336 :M 1.529 .153(complete; the population measurement model may be non-unidimensional because a)J 59 352 :M 1.907 .191(single indicator may be caused by multiple latents, cause other indicators, or have)J 59 368 :M (correlated errors with other variables.)S 81 384 :M .191 .019(Given this input, if the population measurement model is unidimensional, it entails a)J 59 400 :M .499 .05(characteristic set of vanishing tetrad differences,)J 297 400 :M f4_12 sf .56 .056( regardless of the population structural)J 59 416 :M 3.815 .381(model )J 96 416 :M f3_12 sf 3.26 .326(\(Scheines, 1993\). For example, if the population measurement model is)J 59 432 :M .643 .064(unidimensional, and X)J 171 435 :M f3_7 sf (1)S 175 432 :M f3_12 sf 1.071 .107(, X)J 191 435 :M f3_7 sf (2)S 195 432 :M f3_12 sf .866 .087(, and X)J f3_7 sf 0 3 rm (3)S 0 -3 rm 236 432 :M f3_12 sf .857 .086( measure a single latent, then )J 386 432 :M f1_12 sf (r)S 393 435 :M f7_7 sf .019(X1,X2)A 414 432 :M f7_12 sf .609 .061( )J f1_12 sf (r)S 426 435 :M f7_7 sf .019(X3,X4)A 447 432 :M f7_12 sf 1.149 .115<20D020>J 463 432 :M f1_12 sf (r)S 470 435 :M f7_7 sf .019(X1,X3)A 59 448 :M f1_12 sf (r)S 66 451 :M f7_7 sf .122 .012(X2,X4 )J f3_12 sf 0 -3 rm .163 .016(is entailed to be zero regardless of the population structural model. This means that)J 0 3 rm 59 464 :M .214 .021(Purify can test whether the specified measurement model is truly unidimensional without)J 59 480 :M 1.384 .138(knowing the structural model. If the characteristic set of vanishing tetrad differences)J 59 496 :M 1.134 .113(entailed by a unidimensional measurement model is judged to hold in the population,)J 59 512 :M -.005(Purify concludes that the measurement model specified is truly unidimensional, and halts.)A 59 528 :M 2.274 .227(If the population measurement model is the one in Figure 20)J 385 528 :M 2.436 .244(, some of the tetrad)J 59 544 :M 1.176 .118(differences entailed by the initially specified model in )J 337 544 :M 1.174 .117(Figure 21, e.g., )J f1_12 sf (r)S 425 547 :M f7_7 sf .019(X1,X2)A 446 544 :M f7_12 sf .825 .082( )J f1_12 sf (r)S 459 547 :M f7_7 sf .019(X3,X4)A 480 544 :M f7_12 sf 1.106 .111<20D0>J 59 560 :M f1_12 sf (r)S 66 563 :M f7_7 sf .019(X1,X3)A 87 560 :M f7_12 sf 2.78 .278( )J 94 560 :M f1_12 sf (r)S 101 563 :M f7_7 sf .019(X2,X4)A 122 560 :M f3_12 sf 1.269 .127(, are not entailed to vanish, and with a representative sample Purify will)J 59 576 :M 2.032 .203(conclude that the population measurement model among the given set of indicator)J 59 592 :M 1.622 .162(variables is not unidimensional. Purify then begins to search for a submodel that is)J 59 608 :M .098 .01(unidimensional by sequentially eliminating indicators. In general, searching all subsets of)J 59 624 :M 2.572 .257(the given measured indicators for a set of indicators that form a unidimensional)J 59 640 :M .175 .017(measurement model would take too long, due to the enormous number of subsets. But by)J 59 656 :M (examining )S 113 656 :M f4_12 sf (which)S 142 656 :M f3_12 sf -.005( vanishing tetrad differences do not hold in the population, the algorithm)A 59 672 :M 3.466 .347(can greatly narrow the search, making it feasible to handle initially specified)J endp %%Page: 33 33 %%BeginPageSetup initializepage (peter; page: 33 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (33)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .748 .075(measurement models with more than 50 measured variables in minutes. In this case on)J 59 70 :M .778 .078(simulated data it correctly removes X)J f3_7 sf 0 3 rm (1)S 0 -3 rm 250 70 :M f3_12 sf .857 .086(, X)J f3_7 sf 0 3 rm (7)S 0 -3 rm 270 70 :M f3_12 sf 1.15 .115(, X)J 287 73 :M f3_7 sf .148(12)A f3_12 sf 0 -3 rm .548 .055(, and X)J 0 3 rm f3_7 sf .148(13)A f3_12 sf 0 -3 rm .927 .093( from the measurement model,)J 0 3 rm 59 86 :M 2.469 .247(leaving a set of indicators that have a measurement model correctly specified as)J 59 102 :M (unidimensional.)S 59 130 :M f4_12 sf (5)S 65 130 :M (.)S 68 130 :M (2)S 74 130 :M (.)S 77 130 :M (2)S 83 130 :M ( )S 90 130 :M (MIMbuild)S 81 149 :M f3_12 sf 2.582 .258(In many studies the theoretical question addressed cannot be reduced to the)J 59 165 :M .49 .049(significance of a single parameter in an otherwise reliably specified model. There might)J 59 181 :M .094 .009(be many latent variables, and the problem of finding a reasonable structural model is then)J 59 197 :M .209 .021(difficult. With just four latent variables there are well over 700 structural models with no)J 59 213 :M .833 .083(correlated errors. Even with substantial background knowledge, this is a large space to)J 59 229 :M 1.966 .197(search. With eight latent variables the space is astronomical. Several strategies for)J 59 245 :M .952 .095(automatic structural model search are possible. One might begin with a null structural)J 59 261 :M .547 .055(model and do a Lagrange Multiplier search limited to structural parameters. To the best)J 59 277 :M .05 .005(of our knowledge no one has studied the behavior of this strategy. One might estimate the)J 59 293 :M .202 .02(correlations among the latent variables and then apply Build to the latents as if they were)J 59 309 :M .58 .058(measured. In our experience, this works well in simulation studies at moderate to large)J 59 325 :M .089 .009(sample sizes, but we do not know how to properly adjust the sample size when testing for)J 59 341 :M 2.125 .213(vanishing partial correlations among latents that are being treated \322as if\323 they are)J 59 357 :M .767 .077(measured. A third alternative takes further advantage of the vanishing tetrad difference)J 59 373 :M (constraint.)S 81 389 :M .075 .008(We have already seen that if the population measurement model is unidimensional, a)J 59 405 :M 1.036 .104(SEM entails a characteristic set of vanishing tetrad differences, regardless of what the)J 59 421 :M .984 .098(population structural model may be. But if the measurement model is unidimensional,)J 59 437 :M .171 .017(there are other tetrad differences which are entailed to vanish for some structural models,)J 59 453 :M .173 .017(but are not entailed to vanish for other structural models. These constraints are extremely)J 59 469 :M .604 .06(easy to compute and test, and the tests are not susceptible to specification error in other)J 59 485 :M .721 .072(parts of the structural model \(Scheines, 1993\). For example, in the model in )J 441 485 :M .601 .06(Figure 22,)J 59 501 :M .35 .035(\(where the population measurement model is unidimensional\) all three tetrad constraints)J 59 517 :M .219 .022(involving one indicator from T)J f3_7 sf 0 3 rm (1)S 0 -3 rm 213 517 :M f3_12 sf .287 .029(, two from T)J 275 520 :M f3_7 sf (2)S 279 517 :M f3_12 sf .285 .029(, and one from T)J 361 520 :M f3_7 sf (3)S 365 517 :M f3_12 sf .262 .026( are entailed by the model)J 59 533 :M .448 .045(if and only if there is no edge between T)J f3_7 sf 0 3 rm (1)S 0 -3 rm 263 533 :M f3_12 sf .531 .053( and T)J 295 536 :M f3_7 sf (3)S 299 533 :M f3_12 sf .411 .041(. The MIMbuild algorithm uses tests of)J 59 549 :M .789 .079(vanishing tetrad differences to construct a set of structural models that entail vanishing)J 59 565 :M (partial correlations among latent variables judged to hold in the population.)S 81 581 :M 3.05 .305(The set of structural models that MIMbuild outputs entail the same set of)J 59 597 :M 2.881 .288(unconditional correlations and partial correlations with only one variable in the)J 59 613 :M 1.278 .128(conditioning set. Because it can output models which are not fully partial correlation)J 59 629 :M .74 .074(equivalent or covariance equivalent, MIMbuild represents only a partial solution to the)J 59 645 :M 2.153 .215(RSEM structural model specification problem. A \322?\323 is attached to those parts of)J 59 661 :M -.002(MIMbuild\325s output that might change if second order or higher partial correlations among)A endp %%Page: 34 34 %%BeginPageSetup initializepage (peter; page: 34 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (34)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .937 .094(latent variables could be tested. Section 8.4 of the appendix details the sense in which)J 59 70 :M (MIMbuild is a correct estimator of structural models.)S 236 114 :M f4_12 sf (6)S 242 114 :M (.)S 245 114 :M 4 0 rm ( )S 252 114 :M (Applications)S 81 158 :M f3_12 sf .193 .019(The TETRAD II procedures have been used to study job satisfaction among military)J 59 174 :M .64 .064(personnel \(Callahan & Sorensen, 1992\), to develop psychiatric measures \(Prigerson, et.)J 59 190 :M .487 .049(al., 1995\), to predict survival and death in pneumonia patients \(Cooper, et. al., 1995\), to)J 59 206 :M 1.506 .151(study mechanisms in plant biology \(Shipley, forthcoming\), to study dropout rates in)J 59 222 :M .756 .076(American universities \(Druzdzel & Glymour, 1994\), even to recalibrate instruments on)J 59 238 :M .96 .096(orbiting satellites \(Waldemark & Norqvist, 1995\). In this section we will illustrate the)J 59 254 :M .436 .044(application of the search procedures to three data sets, two published and one simulated.)J 59 270 :M .494 .049(In the case of the empirical examples we do not mean to endorse the assumptions made)J 59 286 :M .31 .031(by the researchers who used the data sets, or the scales they constructed. In the first case)J 59 302 :M .208 .021(our intent is to show how the search procedures implemented in TETRAD II can be used)J 59 318 :M 1.007 .101(to find plausible alternatives to a published model. The existence of these alternatives)J 59 334 :M .271 .027(weakens the evidential support for conclusions published, but it is not our intent to claim)J 59 350 :M 1.102 .11(that the alternatives found by TETRAD II are correct. In the second case we ran the)J 59 366 :M 1.494 .149(Purify and Search procedures on a published data set with the same results as those)J 59 382 :M 1.027 .103(published, and in the third case we show how the procedures perform on a very large)J 59 398 :M .075 .008(search space. Other applications can be found in \(Scheines, et. al., 1994\), and in \(Spirtes,)J 59 414 :M (et. al., 1993\).)S 59 442 :M f4_12 sf (6)S 65 442 :M (.)S 68 442 :M (1)S 74 442 :M ( )S 81 442 :M (Finding Alternative Models)S 81 464 :M f3_12 sf 1.275 .127(Before giving a proposed hypothesis any great credence, good scientific practice)J 59 480 :M .142 .014(ought to try to articulate and investigate every serious alternative. A frequent objection to)J 59 496 :M 1.089 .109(causal models in any discipline is that they are arbitrarily selected without any sound)J 59 512 :M 1.593 .159(arguments that would exclude alternative explanations of data. That some cherished)J 59 528 :M .731 .073(causal model cannot be rejected statistically is little reason to believe its causal claims:)J 59 544 :M 1.871 .187(There might be alternatives that also cannot be rejected statistically, but that make)J 59 560 :M .327 .033(contrary causal claims. Published studies may be defective in their general distributional)J 59 576 :M 1.03 .103(assumptions, in their data collection procedures, or in their assumptions about what is)J 59 592 :M .47 .047(influencing what. Here is an illustration of how TETRAD II is meant to be used to help)J 59 608 :M 2.191 .219(search for and articulate alternative causal explanations under varying background)J 59 624 :M .188 .019(assumptions. )J 125 624 :M .242 .024(In a study published in the American Sociological Review, Timberlake and)J 59 640 :M .511 .051(Williams \(1984\) claimed that foreign investment in Third World or "peripheral" nations)J 59 656 :M .221 .022(causes the exclusion of various groups from the political process. In other words, foreign)J endp %%Page: 35 35 %%BeginPageSetup initializepage (peter; page: 35 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (35)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .143 .014(investment inhibits democracy. Their empirical case for this claim rests on fitting a linear)J 59 70 :M (regression.)S 200 102 :M (PO)S 251 102 :M (FI)S 296 102 :M (EN)S 345 102 :M (CV)S 197 119 :M (1.000)S 180 105 1 1 rF 180 105 1 1 rF 181 105 53 1 rF 234 105 1 1 rF 235 105 44 1 rF 279 105 1 1 rF 280 105 48 1 rF 328 105 1 1 rF 329 105 49 1 rF 378 105 1 1 rF 378 105 1 1 rF 180 106 1 16 rF 378 106 1 16 rF 193 135 :M (-0.175)S 247 135 :M (1.000)S 180 122 1 16 rF 378 122 1 16 rF 193 151 :M (-0.480)S 247 151 :M (0.330)S 296 151 :M (1.000)S 180 138 1 16 rF 378 138 1 16 rF 197 167 :M (0.868)S 243 167 :M (-0.391)S 292 167 :M (-0.430)S 334 167 :M (1.000)S 180 154 1 16 rF 180 170 1 1 rF 180 170 1 1 rF 181 170 53 1 rF 234 170 1 1 rF 235 170 44 1 rF 279 170 1 1 rF 280 170 48 1 rF 328 170 1 1 rF 329 170 49 1 rF 378 154 1 16 rF 378 170 1 1 rF 378 170 1 1 rF 166 196 :M f0_12 sf (Table 1. Political Repression Data \(N = 72\))S 81 224 :M f3_12 sf .775 .078(They develop measures of political exclusion \(PO\), foreign investment penetration)J 59 240 :M .097 .01(\(FI\), energy development \(EN\), civil liberties \(CV\) \(measured on an ordered scale from 1)J 59 256 :M .087 .009(to 7, with lower values indicating )J 224 256 :M f4_12 sf .021(greater)A f3_12 sf .073 .007( civil liberties.\) We show the correlations given)J 59 272 :M (by Timberlake and Williams for these variables on 72 "non-core" countries in Table 1.)S 81 288 :M 1.092 .109(An apparent embarrassment to their claim is that political exclusion is negatively)J 59 304 :M .893 .089(correlated with foreign investment; further, foreign investment is negatively correlated)J 59 320 :M 1.342 .134(with the civil liberties scale \(and hence because of their reverse ordering of the civil)J 59 336 :M 1.835 .184(liberties scale,)J 133 336 :M f4_12 sf 2.117 .212( positively )J f3_12 sf 2.697 .27(correlated with civil liberties\). To defeat this objection,)J 59 352 :M .823 .082(Timberlake and Williams regress PO on the other variables on the assumption that the)J 59 368 :M .401 .04(coefficient relating FI to PO is a superior measure of FI's causal influence on PO than is)J 59 384 :M (their simple correlation. A regression on the correlations above yields:)S 81 416 :M f10_12 sf -.195(PO = .227*)A f11_12 sf -.195(FI)A f10_12 sf -.195( - .176*)A f11_12 sf -.195(EN)A f10_12 sf -.195( + .880*)A f11_12 sf -.195(CV)A f10_12 sf -.195( + )A f1_12 sf (e)S 81 432 :M f10_12 sf -.227( )A 137 432 :M -.205(\(.058\) \(.059\) \(.060\))A 81 448 :M -.227( )A 137 448 :M -.205(3.941 -2.985 14.604)A 81 480 :M f3_12 sf .105 .01(You can see that the crucial coefficient is positive and highly significant. Timberlake)J 59 496 :M .784 .078(and Williams took this as evidence to support the claim that foreign investment causes)J 59 512 :M (more political exclusion. They do not explicitly consider any alternative models.)S 81 528 :M 1.004 .1(But a regression model is only one among many that might describe the relations)J 59 544 :M .355 .036(among these four variables. To search for alternatives, we again use TETRAD II's Build)J 59 560 :M .699 .07(module, again without considering whether the linearity and normality assumptions are)J 59 576 :M (warranted.)S 110 571 :M f3_7 sf (1)S 113 571 :M (6)S 116 576 :M f3_12 sf 1.485 .149( Without assuming that all common causes are included in the variables)J 59 592 :M 1.189 .119(measured, and using a significance level \()J f1_12 sf (a)S 280 592 :M f3_12 sf 1.277 .128(\) of .05 for its statistical hypothesis tests,)J 59 608 :M (Build's output is the PAG in Figure 23.)S 59 650 :M ( )S 59 647.48 -.48 .48 203.48 647 .48 59 647 @a 81 659 :M f3_6 sf (1)S 84 659 :M (6)S 87 663 :M f3_10 sf .43 .043(Because our aim is to illustrate the use of TETRAD II)J f0_10 sf .076 .008( )J 313 663 :M f3_10 sf .372 .037(in finding alterantives to a given model, the)J 59 675 :M .306 .031(correctness of the distribution and linearity assumptions made by Timberlake and Williams is not at issue.)J 59 687 :M (We note, however, that we were unable to reproduce their correlation matrix from the sources they cite.)S endp %%Page: 36 36 %%BeginPageSetup initializepage (peter; page: 36 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (36)S gR gS 165 46 16 14 rC 165 55 :M f3_12 sf (F)S 172 55 :M (I)S gR gS 237 47 16 13 rC 237 56 :M f3_12 sf (E)S 244 56 :M (N)S gR gS 305 47 20 20 rC 305 56 :M f3_12 sf (P)S 312 56 :M (O)S gR gS 368 47 22 20 rC 368 56 :M f3_12 sf ( )S 371 56 :M (C)S 379 56 :M (V)S gR gS 158 41 234 52 rC np 362 51 :M 353 55 :L 353 55 :L 353 54 :L 353 54 :L 353 54 :L 353 54 :L 353 54 :L 353 54 :L 353 53 :L 353 53 :L 353 53 :L 353 53 :L 353 53 :L 353 53 :L 353 52 :L 352 52 :L 352 52 :L 352 52 :L 352 52 :L 352 52 :L 352 51 :L 352 51 :L 352 51 :L 352 51 :L 352 51 :L 352 51 :L 352 50 :L 352 50 :L 352 50 :L 352 50 :L 352 50 :L 352 50 :L 352 49 :L 352 49 :L 353 49 :L 353 49 :L 353 49 :L 353 49 :L 353 48 :L 353 48 :L 353 48 :L 353 48 :L 353 48 :L 353 48 :L 353 47 :L 353 47 :L 353 47 :L 353 47 :L 362 51 :L 362 51 :L eofill 333 52 -1 1 355 51 1 333 51 @a 158.5 41.5 24 21 rS 229.5 42.5 24 21 rS 299.5 43.5 25 20 rS 365.5 42.5 26 21 rS np 377 63 :M 381 72 :L 381 73 :L 381 73 :L 381 73 :L 381 73 :L 380 73 :L 380 73 :L 380 73 :L 380 73 :L 380 73 :L 380 73 :L 379 73 :L 379 73 :L 379 73 :L 379 73 :L 379 73 :L 379 73 :L 378 73 :L 378 73 :L 378 73 :L 378 73 :L 378 73 :L 378 73 :L 377 73 :L 377 73 :L 377 73 :L 377 73 :L 377 73 :L 377 73 :L 376 73 :L 376 73 :L 376 73 :L 376 73 :L 376 73 :L 376 73 :L 375 73 :L 375 73 :L 375 73 :L 375 73 :L 375 73 :L 375 73 :L 374 73 :L 374 73 :L 374 73 :L 374 73 :L 374 73 :L 374 73 :L 373 73 :L 377 63 :L 377 63 :L eofill -1 -1 378 79 1 1 377 72 @b 0 90 118 32 318.5 76.5 @n 90 180 314 46 327.5 69.5 @n 331 52 2.5 @e 170 66 2.5 @e 261 53 -1 1 289 52 1 261 52 @a 259 53 2.5 @e 292 53 2.5 @e gR gS 0 0 552 730 rC 189 118 :M f0_12 sf (Figure )S 226 118 :M (23. Build output at )S f2_12 sf (a)S 333 118 :M f0_12 sf ( = .05)S 59 162 :M f3_12 sf .662 .066(Since this structure entails that foreign investment \(FI\) and political exclusion \(PO\) are)J 59 178 :M (uncorrelated, we increase )S f1_12 sf (a )S 195 178 :M f3_12 sf (to .15, at which point Build produces the PAG in )S 433 178 :M (Figure 24.)S 165 202 16 14 rC 165 211 :M (F)S 172 211 :M (I)S gR gS 237 203 16 13 rC 237 212 :M f3_12 sf (E)S 244 212 :M (N)S gR gS 305 203 20 20 rC 305 212 :M f3_12 sf (P)S 312 212 :M (O)S gR gS 368 203 22 20 rC 368 212 :M f3_12 sf ( )S 371 212 :M (C)S 379 212 :M (V)S gR gS 158 197 234 54 rC np 362 207 :M 353 211 :L 353 211 :L 353 210 :L 353 210 :L 353 210 :L 353 210 :L 353 210 :L 353 210 :L 353 209 :L 353 209 :L 353 209 :L 353 209 :L 353 209 :L 353 209 :L 353 208 :L 352 208 :L 352 208 :L 352 208 :L 352 208 :L 352 208 :L 352 207 :L 352 207 :L 352 207 :L 352 207 :L 352 207 :L 352 207 :L 352 206 :L 352 206 :L 352 206 :L 352 206 :L 352 206 :L 352 206 :L 352 205 :L 352 205 :L 353 205 :L 353 205 :L 353 205 :L 353 205 :L 353 204 :L 353 204 :L 353 204 :L 353 204 :L 353 204 :L 353 204 :L 353 203 :L 353 203 :L 353 203 :L 353 203 :L 362 207 :L 362 207 :L eofill 333 208 -1 1 355 207 1 333 207 @a 158.5 197.5 24 21 rS 229.5 198.5 24 21 rS 299.5 199.5 25 20 rS 365.5 198.5 26 21 rS np 377 219 :M 381 228 :L 381 229 :L 381 229 :L 381 229 :L 381 229 :L 380 229 :L 380 229 :L 380 229 :L 380 229 :L 380 229 :L 380 229 :L 379 229 :L 379 229 :L 379 229 :L 379 229 :L 379 229 :L 379 229 :L 378 229 :L 378 229 :L 378 229 :L 378 229 :L 378 229 :L 378 229 :L 377 229 :L 377 229 :L 377 229 :L 377 229 :L 377 229 :L 377 229 :L 376 229 :L 376 229 :L 376 229 :L 376 229 :L 376 229 :L 376 229 :L 375 229 :L 375 229 :L 375 229 :L 375 229 :L 375 229 :L 375 229 :L 374 229 :L 374 229 :L 374 229 :L 374 229 :L 374 229 :L 374 229 :L 373 229 :L 377 219 :L 377 219 :L eofill -1 -1 378 235 1 1 377 228 @b 0 90 204 34 275.5 232.5 @n 331 208 2.5 @e 261 209 -1 1 289 208 1 261 208 @a 259 209 2.5 @e 292 209 2.5 @e 5 4 224 210.5 @f np 179 221 :M 177 229 :L 177 229 :L 177 229 :L 177 229 :L 177 229 :L 178 229 :L 178 229 :L 178 229 :L 178 229 :L 178 229 :L 178 229 :L 178 229 :L 178 229 :L 178 229 :L 178 229 :L 178 229 :L 179 229 :L 179 229 :L 179 229 :L 179 229 :L 179 229 :L 179 229 :L 179 229 :L 179 229 :L 179 229 :L 179 229 :L 180 229 :L 180 229 :L 180 229 :L 180 229 :L 180 229 :L 180 229 :L 180 229 :L 180 229 :L 180 229 :L 181 229 :L 181 229 :L 181 229 :L 181 229 :L 181 229 :L 181 229 :L 181 229 :L 181 229 :L 181 229 :L 181 229 :L 182 229 :L 182 229 :L 182 229 :L 179 221 :L 179 221 :L eofill 178 231 -1 1 181 234 1 178 230 @a 90 180 198 30 281.5 234.5 @n 190 211 -1 1 220 210 1 190 210 @a np 191 212 :M 191 208 :L 187 210 :L 191 212 :L eofill -1 -1 192 213 1 1 191 208 @b -1 -1 188 211 1 1 191 208 @b 187 211 -1 1 192 212 1 187 210 @a gR gS 0 0 552 730 rC 189 276 :M f0_12 sf (Figure )S 226 276 :M (24. Build output at )S f2_12 sf (a)S 333 276 :M f0_12 sf ( = .15)S 59 320 :M f3_12 sf .067 .007(Because all of the connections in the PAG involving FI have arrowheads directed into FI,)J 59 336 :M .353 .035(these data indicate that foreign investment is not a cause of any of the other variables. A)J 59 352 :M .566 .057(large number of causal models are members of the equivalence class represented by the)J 59 368 :M .809 .081(output in Figure 24)J 155 368 :M .89 .089(. The model in Figure 25 is one of the simplest in this class, and is)J 59 384 :M (plausible besides.)S 165 408 16 14 rC 165 417 :M (F)S 172 417 :M (I)S gR gS 237 409 16 13 rC 237 418 :M f3_12 sf (E)S 244 418 :M (N)S gR gS 305 409 20 20 rC 305 418 :M f3_12 sf (P)S 312 418 :M (O)S gR gS 368 409 22 20 rC 368 418 :M f3_12 sf ( )S 371 418 :M (C)S 379 418 :M (V)S gR gS 158 403 234 54 rC 158.5 403.5 24 21 rS 229.5 404.5 24 21 rS 299.5 405.5 25 20 rS 365.5 404.5 26 21 rS 0 90 204 34 275.5 438.5 @n np 179 427 :M 177 435 :L 177 435 :L 177 435 :L 177 435 :L 177 435 :L 178 435 :L 178 435 :L 178 435 :L 178 435 :L 178 435 :L 178 435 :L 178 435 :L 178 435 :L 178 435 :L 178 435 :L 178 435 :L 179 435 :L 179 435 :L 179 435 :L 179 435 :L 179 435 :L 179 435 :L 179 435 :L 179 435 :L 179 435 :L 179 435 :L 180 435 :L 180 435 :L 180 435 :L 180 435 :L 180 435 :L 180 435 :L 180 435 :L 180 435 :L 180 435 :L 181 435 :L 181 435 :L 181 435 :L 181 435 :L 181 435 :L 181 435 :L 181 435 :L 181 435 :L 181 435 :L 181 435 :L 182 435 :L 182 435 :L 182 435 :L 179 427 :L 179 427 :L eofill 178 437 -1 1 181 440 1 178 436 @a 90 180 198 30 281.5 440.5 @n 190 417 -1 1 220 416 1 190 416 @a np 191 418 :M 191 414 :L 187 416 :L 191 418 :L eofill -1 -1 192 419 1 1 191 414 @b -1 -1 188 417 1 1 191 414 @b 187 417 -1 1 192 418 1 187 416 @a 261 416 -1 1 291 415 1 261 415 @a np 290 413 :M 290 417 :L 293 415 :L 290 413 :L eofill -1 -1 291 418 1 1 290 413 @b -1 -1 291 418 1 1 293 415 @b 290 414 -1 1 294 415 1 290 413 @a 326 416 -1 1 360 415 1 326 415 @a np 359 413 :M 359 417 :L 362 415 :L 359 413 :L eofill -1 -1 360 418 1 1 359 413 @b -1 -1 360 418 1 1 362 415 @b 359 414 -1 1 363 415 1 359 413 @a -1 -1 377 441 1 1 378 432 @b np 375 432 :M 380 433 :L 378 429 :L 375 432 :L eofill 375 433 -1 1 381 433 1 375 432 @a 378 430 -1 1 381 433 1 378 429 @a -1 -1 376 433 1 1 378 429 @b gR gS 0 0 552 730 rC 116 482 :M f0_12 sf (Figure )S 153 482 :M (25. An Alternative to Timberlake and William\325s Model)S 81 510 :M f3_12 sf .838 .084(This model asserts that EN \(a measure of economic development\) causes both the)J 59 526 :M .167 .017(level of foreign investment and the level of political exclusion. Political exclusion causes)J 59 542 :M 1.675 .168(the lack of civil liberties, and there is some unmeasured common cause connecting)J 59 558 :M .576 .058(foreign investment and civil liberties \(or in other terms, that their errors are correlated\).)J 59 574 :M .075 .008(Estimating this model with )J 193 574 :M .079 .008(EQS yields a )J f1_12 sf (c)S 266 571 :M f3_10 sf (2)S f3_12 sf 0 3 rm .08 .008( = .136 with 2 degrees of freedom, with p\()J 0 -3 rm f1_12 sf 0 3 rm (c)S 0 -3 rm 482 571 :M f3_10 sf (2)S f3_12 sf 0 3 rm <29>S 0 -3 rm 59 590 :M .921 .092(= .934. We give the coefficients with their standard errors and t-statistics in Figure 26)J 59 606 :M (below.)S endp %%Page: 37 37 %%BeginPageSetup initializepage (peter; page: 37 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (37)S gR gS 160 75 16 16 rC 160 84 :M f3_12 sf (F)S 167 84 :M (I)S gR gS 232 76 20 15 rC 232 85 :M f3_12 sf (E)S 239 85 :M (N)S gR gS 301 76 22 16 rC 301 85 :M f3_12 sf (P)S 308 85 :M (O)S gR gS 373 77 23 16 rC 373 86 :M f3_12 sf (C)S 381 86 :M (V)S gR gS 153 41 243 117 rC np 366 80 :M 357 84 :L 357 83 :L 357 83 :L 356 83 :L 356 83 :L 356 83 :L 356 83 :L 356 82 :L 356 82 :L 356 82 :L 356 82 :L 356 82 :L 356 82 :L 356 81 :L 356 81 :L 356 81 :L 356 81 :L 356 81 :L 356 81 :L 356 80 :L 356 80 :L 356 80 :L 356 80 :L 356 80 :L 356 80 :L 356 79 :L 356 79 :L 356 79 :L 356 79 :L 356 79 :L 356 79 :L 356 78 :L 356 78 :L 356 78 :L 356 78 :L 356 78 :L 356 78 :L 356 77 :L 356 77 :L 356 77 :L 356 77 :L 356 77 :L 356 77 :L 356 76 :L 356 76 :L 356 76 :L 357 76 :L 357 76 :L 366 80 :L 366 80 :L eofill 328 81 -1 1 358 80 1 328 80 @a 154.5 70.5 24 21 rS 225.5 71.5 24 21 rS 294.5 72.5 25 20 rS 369.5 72.5 26 21 rS np 291 81 :M 282 85 :L 282 84 :L 282 84 :L 282 84 :L 282 84 :L 282 84 :L 281 84 :L 281 83 :L 281 83 :L 281 83 :L 281 83 :L 281 83 :L 281 83 :L 281 82 :L 281 82 :L 281 82 :L 281 82 :L 281 82 :L 281 82 :L 281 81 :L 281 81 :L 281 81 :L 281 81 :L 281 81 :L 281 81 :L 281 80 :L 281 80 :L 281 80 :L 281 80 :L 281 80 :L 281 80 :L 281 79 :L 281 79 :L 281 79 :L 281 79 :L 281 79 :L 281 79 :L 281 78 :L 281 78 :L 281 78 :L 281 78 :L 281 78 :L 281 78 :L 282 77 :L 282 77 :L 282 77 :L 282 77 :L 282 77 :L 291 81 :L 291 81 :L eofill 257 82 -1 1 283 81 1 257 81 @a np 181 81 :M 190 77 :L 190 77 :L 190 77 :L 191 77 :L 191 77 :L 191 77 :L 191 78 :L 191 78 :L 191 78 :L 191 78 :L 191 78 :L 191 78 :L 191 79 :L 191 79 :L 191 79 :L 191 79 :L 191 79 :L 191 79 :L 191 80 :L 191 80 :L 191 80 :L 191 80 :L 191 80 :L 191 80 :L 191 81 :L 191 81 :L 191 81 :L 191 81 :L 191 81 :L 191 81 :L 191 82 :L 191 82 :L 191 82 :L 191 82 :L 191 82 :L 191 82 :L 191 83 :L 191 83 :L 191 83 :L 191 83 :L 191 83 :L 191 83 :L 191 84 :L 191 84 :L 191 84 :L 191 84 :L 190 84 :L 190 84 :L 181 81 :L 181 81 :L eofill 189 82 -1 1 219 81 1 189 81 @a 188 107 -1 1 209 111 1 188 106 @a 208 112 -1 1 242 113 1 208 111 @a 241 114 -1 1 282 114 1 241 113 @a -1 -1 282 115 1 1 318 113 @b -1 -1 319 114 1 1 342 110 @b -1 -1 343 111 1 1 354 106 @b 188 107 -1 1 190 107 1 188 106 @a 189 108 -1 1 193 107 1 189 107 @a 192 108 -1 1 195 108 1 192 107 @a 194 109 -1 1 198 108 1 194 108 @a 197 109 -1 1 200 109 1 197 108 @a 199 110 -1 1 202 109 1 199 109 @a 201 110 -1 1 205 110 1 201 109 @a 204 111 -1 1 207 110 1 204 110 @a 206 111 -1 1 209 110 1 206 110 @a 208 111 -1 1 212 111 1 208 110 @a 211 112 -1 1 214 111 1 211 111 @a 213 112 -1 1 216 111 1 213 111 @a 215 112 -1 1 218 111 1 215 111 @a 217 112 -1 1 220 112 1 217 111 @a 219 113 -1 1 222 112 1 219 112 @a 221 113 -1 1 224 112 1 221 112 @a 223 113 -1 1 227 112 1 223 112 @a 226 113 -1 1 229 112 1 226 112 @a 228 113 -1 1 231 112 1 228 112 @a 230 113 -1 1 233 113 1 230 112 @a 232 114 -1 1 235 113 1 232 113 @a 234 114 -1 1 237 113 1 234 113 @a 236 114 -1 1 240 113 1 236 113 @a 239 114 -1 1 242 113 1 239 113 @a 241 114 -1 1 244 113 1 241 113 @a 243 114 -1 1 247 113 1 243 113 @a 246 114 -1 1 249 113 1 246 113 @a 248 114 -1 1 251 113 1 248 113 @a 250 114 -1 1 254 113 1 250 113 @a 253 114 -1 1 256 113 1 253 113 @a 255 114 -1 1 259 114 1 255 113 @a 258 115 -1 1 261 114 1 258 114 @a 259 115 -1 1 264 114 1 259 114 @a 262 115 -1 1 266 114 1 262 114 @a 264 115 -1 1 269 114 1 264 114 @a 267 115 -1 1 271 114 1 267 114 @a 269 115 -1 1 274 114 1 269 114 @a 272 115 -1 1 276 114 1 272 114 @a 274 115 -1 1 278 114 1 274 114 @a 277 115 -1 1 280 114 1 277 114 @a 279 115 -1 1 283 114 1 279 114 @a 282 115 -1 1 285 114 1 282 114 @a 284 115 -1 1 288 114 1 284 114 @a 287 115 -1 1 290 114 1 287 114 @a 289 115 -1 1 292 114 1 289 114 @a 291 115 -1 1 295 114 1 291 114 @a 294 115 -1 1 297 114 1 294 114 @a 296 115 -1 1 299 114 1 296 114 @a 298 115 -1 1 302 114 1 298 114 @a 301 115 -1 1 304 114 1 301 114 @a -1 -1 304 115 1 1 305 113 @b 305 114 -1 1 308 113 1 305 113 @a 307 114 -1 1 310 113 1 307 113 @a 309 114 -1 1 312 113 1 309 113 @a 311 114 -1 1 314 113 1 311 113 @a 313 114 -1 1 316 113 1 313 113 @a 315 114 -1 1 318 113 1 315 113 @a 317 114 -1 1 320 113 1 317 113 @a 319 114 -1 1 322 113 1 319 113 @a -1 -1 322 114 1 1 323 112 @b 323 113 -1 1 325 112 1 323 112 @a 324 113 -1 1 327 112 1 324 112 @a 326 113 -1 1 329 112 1 326 112 @a 328 113 -1 1 330 112 1 328 112 @a 329 113 -1 1 332 112 1 329 112 @a -1 -1 332 113 1 1 332 111 @b 332 112 -1 1 335 111 1 332 111 @a 334 112 -1 1 336 111 1 334 111 @a 335 112 -1 1 338 111 1 335 111 @a -1 -1 338 112 1 1 338 110 @b 338 111 -1 1 341 110 1 338 110 @a 340 111 -1 1 342 110 1 340 110 @a -1 -1 342 111 1 1 343 109 @b 343 110 -1 1 345 109 1 343 109 @a 344 110 -1 1 347 109 1 344 109 @a -1 -1 347 110 1 1 347 108 @b 347 109 -1 1 350 108 1 347 108 @a 349 109 -1 1 351 108 1 349 108 @a -1 -1 351 109 1 1 352 107 @b 352 108 -1 1 354 107 1 352 107 @a -1 -1 354 108 1 1 354 106 @b np 171 99 :M 181 99 :L 181 99 :L 181 99 :L 181 99 :L 181 99 :L 181 99 :L 181 100 :L 181 100 :L 181 100 :L 181 100 :L 181 100 :L 181 100 :L 181 101 :L 181 101 :L 181 101 :L 181 101 :L 181 101 :L 181 102 :L 181 102 :L 181 102 :L 181 102 :L 181 102 :L 181 102 :L 181 103 :L 181 103 :L 181 103 :L 181 103 :L 180 103 :L 180 103 :L 180 103 :L 180 104 :L 180 104 :L 180 104 :L 180 104 :L 180 104 :L 180 104 :L 180 105 :L 180 105 :L 180 105 :L 179 105 :L 179 105 :L 179 105 :L 179 105 :L 179 106 :L 179 106 :L 179 106 :L 179 106 :L 179 106 :L 171 99 :L 171 99 :L eofill 179 104 -1 1 189 106 1 179 103 @a np 375 98 :M 368 104 :L 367 104 :L 367 104 :L 367 104 :L 367 104 :L 367 104 :L 367 104 :L 367 103 :L 367 103 :L 367 103 :L 366 103 :L 366 103 :L 366 103 :L 366 103 :L 366 102 :L 366 102 :L 366 102 :L 366 102 :L 366 102 :L 366 102 :L 366 102 :L 366 101 :L 365 101 :L 365 101 :L 365 101 :L 365 101 :L 365 101 :L 365 100 :L 365 100 :L 365 100 :L 365 100 :L 365 100 :L 365 100 :L 365 99 :L 365 99 :L 365 99 :L 365 99 :L 365 99 :L 365 99 :L 365 98 :L 365 98 :L 365 98 :L 365 98 :L 365 98 :L 365 98 :L 365 97 :L 365 97 :L 365 97 :L 375 98 :L 375 98 :L eofill -1 -1 354 107 1 1 366 101 @b 192 43 28 36 rC 192 65 :M f3_10 sf <28>S 195 65 :M (.)S 198 65 :M (1)S 203 65 :M (0)S 208 65 :M (0)S 213 65 :M <29>S 192 77 :M (3)S 197 77 :M (.)S 200 77 :M (1)S 205 77 :M (3)S 210 77 :M (2)S 192 43 27 12 rC 192 53 :M f0_10 sf (.)S 196 53 :M (3)S 202 53 :M (1)S 208 53 :M (3)S gR gS 260 43 31 12 rC 260 53 :M f0_10 sf (-)S 264 53 :M (.)S 268 53 :M (4)S 274 53 :M (7)S 280 53 :M (9)S gR gS 335 41 27 12 rC 335 51 :M f0_10 sf (.)S 339 51 :M (8)S 345 51 :M (6)S 351 51 :M (1)S gR gS 258 120 31 12 rC 258 130 :M f0_10 sf (-)S 262 130 :M (.)S 266 130 :M (2)S 272 130 :M (3)S 278 130 :M (5)S gR gS 262 54 30 24 rC 262 64 :M f3_10 sf <28>S 265 64 :M (.)S 268 64 :M (1)S 273 64 :M (0)S 278 64 :M (4)S 283 64 :M <29>S 262 76 :M (-)S 265 76 :M (4)S 270 76 :M (.)S 273 76 :M (5)S 278 76 :M (9)S 283 76 :M (8)S gR gS 333 54 32 24 rC 333 64 :M f3_10 sf <28>S 336 64 :M (.)S 339 64 :M (0)S 344 64 :M (5)S 349 64 :M (3)S 354 64 :M <29>S 333 76 :M (1)S 338 76 :M (6)S 343 76 :M (.)S 346 76 :M (3)S 351 76 :M (0)S 356 76 :M (4)S gR gS 260 133 30 24 rC 260 143 :M f3_10 sf <28>S 263 143 :M (.)S 266 143 :M (0)S 271 143 :M (6)S 276 143 :M (2)S 281 143 :M <29>S 260 155 :M (-)S 263 155 :M (3)S 268 155 :M (.)S 271 155 :M (7)S 276 155 :M (6)S 281 155 :M (1)S gR gS 0 0 552 730 rC 173 183 :M f0_12 sf (Figure )S 210 183 :M (26. Estimated Alternative Model)S 59 227 :M f3_12 sf .315 .032(The signs of the coefficients suggest that the relation between FI and PO is negative and)J 59 243 :M 2.206 .221(mediated by a common cause, contrary in two ways to Timberlake and Williams')J 59 259 :M 1.296 .13(hypothesis. We do not mean to suggest that this analysis shows our alternative to be)J 59 275 :M .306 .031(correct. At this small a sample size statistical tests have little power against alternatives,)J 59 291 :M .726 .073(so it is difficult to statistically distinguish between two models even when they are not)J 59 307 :M .191 .019(statistically equivalent. Our point is to show how the Build module can be used to search)J 59 323 :M (for plausible alternatives to a given model.)S 59 351 :M f4_12 sf (6)S 65 351 :M (.)S 68 351 :M (2)S 74 351 :M ( )S 81 351 :M (Specifying Measurement Models of Political Democracy)S 81 373 :M f3_12 sf (Bollen)S 113 373 :M 1.003 .1( \(1980\) studied whether a number of measures of political democracy were)J 59 389 :M .852 .085(unidimensional indicators of a common feature of societies. Bollen used the following)J 59 405 :M (measures:)S 193 431 :M (PF)S 239 431 :M (press freedom)S 193 447 :M (FG )S 239 447 :M (freedom of group opposition)S 193 463 :M (GS)S 239 463 :M (government sanctions)S 193 479 :M (FE )S 239 479 :M (fairness of elections)S 193 495 :M (ES)S 239 495 :M (executive selection)S 193 511 :M (LS )S 239 511 :M (legislature selection)S 59 543 :M 1.958 .196(He considered the unidimensional factor model specified in Figure 27)J 425 543 :M 2.439 .244(, where it is)J 59 559 :M (understood that for each of the measured variables there is an error term.)S endp %%Page: 38 38 %%BeginPageSetup initializepage (peter; page: 38 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (38)S gR 1 G gS 182 41 185 111 rC 338 127 26 22 rF 0 G 337.5 126.5 27 23 rS 1 G 305 127 27 22 rF 0 G 304.5 126.5 28 23 rS 1 G 276 127 27 22 rF 0 G 275.5 126.5 28 23 rS 1 G 248 127 27 22 rF 0 G 247.5 126.5 28 23 rS 1 G 216 127 27 22 rF 0 G 215.5 126.5 28 23 rS 1 G 185 126 27 22 rF 0 G 184.5 125.5 28 23 rS 268 47 10 13 rC 268 56 :M f3_12 sf (T)S gR 0 G gS 182 41 185 111 rC np 207 125 :M 210 115 :L 210 115 :L 210 115 :L 210 115 :L 210 116 :L 210 116 :L 211 116 :L 211 116 :L 211 116 :L 211 116 :L 211 116 :L 211 116 :L 211 116 :L 212 116 :L 212 116 :L 212 116 :L 212 116 :L 212 117 :L 212 117 :L 213 117 :L 213 117 :L 213 117 :L 213 117 :L 213 117 :L 213 117 :L 213 117 :L 213 118 :L 214 118 :L 214 118 :L 214 118 :L 214 118 :L 214 118 :L 214 118 :L 214 118 :L 214 119 :L 215 119 :L 215 119 :L 215 119 :L 215 119 :L 215 119 :L 215 119 :L 215 119 :L 215 120 :L 215 120 :L 215 120 :L 216 120 :L 216 120 :L 216 120 :L 207 125 :L 207 125 :L eofill -1 -1 212 120 1 1 261 60 @b np 236 124 :M 236 114 :L 236 114 :L 236 114 :L 237 114 :L 237 114 :L 237 114 :L 237 114 :L 237 114 :L 237 114 :L 238 114 :L 238 114 :L 238 114 :L 238 114 :L 238 114 :L 238 114 :L 239 114 :L 239 114 :L 239 114 :L 239 115 :L 239 115 :L 239 115 :L 240 115 :L 240 115 :L 240 115 :L 240 115 :L 240 115 :L 240 115 :L 240 115 :L 241 115 :L 241 115 :L 241 115 :L 241 116 :L 241 116 :L 241 116 :L 241 116 :L 242 116 :L 242 116 :L 242 116 :L 242 116 :L 242 116 :L 242 117 :L 242 117 :L 243 117 :L 243 117 :L 243 117 :L 243 117 :L 243 117 :L 243 117 :L 236 124 :L 236 124 :L eofill -1 -1 240 118 1 1 265 63 @b np 261 125 :M 258 115 :L 258 115 :L 258 115 :L 259 115 :L 259 115 :L 259 115 :L 259 115 :L 259 115 :L 259 115 :L 260 115 :L 260 115 :L 260 115 :L 260 115 :L 260 115 :L 260 115 :L 261 115 :L 261 115 :L 261 115 :L 261 115 :L 261 115 :L 261 115 :L 262 115 :L 262 115 :L 262 115 :L 262 115 :L 262 115 :L 262 115 :L 263 115 :L 263 115 :L 263 115 :L 263 115 :L 263 115 :L 263 115 :L 264 115 :L 264 115 :L 264 116 :L 264 116 :L 264 116 :L 264 116 :L 265 116 :L 265 116 :L 265 116 :L 265 116 :L 265 116 :L 265 116 :L 266 116 :L 266 116 :L 266 116 :L 261 125 :L 261 125 :L eofill -1 -1 262 118 1 1 269 66 @b np 296 125 :M 290 118 :L 290 118 :L 290 117 :L 290 117 :L 290 117 :L 290 117 :L 290 117 :L 291 117 :L 291 117 :L 291 117 :L 291 117 :L 291 116 :L 291 116 :L 291 116 :L 292 116 :L 292 116 :L 292 116 :L 292 116 :L 292 116 :L 292 116 :L 293 116 :L 293 116 :L 293 116 :L 293 116 :L 293 115 :L 293 115 :L 293 115 :L 294 115 :L 294 115 :L 294 115 :L 294 115 :L 294 115 :L 295 115 :L 295 115 :L 295 115 :L 295 115 :L 295 115 :L 295 115 :L 296 115 :L 296 115 :L 296 115 :L 296 115 :L 296 115 :L 296 115 :L 297 115 :L 297 115 :L 297 115 :L 297 115 :L 296 125 :L 296 125 :L eofill 276 67 -1 1 294 117 1 276 66 @a np 320 128 :M 312 122 :L 312 122 :L 312 122 :L 312 122 :L 312 122 :L 313 122 :L 313 122 :L 313 121 :L 313 121 :L 313 121 :L 313 121 :L 313 121 :L 313 121 :L 314 121 :L 314 121 :L 314 120 :L 314 120 :L 314 120 :L 314 120 :L 314 120 :L 314 120 :L 315 120 :L 315 120 :L 315 120 :L 315 119 :L 315 119 :L 315 119 :L 315 119 :L 316 119 :L 316 119 :L 316 119 :L 316 119 :L 316 119 :L 316 119 :L 317 119 :L 317 119 :L 317 119 :L 317 118 :L 317 118 :L 317 118 :L 318 118 :L 318 118 :L 318 118 :L 318 118 :L 318 118 :L 318 118 :L 319 118 :L 319 118 :L 320 128 :L 320 128 :L eofill 280 66 -1 1 317 121 1 280 65 @a np 341 125 :M 332 121 :L 332 121 :L 332 121 :L 332 120 :L 332 120 :L 332 120 :L 333 120 :L 333 120 :L 333 120 :L 333 119 :L 333 119 :L 333 119 :L 333 119 :L 333 119 :L 333 119 :L 333 119 :L 334 119 :L 334 118 :L 334 118 :L 334 118 :L 334 118 :L 334 118 :L 334 118 :L 334 118 :L 335 118 :L 335 117 :L 335 117 :L 335 117 :L 335 117 :L 335 117 :L 335 117 :L 335 117 :L 336 117 :L 336 117 :L 336 116 :L 336 116 :L 336 116 :L 336 116 :L 337 116 :L 337 116 :L 337 116 :L 337 116 :L 337 116 :L 337 116 :L 337 116 :L 338 116 :L 338 116 :L 338 115 :L 341 125 :L 341 125 :L eofill 283 63 -1 1 337 119 1 283 62 @a 194 131 20 13 rC 194 140 :M f3_12 sf (P)S 201 140 :M (F)S gR gS 224 131 19 13 rC 224 140 :M f3_12 sf (F)S 231 140 :M (G)S gR gS 257 131 21 13 rC 257 140 :M f3_12 sf (G)S 266 140 :M (S)S gR gS 285 131 25 13 rC 285 140 :M f3_12 sf (F)S 292 140 :M (E)S gR gS 311 131 22 13 rC 311 140 :M f3_12 sf (E)S 318 140 :M (S)S gR gS 344 131 19 13 rC 344 140 :M f3_12 sf (L)S 351 140 :M (S)S gR .75 lw gS 182 41 185 111 rC 16 19 272.5 52 @f gR gS 0 0 552 730 rC 117 177 :M f0_12 sf (Figure )S 154 177 :M (27. Initial Measurement Model of Political Democracy)S 81 205 :M f3_12 sf .358 .036(Bollen estimated this model with )J 246 205 :M .374 .037(LISREL and found that the data reject it.)J 446 200 :M f3_7 sf (1)S 449 200 :M (7)S 452 205 :M f3_12 sf .346 .035( Instead)J 59 221 :M .554 .055(of attempting to locate and discard the impure indicators, Bollen elaborated his original)J 59 237 :M .099 .01(model by correlating error terms \()J 223 237 :M .105 .01(Figure 28\). When estimated with EQS, this model has a)J 59 253 :M f1_12 sf (c)S 66 250 :M f3_10 sf .09(2)A f3_12 sf 0 3 rm .284 .028( of 6.009 based on 6 degrees of freedom, with p\()J 0 -3 rm f1_12 sf 0 3 rm (c)S 0 -3 rm 315 250 :M f3_10 sf .076(2)A f3_12 sf 0 3 rm .279 .028(\) = 0.42218. The Search module of)J 0 -3 rm 59 269 :M 1.36 .136(TETRAD II, which uses vanishing tetrad differences to search for elaborations of an)J 59 285 :M (initial model, arrives at a set of factor models which contains Bollen\325s model and others.)S 1 G 184 304 182 140 rC 338 374 26 21 rF 0 G 1 lw 337.5 373.5 27 22 rS 1 G 305 374 27 21 rF 0 G 304.5 373.5 28 22 rS 1 G 276 374 27 21 rF 0 G 275.5 373.5 28 22 rS 1 G 248 374 27 21 rF 0 G 247.5 373.5 28 22 rS 1 G 216 374 27 21 rF 0 G 215.5 373.5 28 22 rS 1 G 185 373 27 21 rF 0 G 184.5 372.5 28 22 rS 269 307 17 19 rC 269 319 :M f8_12 sf (T)S gR gS 184 304 182 140 rC np 207 368 :M 211 359 :L 211 359 :L 212 359 :L 212 359 :L 212 360 :L 212 360 :L 212 360 :L 212 360 :L 213 360 :L 213 360 :L 213 360 :L 213 360 :L 213 360 :L 213 360 :L 213 361 :L 213 361 :L 214 361 :L 214 361 :L 214 361 :L 214 361 :L 214 361 :L 214 361 :L 214 362 :L 214 362 :L 215 362 :L 215 362 :L 215 362 :L 215 362 :L 215 362 :L 215 362 :L 215 363 :L 215 363 :L 215 363 :L 215 363 :L 216 363 :L 216 363 :L 216 364 :L 216 364 :L 216 364 :L 216 364 :L 216 364 :L 216 364 :L 216 364 :L 216 365 :L 216 365 :L 216 365 :L 216 365 :L 216 365 :L 207 368 :L 207 368 :L 1 lw eofill -1 -1 213 365 1 1 264 322 @b np 233 370 :M 235 360 :L 235 360 :L 235 360 :L 235 360 :L 236 361 :L 236 361 :L 236 361 :L 236 361 :L 236 361 :L 236 361 :L 237 361 :L 237 361 :L 237 361 :L 237 361 :L 237 361 :L 237 361 :L 238 361 :L 238 361 :L 238 362 :L 238 362 :L 238 362 :L 238 362 :L 238 362 :L 239 362 :L 239 362 :L 239 362 :L 239 362 :L 239 362 :L 239 363 :L 239 363 :L 239 363 :L 240 363 :L 240 363 :L 240 363 :L 240 363 :L 240 363 :L 240 364 :L 240 364 :L 240 364 :L 241 364 :L 241 364 :L 241 364 :L 241 364 :L 241 364 :L 241 365 :L 241 365 :L 241 365 :L 241 365 :L 233 370 :L 233 370 :L eofill -1 -1 238 366 1 1 267 324 @b np 264 371 :M 261 361 :L 261 361 :L 261 361 :L 262 361 :L 262 361 :L 262 361 :L 262 361 :L 262 361 :L 262 361 :L 263 361 :L 263 361 :L 263 361 :L 263 361 :L 263 361 :L 263 361 :L 264 361 :L 264 361 :L 264 361 :L 264 361 :L 264 361 :L 264 361 :L 265 361 :L 265 361 :L 265 361 :L 265 361 :L 265 361 :L 265 361 :L 266 361 :L 266 361 :L 266 361 :L 266 361 :L 266 361 :L 266 362 :L 267 362 :L 267 362 :L 267 362 :L 267 362 :L 267 362 :L 267 362 :L 268 362 :L 268 362 :L 268 362 :L 268 362 :L 268 362 :L 268 362 :L 269 362 :L 269 363 :L 269 363 :L 264 371 :L 264 371 :L eofill -1 -1 265 365 1 1 270 326 @b np 291 372 :M 285 365 :L 285 365 :L 285 365 :L 285 364 :L 285 364 :L 285 364 :L 285 364 :L 286 364 :L 286 364 :L 286 364 :L 286 364 :L 286 364 :L 286 364 :L 286 363 :L 287 363 :L 287 363 :L 287 363 :L 287 363 :L 287 363 :L 287 363 :L 288 363 :L 288 363 :L 288 363 :L 288 363 :L 288 363 :L 288 363 :L 289 363 :L 289 362 :L 289 362 :L 289 362 :L 289 362 :L 289 362 :L 290 362 :L 290 362 :L 290 362 :L 290 362 :L 290 362 :L 290 362 :L 291 362 :L 291 362 :L 291 362 :L 291 362 :L 291 362 :L 291 362 :L 292 362 :L 292 362 :L 292 362 :L 292 362 :L 291 372 :L 291 372 :L eofill 275 326 -1 1 289 365 1 275 325 @a np 311 369 :M 303 364 :L 303 364 :L 303 363 :L 303 363 :L 303 363 :L 303 363 :L 304 363 :L 304 363 :L 304 363 :L 304 363 :L 304 362 :L 304 362 :L 304 362 :L 304 362 :L 304 362 :L 305 362 :L 305 362 :L 305 362 :L 305 361 :L 305 361 :L 305 361 :L 305 361 :L 306 361 :L 306 361 :L 306 361 :L 306 361 :L 306 361 :L 306 361 :L 306 360 :L 307 360 :L 307 360 :L 307 360 :L 307 360 :L 307 360 :L 307 360 :L 307 360 :L 308 360 :L 308 360 :L 308 360 :L 308 360 :L 308 360 :L 308 360 :L 309 359 :L 309 359 :L 309 359 :L 309 359 :L 309 359 :L 309 359 :L 311 369 :L 311 369 :L eofill 281 326 -1 1 308 363 1 281 325 @a np 342 371 :M 333 368 :L 333 368 :L 333 368 :L 333 368 :L 333 367 :L 333 367 :L 333 367 :L 333 367 :L 333 367 :L 333 367 :L 333 367 :L 333 366 :L 333 366 :L 334 366 :L 334 366 :L 334 366 :L 334 366 :L 334 365 :L 334 365 :L 334 365 :L 334 365 :L 334 365 :L 334 365 :L 335 365 :L 335 365 :L 335 364 :L 335 364 :L 335 364 :L 335 364 :L 335 364 :L 335 364 :L 335 364 :L 336 364 :L 336 363 :L 336 363 :L 336 363 :L 336 363 :L 336 363 :L 336 363 :L 337 363 :L 337 363 :L 337 363 :L 337 363 :L 337 362 :L 337 362 :L 337 362 :L 338 362 :L 338 362 :L 342 371 :L 342 371 :L eofill 285 323 -1 1 337 367 1 285 322 @a 190 377 18 13 rC 190 386 :M f3_12 sf (P)S 197 386 :M (F)S gR gS 224 377 20 13 rC 224 386 :M f3_12 sf (F)S 231 386 :M (G)S gR gS 257 377 19 13 rC 257 386 :M f3_12 sf (G)S 266 386 :M (S)S gR gS 285 377 18 13 rC 285 386 :M f3_12 sf (F)S 292 386 :M (E)S gR gS 311 377 20 13 rC 311 386 :M f3_12 sf (E)S 318 386 :M (S)S gR gS 344 377 19 13 rC 344 386 :M f3_12 sf (L)S 351 386 :M (S)S gR gS 195 409 165 18 rC 195 422 :M f1_12 sf ( )S 198 422 :M (e)S 203 422 :M (1)S 209 422 :M ( )S 212 422 :M ( )S 215 422 :M ( )S 218 422 :M ( )S 221 422 :M ( )S 224 422 :M ( )S 227 422 :M (e)S 232 422 :M (2)S 238 422 :M ( )S 241 422 :M ( )S 244 422 :M ( )S 247 422 :M ( )S 250 422 :M ( )S 253 422 :M ( )S 256 422 :M (e)S 261 422 :M (3)S 267 422 :M ( )S 270 422 :M ( )S 273 422 :M ( )S 276 422 :M ( )S 279 422 :M ( )S 282 422 :M ( )S 285 422 :M (e)S 290 422 :M (4)S 296 422 :M ( )S 299 422 :M ( )S 302 422 :M ( )S 305 422 :M ( )S 308 422 :M ( )S 311 422 :M ( )S 314 422 :M ( )S 317 422 :M (e)S 322 422 :M (5)S 328 422 :M ( )S 331 422 :M ( )S 334 422 :M ( )S 337 422 :M ( )S 340 422 :M ( )S 343 422 :M (e)S 348 422 :M (6)S gR gS 184 304 182 140 rC np 203 395 :M 207 404 :L 207 404 :L 207 404 :L 206 404 :L 206 404 :L 206 404 :L 206 404 :L 206 404 :L 206 404 :L 205 404 :L 205 405 :L 205 405 :L 205 405 :L 205 405 :L 205 405 :L 204 405 :L 204 405 :L 204 405 :L 204 405 :L 204 405 :L 204 405 :L 203 405 :L 203 405 :L 203 405 :L 203 405 :L 203 405 :L 203 405 :L 202 405 :L 202 405 :L 202 405 :L 202 405 :L 202 405 :L 202 405 :L 201 405 :L 201 405 :L 201 405 :L 201 405 :L 201 405 :L 200 405 :L 200 404 :L 200 404 :L 200 404 :L 200 404 :L 200 404 :L 199 404 :L 199 404 :L 199 404 :L 199 404 :L 203 395 :L 203 395 :L 1 lw eofill -1 -1 203 411 1 1 202 403 @b np 234 395 :M 238 404 :L 238 404 :L 237 404 :L 237 404 :L 237 404 :L 237 404 :L 237 404 :L 237 404 :L 236 404 :L 236 404 :L 236 405 :L 236 405 :L 236 405 :L 236 405 :L 235 405 :L 235 405 :L 235 405 :L 235 405 :L 235 405 :L 235 405 :L 234 405 :L 234 405 :L 234 405 :L 234 405 :L 234 405 :L 234 405 :L 233 405 :L 233 405 :L 233 405 :L 233 405 :L 233 405 :L 233 405 :L 232 405 :L 232 405 :L 232 405 :L 232 405 :L 232 405 :L 232 405 :L 231 405 :L 231 404 :L 231 404 :L 231 404 :L 231 404 :L 231 404 :L 230 404 :L 230 404 :L 230 404 :L 230 404 :L 234 395 :L 234 395 :L eofill -1 -1 234 411 1 1 233 403 @b np 265 395 :M 269 404 :L 268 404 :L 268 404 :L 268 404 :L 268 404 :L 268 404 :L 268 404 :L 267 404 :L 267 404 :L 267 404 :L 267 405 :L 267 405 :L 267 405 :L 266 405 :L 266 405 :L 266 405 :L 266 405 :L 266 405 :L 266 405 :L 265 405 :L 265 405 :L 265 405 :L 265 405 :L 265 405 :L 265 405 :L 264 405 :L 264 405 :L 264 405 :L 264 405 :L 264 405 :L 264 405 :L 263 405 :L 263 405 :L 263 405 :L 263 405 :L 263 405 :L 262 405 :L 262 405 :L 262 405 :L 262 404 :L 262 404 :L 262 404 :L 261 404 :L 261 404 :L 261 404 :L 261 404 :L 261 404 :L 261 404 :L 265 395 :L 265 395 :L eofill -1 -1 265 411 1 1 264 403 @b np 292 395 :M 296 404 :L 296 404 :L 296 404 :L 296 404 :L 296 404 :L 296 404 :L 295 404 :L 295 404 :L 295 404 :L 295 404 :L 295 405 :L 295 405 :L 294 405 :L 294 405 :L 294 405 :L 294 405 :L 294 405 :L 294 405 :L 293 405 :L 293 405 :L 293 405 :L 293 405 :L 293 405 :L 293 405 :L 292 405 :L 292 405 :L 292 405 :L 292 405 :L 292 405 :L 292 405 :L 291 405 :L 291 405 :L 291 405 :L 291 405 :L 291 405 :L 291 405 :L 290 405 :L 290 405 :L 290 405 :L 290 404 :L 290 404 :L 290 404 :L 289 404 :L 289 404 :L 289 404 :L 289 404 :L 289 404 :L 289 404 :L 292 395 :L 292 395 :L eofill -1 -1 293 411 1 1 292 403 @b np 321 395 :M 325 404 :L 325 404 :L 325 404 :L 325 404 :L 325 404 :L 324 404 :L 324 404 :L 324 404 :L 324 404 :L 324 404 :L 324 405 :L 323 405 :L 323 405 :L 323 405 :L 323 405 :L 323 405 :L 323 405 :L 322 405 :L 322 405 :L 322 405 :L 322 405 :L 322 405 :L 322 405 :L 321 405 :L 321 405 :L 321 405 :L 321 405 :L 321 405 :L 321 405 :L 320 405 :L 320 405 :L 320 405 :L 320 405 :L 320 405 :L 320 405 :L 319 405 :L 319 405 :L 319 405 :L 319 405 :L 319 404 :L 319 404 :L 318 404 :L 318 404 :L 318 404 :L 318 404 :L 318 404 :L 318 404 :L 317 404 :L 321 395 :L 321 395 :L eofill -1 -1 322 411 1 1 321 403 @b np 348 394 :M 352 403 :L 352 403 :L 352 403 :L 352 403 :L 352 403 :L 351 403 :L 351 403 :L 351 403 :L 351 403 :L 351 403 :L 351 404 :L 350 404 :L 350 404 :L 350 404 :L 350 404 :L 350 404 :L 349 404 :L 349 404 :L 349 404 :L 349 404 :L 349 404 :L 349 404 :L 348 404 :L 348 404 :L 348 404 :L 348 404 :L 348 404 :L 348 404 :L 347 404 :L 347 404 :L 347 404 :L 347 404 :L 347 404 :L 347 404 :L 346 404 :L 346 404 :L 346 404 :L 346 404 :L 346 404 :L 346 403 :L 345 403 :L 345 403 :L 345 403 :L 345 403 :L 345 403 :L 345 403 :L 344 403 :L 344 403 :L 348 394 :L 348 394 :L eofill -1 -1 349 410 1 1 348 402 @b 90 180 36 18 250.5 424.5 @n 0 90 32 18 248.5 424.5 @n 90 180 26 14 334.5 424.5 @n 0 90 26 14 332.5 424.5 @n 90 180 134 36 297.5 424.5 @n 0 90 108 38 296.5 423.5 @n .75 lw 21 19 273 314 @f gR gS 0 0 552 730 rC 115 469 :M f0_12 sf (Figure )S 152 469 :M (28. Bollen\325s Respecification of the Measurement Model)S 81 497 :M f3_12 sf .252 .025(Although Bollen's final measurement model of democracy fits the data well, it is not)J 59 513 :M 1.75 .175(unidimensional. To find a unidimensional submodel, we can run Purify on Bollen's)J 59 529 :M 2.464 .246(original model \(Figure 27\) and data. Giving the initial model and the measured)J 59 545 :M 1.576 .158(covariances to Purify, FG and LS are identified as impure indicators and discarded,)J 59 561 :M (resulting in the measurement model we picture in )S 300 561 :M (Figure 29.)S 59 671 :M ( )S 59 668.48 -.48 .48 203.48 668 .48 59 668 @a 81 683 :M f3_6 sf (1)S 84 683 :M (7)S 87 687 :M f3_10 sf (EQS yields a )S 142 687 :M f1_10 sf (c)S f3_10 sf 0 -3 rm (2)S 0 3 rm ( = 42.076 based on 9 degrees of freedom, with p\()S 349 687 :M f1_10 sf (c)S f3_10 sf 0 -3 rm (2)S 0 3 rm (\) < 0.001.)S endp %%Page: 39 39 %%BeginPageSetup initializepage (peter; page: 39 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (39)S gR 1 G gS 211 41 128 108 rC 310 125 27 22 rF 0 G 309.5 124.5 28 23 rS 1 G 281 125 27 22 rF 0 G 280.5 124.5 28 23 rS 1 G 254 125 26 22 rF 0 G 253.5 124.5 27 23 rS 1 G 213 125 27 22 rF 0 G 212.5 124.5 28 23 rS 273 45 16 20 rC 273 57 :M f8_12 sf (T)S gR 0 G gS 211 41 128 108 rC np 231 122 :M 232 112 :L 233 112 :L 233 112 :L 233 112 :L 233 112 :L 233 112 :L 233 112 :L 234 112 :L 234 112 :L 234 112 :L 234 112 :L 234 112 :L 234 113 :L 235 113 :L 235 113 :L 235 113 :L 235 113 :L 235 113 :L 235 113 :L 236 113 :L 236 113 :L 236 113 :L 236 113 :L 236 113 :L 236 113 :L 236 114 :L 237 114 :L 237 114 :L 237 114 :L 237 114 :L 237 114 :L 237 114 :L 237 114 :L 238 114 :L 238 115 :L 238 115 :L 238 115 :L 238 115 :L 238 115 :L 238 115 :L 238 115 :L 239 115 :L 239 116 :L 239 116 :L 239 116 :L 239 116 :L 239 116 :L 239 116 :L 231 122 :L 231 122 :L eofill -1 -1 236 116 1 1 269 61 @b np 266 123 :M 263 113 :L 264 113 :L 264 113 :L 264 113 :L 264 113 :L 264 113 :L 264 113 :L 265 113 :L 265 113 :L 265 113 :L 265 113 :L 265 113 :L 265 113 :L 266 113 :L 266 113 :L 266 113 :L 266 113 :L 266 113 :L 266 113 :L 267 113 :L 267 113 :L 267 113 :L 267 113 :L 267 113 :L 268 113 :L 268 113 :L 268 113 :L 268 113 :L 268 113 :L 268 113 :L 269 113 :L 269 113 :L 269 113 :L 269 113 :L 269 113 :L 269 114 :L 270 114 :L 270 114 :L 270 114 :L 270 114 :L 270 114 :L 270 114 :L 270 114 :L 271 114 :L 271 114 :L 271 114 :L 271 114 :L 271 114 :L 266 123 :L 266 123 :L eofill -1 -1 267 116 1 1 275 64 @b np 302 123 :M 295 116 :L 295 116 :L 295 115 :L 296 115 :L 296 115 :L 296 115 :L 296 115 :L 296 115 :L 296 115 :L 296 115 :L 297 115 :L 297 114 :L 297 114 :L 297 114 :L 297 114 :L 297 114 :L 297 114 :L 298 114 :L 298 114 :L 298 114 :L 298 114 :L 298 114 :L 298 114 :L 299 114 :L 299 113 :L 299 113 :L 299 113 :L 299 113 :L 299 113 :L 300 113 :L 300 113 :L 300 113 :L 300 113 :L 300 113 :L 300 113 :L 301 113 :L 301 113 :L 301 113 :L 301 113 :L 301 113 :L 301 113 :L 302 113 :L 302 113 :L 302 113 :L 302 113 :L 302 113 :L 302 113 :L 303 113 :L 302 123 :L 302 123 :L eofill 282 64 -1 1 299 115 1 282 63 @a np 326 126 :M 318 120 :L 318 120 :L 318 120 :L 318 120 :L 318 120 :L 318 120 :L 318 120 :L 318 119 :L 318 119 :L 319 119 :L 319 119 :L 319 119 :L 319 119 :L 319 119 :L 319 119 :L 319 118 :L 319 118 :L 320 118 :L 320 118 :L 320 118 :L 320 118 :L 320 118 :L 320 118 :L 320 118 :L 321 117 :L 321 117 :L 321 117 :L 321 117 :L 321 117 :L 321 117 :L 321 117 :L 322 117 :L 322 117 :L 322 117 :L 322 117 :L 322 117 :L 322 117 :L 323 116 :L 323 116 :L 323 116 :L 323 116 :L 323 116 :L 323 116 :L 324 116 :L 324 116 :L 324 116 :L 324 116 :L 324 116 :L 326 126 :L 326 126 :L eofill 285 62 -1 1 322 119 1 285 61 @a 221 130 19 13 rC 221 139 :M f3_12 sf (P)S 228 139 :M (F)S gR gS 262 129 20 13 rC 262 138 :M f3_12 sf (G)S 271 138 :M (S)S gR gS 291 129 18 13 rC 291 138 :M f3_12 sf (F)S 298 138 :M (E)S gR gS 317 129 21 13 rC 317 138 :M f3_12 sf (E)S 324 138 :M (S)S gR .75 lw gS 211 41 128 108 rC 17 21 277 52 @f gR gS 0 0 552 730 rC 177 174 :M f0_12 sf (Figure )S 214 174 :M (29. Sub-model found by Purify)S 59 202 :M f3_12 sf ( )S 62 202 :M 1.694 .169(Estimating the resulting unidimensional measurement model \()J 381 202 :M 2.083 .208(Figure 29\) with EQS)J 59 218 :M (yields a )S 99 218 :M f1_12 sf (c)S 106 215 :M f3_10 sf (2)S f3_12 sf 0 3 rm ( = 1.687 based on 2 degrees of freedom, with p\()S 0 -3 rm f1_12 sf 0 3 rm (c)S 0 -3 rm 348 215 :M f3_10 sf (2)S f3_12 sf 0 3 rm (\) = 0.43013.)S 0 -3 rm 59 246 :M f4_12 sf (6)S 65 246 :M (.)S 68 246 :M (3)S 74 246 :M ( )S 81 246 :M (A Large Search Space: The Alarm Network)S 81 268 :M f3_12 sf .613 .061(By interviewing several medical experts, Beinlich, et. al., \(1989\) developed a large)J 59 284 :M .927 .093(causal model of the probabilistic relations in emergency medicine \()J 397 284 :M .803 .08(Figure 30\).)J 451 279 :M f3_7 sf (1)S 454 279 :M (8)S 457 284 :M f3_12 sf 1.071 .107( Using)J 59 300 :M .701 .07(the directed graph associated with this model \()J 291 300 :M .676 .068(Figure 30\), called the ALARM network,)J 59 316 :M .098 .01(linear coefficients with values between .1 and .9 were randomly assigned to each directed)J 59 332 :M .344 .034(edge in the graph. Using a standard joint normal distribution \(mean 0, variance 1\) on the)J 59 348 :M .158 .016(exogenous variables, three sets of simulated data were generated, each with a sample size)J 59 364 :M .216 .022(of 2,000. The covariance matrix and sample size were given to the TETRAD II program.)J 59 380 :M .191 .019(No information about the orientation of the variables was given to the program. With 37)J 59 396 :M .79 .079(variables, the space of possible models is astronomical,)J 335 391 :M f3_7 sf (1)S 338 391 :M (9)S 341 396 :M f3_12 sf .893 .089( yet the program required less)J 59 412 :M 1.179 .118(than fifteen seconds to return a pattern on a Decstation 3100. In each trial the output)J 59 428 :M .404 .04(pattern omitted two edges in the ALARM network; in one of the cases it also added one)J 59 444 :M (edge that was not present in the ALARM network.)S 59 614 :M ( )S 59 611.48 -.48 .48 203.48 611 .48 59 611 @a 81 623 :M f3_6 sf (1)S 84 623 :M (8)S 87 627 :M f3_10 sf .227 .023( Beinlich\325s network was over discrete variables, and we have run Build on a discrete version of this)J 59 639 :M (network with results similar to those we report here for a SEM interpretation of the structure.)S 81 647 :M f3_6 sf (1)S 84 647 :M (9)S 87 651 :M f3_10 sf .161 .016( With 37 variables there are 666 pairs of variables. Assuming that each pair has either 1\) no)J 59 663 :M .151 .015(edge between therm, or 2\) an edge from X to Y, or 3\) an edge from Y to X, the number of possible models)J 59 675 :M .479 .048(is 3)J 74 671 :M f3_6 sf .083(666)A f3_10 sf 0 4 rm .393 .039(. The actual number to search is smaller, because some of these models will contain cycles, but the)J 0 -4 rm 59 687 :M (space remaining is still far too big to search by evaluating each member.)S endp %%Page: 40 40 %%BeginPageSetup initializepage (peter; page: 40 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (40)S gR gS 59 41 443 227 rC 89.5 120.5 19 18 rS 93 133 :M f3_12 sf (6)S 132.5 120.5 19 18 rS 136 133 :M (5)S 175.5 120.5 19 18 rS 179 133 :M (4)S 218.5 120.5 19 18 rS 222 133 :M (27)S 261.5 120.5 19 18 rS 265 133 :M (11)S 304.5 120.5 19 18 rS 308 133 :M (32)S 347.5 120.5 19 18 rS 351 133 :M (34)S 390.5 120.5 19 18 rS 394 133 :M (35)S 433.5 120.5 19 18 rS 437 133 :M (36)S 476.5 120.5 19 18 rS 480 133 :M (37)S 176.5 80.5 19 18 rS 180 93 :M (19)S 218.5 80.5 19 18 rS 222 93 :M (20)S 261.5 80.5 19 18 rS 265 93 :M (31)S 371.5 81.5 19 18 rS 375 94 :M (15)S 414.5 81.5 19 18 rS 418 94 :M (23)S 479.5 82.5 19 18 rS 483 95 :M (16)S 220 57 :M (10)S 265 57 :M (21)S 365.5 45.5 19 18 rS 369 58 :M (22)S 477.5 47.5 19 18 rS 481 60 :M (13)S 65.5 162.5 20 18 rS 70 175 :M (17)S 161.5 163.5 19 18 rS 165 176 :M (28)S 204.5 163.5 19 18 rS 208 176 :M (29)S 304.5 162.5 19 18 rS 308 175 :M (12)S 434.5 161.5 19 17 rS 438 174 :M (24)S 64.5 205.5 20 18 rS 69 218 :M (25)S 109.5 205.5 20 18 rS 114 218 :M (18)S 155.5 205.5 20 18 rS 159 218 :M (26)S 194.5 206.5 19 18 rS 198 219 :M (7)S 237.5 206.5 19 18 rS 241 219 :M (8)S 280.5 206.5 19 18 rS 284 219 :M (9)S 62.5 247.5 20 18 rS 67 260 :M (1)S 107.5 247.5 20 18 rS 112 260 :M (2)S 153.5 247.5 20 18 rS 157 260 :M (3)S 261.5 247.5 19 18 rS 265 260 :M (30)S 10 156 204 133 129 @k 109 130 -1 1 125 129 1 109 129 @a 10 -24 24 152 129 @k 159 130 -1 1 176 129 1 159 129 @a 10 156 204 219 130 @k 195 131 -1 1 211 130 1 195 130 @a 10 -24 24 237 129 @k 244 130 -1 1 261 129 1 244 129 @a 10 -24 24 281 129 @k 288 130 -1 1 303 129 1 288 129 @a 283 130 -1 1 285 129 1 283 129 @a 10 -24 24 324 130 @k 331 131 -1 1 347 130 1 331 130 @a 10 -24 24 367 130 @k 374 131 -1 1 391 130 1 374 130 @a 10 -24 24 410 130 @k 417 131 -1 1 434 130 1 417 130 @a 10 -24 24 453 130 @k 460 131 -1 1 477 130 1 460 130 @a 349.5 206.5 19 17 rS 353 219 :M (33)S 392.5 206.5 19 17 rS 396 219 :M (14)S 10 -114 -66 187 120 @k -1 -1 187 113 1 1 186 98 @b 10 -114 -66 228 119 @k -1 -1 228 112 1 1 227 98 @b 10 -114 -66 272 120 @k -1 -1 272 113 1 1 271 98 @b 10 21 69 380 99 @k 385 106 -1 1 400 119 1 385 105 @a 10 116 164 478 54 @k -1 -1 435 91 1 1 471 59 @b 10 -66 -18 403 120 @k -1 -1 409 116 1 1 424 100 @b 10 -96 -48 356 120 @k -1 -1 358 113 1 1 373 63 @b 10 230 278 404 120 @k 384 56 -1 1 402 112 1 384 55 @a 10 227 275 383 81 @k 376 64 -1 1 380 73 1 376 63 @a 10 -114 -66 489 120 @k -1 -1 489 113 1 1 488 101 @b 10 102 150 483 66 @k -1 -1 444 121 1 1 477 72 @b 261.5 44.5 19 18 rS 217.5 44.5 19 18 rS 10 -114 -66 271 81 @k -1 -1 271 74 1 1 270 62 @b 10 -24 24 238 53 @k 245 54 -1 1 262 53 1 245 53 @a 10 66 114 316 139 @k -1 -1 316 163 1 1 315 147 @b 10 66 114 445 139 @k -1 -1 445 162 1 1 444 147 @b 10 -2 46 109 131 @k 115 135 -1 1 204 170 1 115 134 @a 10 19 67 99 139 @k 104 145 -1 1 167 204 1 104 144 @a 10 -24 24 87 214 @k 94 215 -1 1 111 214 1 94 214 @a 10 156 204 156 214 @k 131 215 -1 1 148 214 1 131 214 @a 10 -114 -66 76 205 @k -1 -1 76 198 1 1 75 180 @b 10 176 224 156 209 @k 76 181 -1 1 148 206 1 76 180 @a 10 -114 -66 76 247 @k -1 -1 76 240 1 1 75 224 @b 10 184 232 120 248 @k 75 225 -1 1 113 244 1 75 224 @a 10 185 233 165 247 @k 121 224 -1 1 158 243 1 121 223 @a 10 193 241 203 205 @k 171 182 -1 1 197 200 1 171 181 @a 10 -88 -40 204 205 @k -1 -1 208 199 1 1 214 182 @b 10 195 243 246 206 @k 215 182 -1 1 240 201 1 215 181 @a 10 174 222 291 206 @k 215 182 -1 1 283 204 1 215 181 @a 10 8 56 228 138 @k 234 143 -1 1 350 215 1 234 142 @a 10 156 204 392 215 @k 369 216 -1 1 384 215 1 369 215 @a 10 21 69 248 224 @k 253 231 -1 1 271 247 1 253 230 @a 10 109 157 292 224 @k -1 -1 271 248 1 1 286 230 @b 10 156 204 476 50 @k 385 51 -1 1 468 50 1 385 50 @a 10 -114 -66 401 206 @k -1 -1 401 199 1 1 400 139 @b 10 -46 2 282 89 @k -1 -1 289 87 1 1 364 55 @b 10 -88 -40 215 163 @k -1 -1 219 157 1 1 226 138 @b 10 -114 -66 358 207 @k -1 -1 358 200 1 1 357 139 @b gR gS 0 0 552 730 rC 188 293 :M f0_12 sf (Figure )S 225 293 :M (30. The ALARM Network.)S 234 349 :M f4_14 sf (7)S 241 349 :M (.)S 244 349 :M 4 0 rm ( )S 251 349 :M (Conclusion)S 81 377 :M f3_12 sf 2.65 .265(The work we have described is based on assumptions that are implicit, and)J 59 393 :M .211 .021(sometimes explicit, throughout scientific practice. The Causal Independence assumption,)J 59 409 :M 1.043 .104(for example, posits a relation between the absence of causal connection and statistical)J 59 425 :M 1.013 .101(independence that is fundamental to experimental design; the Faithfulness assumption)J 59 441 :M .647 .065(states a preference for explanation by structure over explanation by coincidence that, in)J 59 457 :M .219 .022(various forms, is used in every science. Consequences of these assumptions were worked)J 59 473 :M .258 .026(out for special cases by many social scientists, for example by Simon \(1954\), by Blalock)J 59 489 :M (\(1962\) and by Costner \(1971\).)S 81 505 :M .335 .034(The methods we have described are incomplete, and there is a great deal of research)J 59 521 :M 2.986 .299(that remains to be done and that should lead to improved modeling. Important)J 59 537 :M 1.007 .101(outstanding problems include improving the reliability of model search through better)J 59 553 :M .605 .061(statistical and algorithmic procedures, deriving computationally tractable algorithms for)J 59 569 :M .486 .049(testing covariance equivalence of linear, latent variable models, finding correct methods)J 59 585 :M 1.561 .156(for clustering measured variables that share a latent common cause, completing and)J 59 601 :M .029 .003(implementing known algorithms for predicting the outcomes of interventions from partial)J 59 617 :M .188 .019(causal and distributional specifications, implementing searches for non-recursive models,)J 59 633 :M (and much more.)S 81 649 :M 4.158 .416(We hope the TETRAD II procedures will become a useful part of the)J 59 665 :M .191 .019(methodological toolkit used by quantitative social scientists, that the statistical and social)J endp %%Page: 41 41 %%BeginPageSetup initializepage (peter; page: 41 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (41)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf 1.127 .113(scientific communities will investigate the questions we have raised, and improve the)J 59 70 :M (techniques we have suggested.)S 242 114 :M f4_12 sf (8)S 248 114 :M (.)S 251 114 :M 4 0 rm ( )S 258 114 :M f4_14 sf ( )S 262 114 :M f4_12 sf (Appendix)S 59 154 :M (8)S 65 154 :M (.)S 68 154 :M (1)S 74 154 :M ( )S 81 154 :M (D-Separation)S 81 176 :M f3_12 sf .12 .012(An )J f0_12 sf .3 .03(undirected path)J 182 176 :M f3_12 sf .302 .03( between X)J 238 179 :M f3_7 sf (1)S 242 176 :M f3_12 sf .35 .035( and X)J 275 179 :M f3_7 sf (n)S 279 176 :M f3_12 sf .303 .03( in a graph G is a sequence of vertices such that for each pair of vertices X)J f3_7 sf 0 3 rm (i)S 0 -3 rm 277 192 :M f3_12 sf .658 .066( and X)J 311 195 :M f3_7 sf .149(i+1)A f3_12 sf 0 -3 rm .572 .057( \(1 )J cF f1_12 sf .057A sf .572 .057( i < n\) that are adjacent in the)J 0 3 rm 59 208 :M .46 .046(sequence, either there is an edge X)J f3_7 sf 0 3 rm (i)S 0 -3 rm 232 208 :M f3_12 sf .737 .074( )J 236 208 :M f1_12 sf S 248 208 :M f3_12 sf .67 .067( X)J 261 211 :M f3_7 sf .125(i+1)A f3_12 sf 0 -3 rm .495 .049( or an edge X)J 0 3 rm 338 211 :M f3_7 sf .186(i+1)A f3_12 sf 0 -3 rm .178 .018( )J 0 3 rm 351 208 :M f1_12 sf S 363 208 :M f3_12 sf .526 .053( X)J f3_7 sf 0 3 rm (i)S 0 -3 rm 377 208 :M f3_12 sf .614 .061( in G. A )J 422 208 :M f0_12 sf .335 .033(directed path)J 59 224 :M f3_12 sf .167 .017(between X)J 112 227 :M f3_7 sf (1)S 116 224 :M f3_12 sf .209 .021( and X)J f3_7 sf 0 3 rm (n)S 0 -3 rm 152 224 :M f3_12 sf .213 .021( in a graph G is a sequence of vertices such that for each)J 59 240 :M (pair of vertices X)S 143 243 :M f3_7 sf (i)S 145 240 :M f3_12 sf ( and X)S 177 243 :M f3_7 sf (i+1)S f3_12 sf 0 -3 rm ( \(1 )S cF f1_12 sf S sf ( i < n\) that are adjacent in the sequence, there is an edge X)S 0 3 rm 489 243 :M f3_7 sf (i)S 59 256 :M f1_12 sf S 71 256 :M f3_12 sf .297 .03( X)J 83 259 :M f3_7 sf .087(i+1)A f3_12 sf 0 -3 rm .233 .023( in G. X is a )J 0 3 rm 155 256 :M f0_12 sf .079(descendant)A f3_12 sf .19 .019( of Y in directed graph G if and only if there is a directed)J 59 272 :M .235 .023(path from Y to X or Y = X. In graph G a vertex X)J f3_7 sf 0 3 rm (i)S 0 -3 rm 305 272 :M f3_12 sf .084 .008( is a )J f0_12 sf .287 .029(collider on undirected path )J 472 272 :M f3_12 sf .244 .024(U if)J 59 288 :M 1.297 .13(and only if U contains a subpath X)J f3_7 sf 0 3 rm .314(i-1)A 0 -3 rm 248 288 :M f3_12 sf .37 .037( )J f1_12 sf S 265 288 :M f3_12 sf 1.833 .183( X)J 279 291 :M f3_7 sf (i)S 281 288 :M f3_12 sf .37 .037( )J f1_12 sf S 298 288 :M f3_12 sf 1.833 .183( X)J 312 291 :M f3_7 sf .214(i+1)A f3_12 sf 0 -3 rm 1.147 .115(. Otherwise if X)J 0 3 rm f3_7 sf (i)S 406 288 :M f3_12 sf 1.613 .161( if on U, X)J 466 291 :M f3_7 sf (i)S 468 288 :M f3_12 sf 1.68 .168( is a)J 59 304 :M f0_12 sf .163 .016(noncollider on )J 137 304 :M f3_12 sf .171 .017(U. Following Pearl \(1988\), in a directed acyclic graph G, for disjoint sets)J 59 320 :M .772 .077(of vertices )J f0_12 sf (X)S 124 320 :M f3_12 sf .49 .049(, )J f0_12 sf (Y)S 140 320 :M f3_12 sf .823 .082(, and )J f0_12 sf .976(W)A f3_12 sf .406 .041(, )J f0_12 sf (X)S 197 320 :M f3_12 sf .838 .084( and )J f0_12 sf (Y)S 232 320 :M f3_12 sf 1.197 .12( are )J 256 320 :M f0_12 sf .684 .068(d-separated )J 322 320 :M f3_12 sf 1.349 .135(given )J f0_12 sf .873(W)A f3_12 sf .891 .089( in G if and only if there)J 59 336 :M .863 .086(exists no undirected path U between a member of )J f0_12 sf (X)S 321 336 :M f3_12 sf 1.039 .104( and a member of )J 415 336 :M f0_12 sf (Y)S 424 336 :M f3_12 sf .935 .093(, such that \(i\))J 59 352 :M .979 .098(every collider on U has a descendent in )J 263 352 :M f0_12 sf .978(W)A f3_12 sf .999 .1( and \(ii\) no other vertex on U is in )J 457 352 :M f0_12 sf .535(W)A f3_12 sf .708 .071(. An)J 59 368 :M (illustration of d-separation is given in the directed acyclic graph shown in )S 416 368 :M (Figure 31.)S 152 387 245 70 rC 156 399 :M f4_12 sf (X)S 210 399 :M (U)S 271 399 :M (V)S 10 156 204 201 396 @k 173 397 -1 1 193 396 1 173 396 @a 10 -24 24 228 396 @k 235 397 -1 1 264 396 1 235 396 @a 10 156 204 323 394 @k 295 395 -1 1 315 394 1 295 394 @a 328 399 :M (W)S 382 399 :M (Y)S 10 -24 24 354 396 @k 361 397 -1 1 372 396 1 361 396 @a 213 445 :M (S)S 221 449 :M f3_12 sf (1)S 330 446 :M f4_12 sf (S)S 341 451 :M f3_12 sf (2)S 10 -114 -66 217 432 @k -1 -1 217 425 1 1 216 404 @b 10 -114 -66 337 434 @k -1 -1 337 427 1 1 336 408 @b gR gS 0 0 552 730 rC 261 476 :M f0_12 sf (Figure )S 298 476 :M (31)S 153 498 :M f3_12 sf ({X} and {Y} are d-separated given the empty set)S 153 514 :M ({X} and {Y} are not d-separated given set {S)S f3_7 sf 0 3 rm (1)S 0 -3 rm 376 514 :M f3_12 sf (, S)S 389 517 :M f3_7 sf (2)S 393 514 :M f3_12 sf (})S 153 530 :M ({X} and {Y} are d-separated given the set {S)S 372 533 :M f3_7 sf (1)S 376 530 :M f3_12 sf (, S)S 389 533 :M f3_7 sf (2)S 393 530 :M f3_12 sf (, V})S 59 558 :M f4_12 sf (8)S 65 558 :M (.)S 68 558 :M (2)S 74 558 :M ( )S 81 558 :M (The Build Algorithm with the Assumption of No Correlated Errors)S 81 580 :M f3_12 sf .708 .071(The Build module uses the PC algorithm \(Spirtes, et al., 1993\) when it is assumed)J 59 596 :M .419 .042(that the RSEM that generated the data has no correlated errors or latent common causes.)J 59 612 :M .074 .007(Let )J f0_12 sf .031(Adjacencies)A f3_12 sf .082 .008(\(C,A\) be the set of vertices adjacent to A in a graph C.)J 402 607 :M f3_7 sf (2)S 405 607 :M (0)S 408 612 :M f3_12 sf .092 .009( In the algorithm,)J 59 628 :M 2.098 .21(the graph C is continually updated, so )J 267 628 :M f0_12 sf .342(Adjacencies)A f3_12 sf 1.331 .133(\(C,A\) changes as the algorithm)J 59 674 :M ( )S 59 671.48 -.48 .48 203.48 671 .48 59 671 @a 81 683 :M f3_6 sf (2)S 84 683 :M (0)S 87 687 :M f3_10 sf (Note that C is not defined to be a directed graph, so that edges can be either directed or undirected.)S endp %%Page: 42 42 %%BeginPageSetup initializepage (peter; page: 42 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (42)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf .551 .055(progresses. )J 117 54 :M f0_12 sf (A)S 126 54 :M f3_12 sf .305 .031( \\ )J f0_12 sf .602(B)A f3_12 sf .958 .096( is the set of members of )J 275 54 :M f0_12 sf (A)S 284 54 :M f3_12 sf .816 .082( that are not elements of )J 410 54 :M f0_12 sf .356(B)A f3_12 sf .724 .072(. We adopt the)J 59 70 :M (convention that if )S 147 70 :M f0_12 sf (S)S 154 70 :M f3_12 sf ( is the empty set, then )S 262 70 :M f1_12 sf (r)S 269 73 :M f3_7 sf (X,Y.)S 283 73 :M f0_7 sf (S)S 287 70 :M f3_12 sf ( is )S f1_12 sf (r)S 308 73 :M f3_7 sf (X,Y)S 320 70 :M f3_12 sf (.)S 323 65 :M f3_7 sf (2)S 326 65 :M (1)S 77 102 :M f0_12 sf (PC Algorithm:)S 95 118 :M f3_12 sf (A\) Form the complete undirected graph C on the vertex set )S f0_12 sf (V)S 390 118 :M f3_12 sf (.)S 95 134 :M (B\) n = 0.)S 117 150 :M f0_12 sf (repeat)S 135 166 :M (repeat)S 149 182 :M f3_12 sf .277 .028(select an ordered pair of variables X and Y that are adjacent in C such)J 149 198 :M .099 .01(that the number of vertices in )J f0_12 sf .035(Adjacencies)A f3_12 sf .126 .013(\(C,X\)\\{Y} is greater than or)J 149 214 :M (equal to n;)S 149 230 :M f0_12 sf (repeat)S 167 246 :M f3_12 sf (select a subset )S 239 246 :M f0_12 sf (S)S 246 246 :M f3_12 sf ( of )S 262 246 :M f0_12 sf (Adjacencies)S f3_12 sf (\(C,X\)\\{Y} with n vertices;)S 167 262 :M (if the statistical test fails to reject )S 329 262 :M f1_12 sf (r)S 336 265 :M f3_7 sf (X,Y.)S 350 265 :M f0_7 sf (S)S 354 262 :M f3_12 sf ( = 0, then delete edge)S 167 278 :M (X )S 179 278 :M f1_12 sf S f3_12 sf ( Y from C and set )S 280 278 :M f0_12 sf (Sepset)S 313 278 :M f3_12 sf (\(X,Y\) = )S f0_12 sf (S)S 361 278 :M f3_12 sf ( and )S f0_12 sf (Sepset)S 417 278 :M f3_12 sf (\(Y,X\) = )S f0_12 sf (S)S 465 278 :M f3_12 sf (;)S 149 294 :M f0_12 sf (unti)S 170 294 :M f3_12 sf (l every subset )S 239 294 :M f0_12 sf (S)S 246 294 :M f3_12 sf ( of )S 262 294 :M f0_12 sf (Adjacencies)S f3_12 sf (\(C,X\)\\{Y} with n vertices has)S 149 310 :M ( )S 161 310 :M (been selected or some subset )S 303 310 :M f0_12 sf (S)S 310 310 :M f3_12 sf ( has been found for which )S 438 310 :M f1_12 sf (r)S 445 313 :M f3_7 sf (X,Y.)S 459 313 :M f0_7 sf (S)S 463 310 :M f3_12 sf ( = 0;)S 131 326 :M f0_12 sf 1.137(until)A f3_12 sf 3.806 .381( all ordered pairs of adjacent vertices X and Y such that)J 131 342 :M f0_12 sf (Adjacencies)S f3_12 sf (\(C,X\)\\{Y} has greater than or equal to n vertices have been)S 131 358 :M ( )S 143 358 :M (selected;)S 131 374 :M (n = n + 1;)S 113 390 :M f0_12 sf (until)S f3_12 sf ( for each ordered pair of adjacent vertices X, Y, )S 369 390 :M f0_12 sf (Adjacencies)S f3_12 sf (\(C,X\)\\{Y})S 113 406 :M ( )S 143 406 :M (has less than n vertices.)S 95 438 :M .135 .013(C\) For each triple of vertices X, Y, Z such that the pair X, Y and the pair Y, Z are)J 95 454 :M (each adjacent in C but the pair X, Z are not adjacent in C, orient X )S 417 454 :M f1_12 sf S f3_12 sf ( Y )S 444 454 :M f1_12 sf S f3_12 sf ( Z as)S 95 470 :M (X )S 107 470 :M f1_12 sf S 119 470 :M f3_12 sf ( Y )S 134 470 :M f1_12 sf S 146 470 :M f3_12 sf ( Z if and only if Y is not in )S 278 470 :M f0_12 sf (Sepset)S 311 470 :M f3_12 sf (\(X,Z\).)S 95 502 :M (D\) )S 111 502 :M f0_12 sf (repeat)S 113 518 :M f3_12 sf ( )S 116 518 :M (If X )S 139 518 :M f1_12 sf S 151 518 :M f3_12 sf ( Y )S 166 518 :M f1_12 sf S f3_12 sf ( Z in C, and X and Z are not adjacent in C, then orient as Y )S 465 518 :M f1_12 sf S 477 518 :M f3_12 sf ( Z.)S 99 534 :M ( )S 111 534 :M f0_12 sf (until)S f3_12 sf ( no more edges can be oriented.)S 59 650 :M ( )S 59 647.48 -.48 .48 203.48 647 .48 59 647 @a 81 659 :M f3_6 sf (2)S 84 659 :M (1)S 87 663 :M f3_10 sf .117 .012( The simplified version of the algorithm presented here does not make all of the orientations that are)J 59 675 :M 2.263 .226(theoretically possible, because we have found in practice that additional orientation rules, while)J 59 687 :M (theoretically correct, are in practice unreliable until the sample size is very large.)S endp %%Page: 43 43 %%BeginPageSetup initializepage (peter; page: 43 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (43)S gR gS 64 80 15 12 rC 65 89 :M f3_12 sf (A)S gR gS 125 80 15 12 rC 126 89 :M f3_12 sf (B)S gR gS 173 50 15 12 rC 174 59 :M f3_12 sf (C)S gR gS 175 114 15 12 rC 176 123 :M f3_12 sf (D)S gR gS 219 79 13 12 rC 220 88 :M f3_12 sf (E)S gR gS 63 41 424 447 rC np 118 85 :M 106 88 :L 106 85 :L 106 82 :L 118 85 :L eofill 82 86 -1 1 107 85 1 82 85 @a np 169 60 :M 161 69 :L 159 67 :L 157 64 :L 169 60 :L eofill -1 -1 143 80 1 1 159 67 @b np 170 116 :M 159 111 :L 160 109 :L 162 106 :L 170 116 :L eofill 142 96 -1 1 161 109 1 142 95 @a np 217 80 :M 206 75 :L 208 72 :L 210 70 :L 217 80 :L eofill 191 60 -1 1 209 72 1 191 59 @a np 219 90 :M 212 100 :L 210 98 :L 208 95 :L 219 90 :L eofill -1 -1 191 115 1 1 210 98 @b 120 152 71 12 rC 121 161 :M f3_12 sf (True Graph)S gR gS 293 81 15 12 rC 294 90 :M f3_12 sf (A)S gR gS 365 79 15 12 rC 366 88 :M f3_12 sf (B)S gR gS 417 42 15 12 rC 418 51 :M f3_12 sf (C)S gR gS 415 112 15 12 rC 416 121 :M f3_12 sf (D)S gR gS 465 78 13 12 rC 466 87 :M f3_12 sf (E)S gR gS 63 41 424 447 rC 310 87 -1 1 356 86 1 310 86 @a -1 -1 383 79 1 1 416 53 @b 382 95 -1 1 412 109 1 382 94 @a 433 52 -1 1 465 72 1 433 51 @a -1 -1 432 115 1 1 465 88 @b -1 -1 311 79 1 1 411 48 @b 306 92 -1 1 411 116 1 306 91 @a 391 85 -1 1 461 84 1 391 84 @a -1 -1 425 110 1 1 424 55 @b -1 -1 296 141 1 1 295 94 @b 295 141 -1 1 471 140 1 295 140 @a -1 -1 471 141 1 1 470 94 @b 314 154 165 12 rC 315 163 :M f3_12 sf (Complete Undirected Graph)S gR gS 209 205 106 12 rC 210 214 :M f3_12 sf (No zero correlations.)S gR gS 240 187 31 12 rC 241 196 :M f3_12 sf (n = 0)S gR gS 305 302 15 12 rC 306 311 :M f3_12 sf (A)S gR gS 360 302 15 12 rC 361 311 :M f3_12 sf (B)S gR gS 416 269 15 12 rC 417 278 :M f3_12 sf (C)S gR gS 415 334 15 12 rC 416 343 :M f3_12 sf (D)S gR gS 460 302 13 12 rC 461 311 :M f3_12 sf (E)S gR gS 63 41 424 447 rC 320 309 -1 1 355 308 1 320 308 @a -1 -1 378 304 1 1 412 278 @b 379 318 -1 1 411 335 1 379 317 @a 432 278 -1 1 459 299 1 432 277 @a -1 -1 430 335 1 1 459 316 @b 385 309 -1 1 454 308 1 385 308 @a 65 183 -1 1 485 182 1 65 182 @a 66 228 -1 1 486 227 1 66 227 @a 65 360 -1 1 485 359 1 65 359 @a 297 435 15 12 rC 298 444 :M f3_12 sf (A)S gR gS 362 436 15 12 rC 363 445 :M f3_12 sf (B)S gR gS 416 403 15 12 rC 417 412 :M f3_12 sf (C)S gR gS 415 466 13 12 rC 416 475 :M f3_12 sf (D)S gR gS 63 41 424 447 rC 320 442 -1 1 355 441 1 320 441 @a -1 -1 378 437 1 1 412 411 @b 379 451 -1 1 411 468 1 379 450 @a 432 411 -1 1 459 432 1 432 410 @a -1 -1 430 468 1 1 459 449 @b 457 435 16 12 rC 458 444 :M f3_12 sf (E)S gR gS 63 41 424 447 rC 66 487 -1 1 486 486 1 66 486 @a 239 239 31 12 rC 240 248 :M f3_12 sf (n = 1)S gR gS 78 271 49 84 rC 79 280 :M f3_10 sf (AC.B)S 79 304 :M (AE.B)S 79 328 :M (AD.B)S 79 352 :M (CD.B)S gR gS 71 264 8 15 rC 72 276 :M f1_14 sf (r)S gR gS 71 288 8 15 rC 72 300 :M f1_14 sf (r)S gR gS 71 312 8 15 rC 72 324 :M f1_14 sf (r)S gR gS 70 336 8 15 rC 71 348 :M f1_14 sf (r)S gR gS 103 269 17 12 rC 104 278 :M f3_12 sf (= 0)S gR gS 105 293 17 12 rC 106 302 :M f3_12 sf (= 0)S gR gS 103 317 17 12 rC 104 326 :M f3_12 sf (= 0)S gR gS 104 342 17 12 rC 105 351 :M f3_12 sf (= 0)S gR gS 140 269 93 84 rC 141 278 :M f3_12 sf (Sepset\(A,C\) = {B})S 141 302 :M (Sepset\(A,E\) = {B})S 141 326 :M (Sepset\(A,D\) = {B})S 141 350 :M (Sepset\(C,D\) = {B})S gR gS 243 365 31 12 rC 244 374 :M f3_12 sf (n = 2)S gR gS 81 397 49 24 rC 82 406 :M f3_10 sf (BE.CD)S gR gS 74 390 8 15 rC 75 402 :M f1_14 sf (r)S gR gS 115 396 17 12 rC 116 405 :M f3_12 sf (= 0)S gR gS 143 395 113 24 rC 144 404 :M f3_12 sf (Sepset\(B,E\) = {C,D})S gR gS 0 0 552 730 rC 118 513 :M f0_12 sf (Figure )S 155 513 :M (32: A trace of the adjacency stage of the PC algorithm)S 81 557 :M f3_12 sf .884 .088(Figure 32)J 129 557 :M 1.046 .105( traces the operation of the first two parts of the PC algorithm for input)J 59 573 :M (Faithful to the true graph in Figure 32.)S 81 589 :M .366 .037(Although it does not in this case, stage B\) of the algorithm may continue testing for)J 59 605 :M .238 .024(some steps after the correct undirected graph has been identified. After stage B\) has been)J 59 621 :M .29 .029(completed, the undirected graph at the bottom of )J 300 621 :M .295 .03(Figure 32 is partially oriented in step C)J 59 637 :M (of the PC algorithm. The triples of variables with only two adjacencies among them are:)S endp %%Page: 44 44 %%BeginPageSetup initializepage (peter; page: 44 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (44)S gR gS 0 0 552 730 rC 206 54 :M f3_12 sf (A )S 218 54 :M f1_12 sf S f3_12 sf ( B )S f1_12 sf S f3_12 sf ( C; )S 278 54 :M (A )S 290 54 :M f1_12 sf S f3_12 sf ( B )S f1_12 sf S f3_12 sf ( D;)S 207 70 :M (C )S f1_12 sf S f3_12 sf ( B )S f1_12 sf S f3_12 sf ( D; )S 279 70 :M (B )S f1_12 sf S f3_12 sf ( C )S f1_12 sf S f3_12 sf ( E;)S 207 86 :M (B )S f1_12 sf S f3_12 sf ( D )S 245 86 :M f1_12 sf S f3_12 sf ( E; )S 279 86 :M (C )S f1_12 sf S f3_12 sf ( E )S f1_12 sf S f3_12 sf ( D;)S 59 118 :M .183 .018(E is not in )J 112 118 :M f0_12 sf (Sepset)S 145 118 :M f3_12 sf .198 .02(\(C,D\) so C )J f1_12 sf .177A f3_12 sf .135 .013( E and E )J 258 118 :M f1_12 sf .122A f3_12 sf .152 .015( D collide at E. None of the other triples form)J 59 134 :M (colliders. The final pattern produced by the algorithm is shown in )S 377 134 :M (Figure 33.)S 159 193 21 12 rC 160 202 :M (A)S gR gS 231 193 21 12 rC 232 202 :M f3_12 sf (B)S gR gS 303 154 21 12 rC 304 163 :M f3_12 sf (C)S gR gS 300 229 21 12 rC 301 238 :M f3_12 sf (D)S gR gS 370 194 21 12 rC 371 203 :M f3_12 sf (E)S gR gS 158 153 234 89 rC 177 199 -1 1 223 198 1 177 198 @a -1 -1 249 194 1 1 293 163 @b 247 206 -1 1 293 228 1 247 205 @a np 365 193 :M 353 189 :L 355 186 :L 357 184 :L 365 193 :L eofill 320 164 -1 1 356 186 1 320 163 @a np 365 201 :M 356 209 :L 355 207 :L 353 204 :L 365 201 :L eofill -1 -1 318 229 1 1 355 207 @b gR gS 0 0 552 730 rC 174 267 :M f0_12 sf (Figure )S 211 267 :M (33. Final Pattern Output by PC.)S 81 295 :M f3_12 sf 3.039 .304(The pattern in Figure 33 represents the partial correlation \(and covariance\))J 59 311 :M (equivalence class of RSEMs we show in )S 257 311 :M (Figure 34.)S .75 lw 70 330 409 195 rC 184.5 395.5 20 17 rS 186 397 17 14 rC 186 406 :M ( )S 189 406 :M (D)S gR .75 lw gS 70 330 409 195 rC 246.5 360.5 20 17 rS 248 362 17 14 rC 248 371 :M f3_12 sf ( )S 251 371 :M (E)S gR gS 70 330 409 195 rC 185.5 330.5 20 17 rS 187 332 17 14 rC 187 341 :M f3_12 sf ( )S 190 341 :M (C)S gR gS 70 330 409 195 rC 121.5 361.5 20 17 rS 123 363 17 14 rC 123 372 :M f3_12 sf ( )S 126 372 :M (B)S gR gS 70 330 409 195 rC 70.5 361.5 20 17 rS 72 363 17 14 rC 72 372 :M f3_12 sf ( )S 75 372 :M (A)S gR gS 70 330 409 195 rC 207 343.75 -.75 .75 241.75 364 .75 207 343 @a np 242 360 :M 238 366 :L 243 365 :L 242 360 :L eofill -.75 -.75 238.75 366.75 .75 .75 242 360 @b -.75 -.75 238.75 366.75 .75 .75 243 365 @b 242 360.75 -.75 .75 243.75 365 .75 242 360 @a -.75 -.75 207.75 401.75 .75 .75 241 378 @b np 238 375 :M 242 381 :L 243 376 :L 238 375 :L eofill 238 375.75 -.75 .75 242.75 381 .75 238 375 @a -.75 -.75 242.75 381.75 .75 .75 243 376 @b 238 375.75 -.75 .75 243.75 376 .75 238 375 @a 395.5 396.5 20 17 rS 397 398 17 14 rC 397 407 :M f3_12 sf ( )S 400 407 :M (D)S gR gS 70 330 409 195 rC 457.5 361.5 20 17 rS 459 363 17 14 rC 459 372 :M f3_12 sf ( )S 462 372 :M (E)S gR gS 70 330 409 195 rC 396.5 331.5 20 17 rS 398 333 17 14 rC 398 342 :M f3_12 sf ( )S 401 342 :M (C)S gR gS 70 330 409 195 rC 332.5 362.5 20 17 rS 334 364 17 14 rC 334 373 :M f3_12 sf ( )S 337 373 :M (B)S gR gS 70 330 409 195 rC 291.5 362.5 20 17 rS 293 364 17 14 rC 293 373 :M f3_12 sf ( )S 296 373 :M (A)S gR gS 70 330 409 195 rC 418 344.75 -.75 .75 452.75 365 .75 418 344 @a np 453 361 :M 449 367 :L 454 366 :L 453 361 :L eofill -.75 -.75 449.75 367.75 .75 .75 453 361 @b -.75 -.75 449.75 367.75 .75 .75 454 366 @b 453 361.75 -.75 .75 454.75 366 .75 453 361 @a -.75 -.75 418.75 402.75 .75 .75 452 379 @b np 449 376 :M 453 382 :L 454 377 :L 449 376 :L eofill 449 376.75 -.75 .75 453.75 382 .75 449 376 @a -.75 -.75 453.75 382.75 .75 .75 454 377 @b 449 376.75 -.75 .75 454.75 377 .75 449 376 @a 182.5 503.5 20 17 rS 184 505 17 14 rC 184 514 :M f3_12 sf ( )S 187 514 :M (D)S gR gS 70 330 409 195 rC 244.5 468.5 20 17 rS 246 470 17 14 rC 246 479 :M f3_12 sf ( )S 249 479 :M (E)S gR gS 70 330 409 195 rC 183.5 438.5 20 17 rS 185 440 17 14 rC 185 449 :M f3_12 sf ( )S 188 449 :M (C)S gR gS 70 330 409 195 rC 119.5 469.5 20 17 rS 121 471 17 14 rC 121 480 :M f3_12 sf ( )S 124 480 :M (B)S gR gS 70 330 409 195 rC 73.5 469.5 20 17 rS 75 471 17 14 rC 75 480 :M f3_12 sf ( )S 78 480 :M (A)S gR gS 70 330 409 195 rC 205 451.75 -.75 .75 239.75 472 .75 205 451 @a np 240 468 :M 236 474 :L 241 473 :L 240 468 :L eofill -.75 -.75 236.75 474.75 .75 .75 240 468 @b -.75 -.75 236.75 474.75 .75 .75 241 473 @b 240 468.75 -.75 .75 241.75 473 .75 240 468 @a -.75 -.75 205.75 509.75 .75 .75 239 486 @b np 236 483 :M 240 489 :L 241 484 :L 236 483 :L eofill 236 483.75 -.75 .75 240.75 489 .75 236 483 @a -.75 -.75 240.75 489.75 .75 .75 241 484 @b 236 483.75 -.75 .75 241.75 484 .75 236 483 @a 395.5 506.5 20 17 rS 397 508 17 14 rC 397 517 :M f3_12 sf ( )S 400 517 :M (D)S gR gS 70 330 409 195 rC 457.5 471.5 20 17 rS 459 473 17 14 rC 459 482 :M f3_12 sf ( )S 462 482 :M (E)S gR gS 70 330 409 195 rC 396.5 441.5 20 17 rS 398 443 17 14 rC 398 452 :M f3_12 sf ( )S 401 452 :M (C)S gR gS 70 330 409 195 rC 332.5 472.5 20 17 rS 334 474 17 14 rC 334 483 :M f3_12 sf ( )S 337 483 :M (B)S gR gS 70 330 409 195 rC 292.5 471.5 20 17 rS 294 473 17 14 rC 294 482 :M f3_12 sf ( )S 297 482 :M (A)S gR gS 70 330 409 195 rC 418 454.75 -.75 .75 452.75 475 .75 418 454 @a np 453 471 :M 449 477 :L 454 476 :L 453 471 :L eofill -.75 -.75 449.75 477.75 .75 .75 453 471 @b -.75 -.75 449.75 477.75 .75 .75 454 476 @b 453 471.75 -.75 .75 454.75 476 .75 453 471 @a -.75 -.75 418.75 512.75 .75 .75 452 489 @b np 449 486 :M 453 492 :L 454 487 :L 449 486 :L eofill 449 486.75 -.75 .75 453.75 492 .75 449 486 @a -.75 -.75 453.75 492.75 .75 .75 454 487 @b 449 486.75 -.75 .75 454.75 487 .75 449 486 @a -2 -2 282 522 2 2 280 333 @b 77 420 -2 2 472 418 2 77 418 @a 92 369.75 -.75 .75 115.75 369 .75 92 369 @a np 115 365 :M 115 373 :L 118 369 :L 115 365 :L eofill -.75 -.75 115.75 373.75 .75 .75 115 365 @b -.75 -.75 115.75 373.75 .75 .75 118 369 @b 115 365.75 -.75 .75 118.75 369 .75 115 365 @a -.75 -.75 145.75 364.75 .75 .75 181 345 @b np 178 343 :M 182 349 :L 183 344 :L 178 343 :L eofill 178 343.75 -.75 .75 182.75 349 .75 178 343 @a -.75 -.75 182.75 349.75 .75 .75 183 344 @b 178 343.75 -.75 .75 183.75 344 .75 178 343 @a 143 372.75 -.75 .75 179.75 398 .75 143 372 @a np 180 394 :M 176 400 :L 181 399 :L 180 394 :L eofill -.75 -.75 176.75 400.75 .75 .75 180 394 @b -.75 -.75 176.75 400.75 .75 .75 181 399 @b 180 394.75 -.75 .75 181.75 399 .75 180 394 @a 317 370.75 -.75 .75 330.75 370 .75 317 370 @a np 318 374 :M 318 366 :L 314 370 :L 318 374 :L eofill -.75 -.75 318.75 374.75 .75 .75 318 366 @b -.75 -.75 314.75 370.75 .75 .75 318 366 @b 314 370.75 -.75 .75 318.75 374 .75 314 370 @a -.75 -.75 354.75 367.75 .75 .75 393 344 @b np 390 342 :M 394 348 :L 395 343 :L 390 342 :L eofill 390 342.75 -.75 .75 394.75 348 .75 390 342 @a -.75 -.75 394.75 348.75 .75 .75 395 343 @b 390 342.75 -.75 .75 395.75 343 .75 390 342 @a 355 373.75 -.75 .75 390.75 397 .75 355 373 @a np 391 393 :M 387 399 :L 392 398 :L 391 393 :L eofill -.75 -.75 387.75 399.75 .75 .75 391 393 @b -.75 -.75 387.75 399.75 .75 .75 392 398 @b 391 393.75 -.75 .75 392.75 398 .75 391 393 @a 98 477.75 -.75 .75 118.75 477 .75 98 477 @a np 99 481 :M 99 473 :L 95 477 :L 99 481 :L eofill -.75 -.75 99.75 481.75 .75 .75 99 473 @b -.75 -.75 95.75 477.75 .75 .75 99 473 @b 95 477.75 -.75 .75 99.75 481 .75 95 477 @a 145 485.75 -.75 .75 180.75 509 .75 145 485 @a np 144 488 :M 148 482 :L 143 483 :L 144 488 :L eofill -.75 -.75 144.75 488.75 .75 .75 148 482 @b -.75 -.75 143.75 483.75 .75 .75 148 482 @b 143 483.75 -.75 .75 144.75 488 .75 143 483 @a -.75 -.75 142.75 476.75 .75 .75 179 452 @b np 176 450 :M 180 456 :L 181 451 :L 176 450 :L eofill 176 450.75 -.75 .75 180.75 456 .75 176 450 @a -.75 -.75 180.75 456.75 .75 .75 181 451 @b 176 450.75 -.75 .75 181.75 451 .75 176 450 @a 317 480.75 -.75 .75 329.75 480 .75 317 480 @a np 318 484 :M 318 476 :L 314 480 :L 318 484 :L eofill -.75 -.75 318.75 484.75 .75 .75 318 476 @b -.75 -.75 314.75 480.75 .75 .75 318 476 @b 314 480.75 -.75 .75 318.75 484 .75 314 480 @a -.75 -.75 356.75 477.75 .75 .75 391 455 @b np 359 479 :M 355 473 :L 354 478 :L 359 479 :L eofill 355 473.75 -.75 .75 359.75 479 .75 355 473 @a -.75 -.75 354.75 478.75 .75 .75 355 473 @b 354 478.75 -.75 .75 359.75 479 .75 354 478 @a 355 482.75 -.75 .75 390.75 507 .75 355 482 @a np 391 503 :M 387 509 :L 392 508 :L 391 503 :L eofill -.75 -.75 387.75 509.75 .75 .75 391 503 @b -.75 -.75 387.75 509.75 .75 .75 392 508 @b 391 503.75 -.75 .75 392.75 508 .75 391 503 @a gR gS 0 0 552 730 rC 69 550 :M f0_12 sf (Figure )S 106 550 :M (34. Equivalence Class of RSEMs represented by the Pattern in Figure 33)S 477 550 :M (.)S 59 590 :M f4_12 sf (8)S 65 590 :M (.)S 68 590 :M (3)S 74 590 :M ( )S 81 590 :M (Purify)S 81 612 :M f3_12 sf .235 .023(We say that an indicator X )J f0_12 sf .107(measures)A 263 612 :M f3_12 sf .33 .033( T in an RSEM if there is a directed edge from)J 59 628 :M .558 .056(T to X in the directed graph associated with the RSEM. If G is the directed graph of an)J 59 644 :M 1.323 .132(RSEM with latent variables )J f0_12 sf .522(T)A f3_12 sf 1.223 .122( and a measurement model with indicators )J 432 644 :M f0_12 sf (X)S 441 644 :M f3_12 sf 1.432 .143( such that)J 59 660 :M .271 .027(every X)J f3_7 sf 0 3 rm (i )S 0 -3 rm 101 660 :M f1_12 sf S 110 660 :M f3_12 sf .059 .006( )J f0_12 sf (X)S 122 660 :M f3_12 sf .177 .018( measures some latent T in )J 255 660 :M f0_12 sf .084(T)A f3_12 sf .143 .014(, then X)J f3_7 sf 0 3 rm (i)S 0 -3 rm 303 660 :M f3_12 sf .065 .007( is a )J f0_12 sf .338 .034(pure indicator)J 400 660 :M f3_12 sf .192 .019( in G if and only if)J endp %%Page: 45 45 %%BeginPageSetup initializepage (peter; page: 45 of 53)setjob %%EndPageSetup gS 0 0 552 730 rC 258 709 34 16 rC 280 722 :M f3_12 sf (45)S gR gS 0 0 552 730 rC 59 54 :M f3_12 sf (X)S 68 57 :M f3_7 sf (i)S 70 54 :M f3_12 sf 1.408 .141( is d-separated from every other indicator by T. A measurement model is pure, or)J 59 70 :M f0_12 sf (unidimensional)S 138 70 :M f3_12 sf (, if and only if all of its indicators are pure.)S 59 114 :M f4_12 sf (8)S 65 114 :M (.)S 68 114 :M (4)S 74 114 :M ( )S 81 114 :M (MIMbuild)S 59 136 :M f3_12 sf .322 .032(MIMbuild takes as input a unidimensional measurement model and covariance data over)J 59 152 :M .247 .025(the indicators in this model, and outputs a modified pattern )J f1_12 sf .128(P)A f3_12 sf .275 .028(, where the adjacencies can)J 59 168 :M 1.313 .131(be either: )J f1_12 sf S 124 168 :M f3_12 sf .842 .084(, )J f1_12 sf 2.583A f3_12 sf 1.285 .128(, ?)J f1_12 sf S 181 168 :M f3_12 sf 2.058 .206(?, or ?)J 222 168 :M f1_12 sf 1.247A f3_12 sf 1.619 .162(?. The edges labeled with a \322?\323 indicate that the)J 59 184 :M .392 .039(MIMbuild algorithm cannot determine if there is an edge in the population graph or not.)J 59 200 :M .155 .015(Suppose that G is the directed graph of an RSEM with no correlated errors that has latent)J 59 216 :M 1.448 .145(variables )J 107 216 :M f0_12 sf .809(T)A f3_12 sf 1.655 .165( and indicators )J 198 216 :M f0_12 sf <58D5>S 211 216 :M f3_12 sf 1.329 .133(. Then if )J f1_12 sf 1.113(P)A f3_12 sf 1.987 .199( is the output of MIMbuild on a correctly)J 59 232 :M 2.058 .206(specified unidimensional measurement model for )J f0_12 sf .665(T)A f3_12 sf .775 .078( and )J f0_12 sf .881 .088(X )J 365 232 :M f2_12 sf S 374 232 :M f0_12 sf 2.452 .245<2058D5>J 393 232 :M f3_12 sf 1.765 .177(, and the statistical)J 59 248 :M .359 .036(decisions about vanishing tetrad differences among )J f0_12 sf (X)S 321 248 :M f3_12 sf .472 .047( made on a sample from a Faithful)J 59 264 :M (parameterization of G are correct, then)S 77 296 :M (1)S 83 296 :M (.)S 86 296 :M ( )S 95 296 :M (If T)S f3_7 sf 0 3 rm (i)S 0 -3 rm 115 296 :M f3_12 sf ( and T)S 146 299 :M f3_7 sf (j)S 148 296 :M f3_12 sf ( are not adjacent in )S f1_12 sf (P)S f3_12 sf (, then they are not adjacent in G.)S 77 313 :M (2)S 83 313 :M (.)S 86 313 :M ( )S 95 313 :M (If T)S f3_7 sf 0 3 rm (i)S 0 -3 rm 115 313 :M f3_12 sf ( and T)S 146 316 :M f3_7 sf (j)S 148 313 :M f3_12 sf ( are adjacent in )S 224 313 :M f1_12 sf (P)S f3_12 sf ( and the edge is not labeled with a "?", then T)S f3_7 sf 0 3 rm (i)S 0 -3 rm 95 326 :M f3_12 sf (and T)S 123 329 :M f3_7 sf (j)S 125 326 :M f3_12 sf ( are adjacent in G.)S 77 342 :M (3)S 83 342 :M (.)S 86 342 :M ( )S 95 342 :M (If T)S f3_7 sf 0 3 rm (i)S 0 -3 rm 115 342 :M f3_12 sf ( )S f1_12 sf S 130 342 :M f3_12 sf ( T)S f3_7 sf 0 3 rm (j)S 0 -3 rm 142 342 :M f3_12 sf ( is in )S f1_12 sf (P)S f3_12 sf (, then T)S 214 345 :M f3_7 sf (j)S 216 342 :M f3_12 sf ( is not an ancestor of T)S 326 345 :M f3_7 sf (i)S 328 342 :M f3_12 sf ( in G.)S 77 359 :M (4)S 83 359 :M (.)S 86 359 :M ( )S 95 359 :M (If T)S f3_7 sf 0 3 rm (i)S 0 -3 rm 115 359 :M f3_12 sf ( )S f1_12 sf S 130 359 :M f3_12 sf ( T)S f3_7 sf 0 3 rm (j)S 0 -3 rm 142 359 :M f3_12 sf ( is in )S f1_12 sf (P)S f3_12 sf ( and the edge between T)S 294 362 :M f3_7 sf (i)S 296 359 :M f3_12 sf ( and T)S 327 362 :M f3_7 sf (j)S 329 359 :M f3_12 sf ( is not labeled with a "?",)S 95 376 :M (then T)S 126 379 :M f3_7 sf (i)S 128 376 :M f3_12 sf ( )S f1_12 sf S 143 376 :M f3_12 sf ( T)S f3_7 sf 0 3 rm (j)S 0 -3 rm 155 376 :M f3_12 sf ( is in G.)S 240 437 :M f4_12 sf (9)S 246 437 :M (.)S 249 437 :M 4 0 rm ( )S 256 437 :M (References)S 59 465 :M f3_12 sf .513 .051(Anderson, J., & Gerbing, D. \(1988\). 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