Testing Equality of ProportionsExplained Latent Variance in Structural

Equation ModelsTenko Raykov

Department of PsychologyFordham University

A covariance structure modeling method to test equality in proportions explained vari-ance in studied unobserved dimensions by means of latent predictors is outlined. Theprocedure is applicable with multiple-indicator, structural equation models where ofinterest is to compare the predictive power of sets of latent independent variables forgiven constructs. The approach accounts for measurement error in all manifest vari-ables and is illustrated with data from a cognitive intervention study.

Examination of explanatory power of sets of predictors is frequently of special inter-est in behavioral, social, and educational research. One of the most attractive fea-tures of the structural equation modeling (SEM) methodology for purposes of ana-lyzing behavioral phenomena is the explicit consideration of error of measurementassociated with independent variables. This allows one to obtain optimal estimatesof regression coefficients for latent dimensions under investigation, which are freeofcontaminatingeffectsof their less-than-perfectmeasurement (e.g.,Bollen,1989).Although current SEM programs routinely provide indexes of explained constructvariance, examining predictive power of possibly distinct sets of latent independentvariables across different latent dependent variables seems to have attracted limitedinterest. In particular, no procedure is available for testing identity in proportions ex-plained variance in two or more unobserved variables in terms of possibly differentfallible predictors within a given structural equation model.

This article intends to fill this gap. A covariance structure modeling method isoutlined that permits testing equality in proportions explained latent constructvariance by sets of predictors measured with error. The approach can be used withmultiple-indicator models where one is concerned with comparing explanatorypower of latent independent variables for focused constructs. This concern can beof relevance in studies aimed at identifying regressor sets with optimal predictiveability as well as when in a model choice context it is relevant to improve explana-

STRUCTURAL EQUATION MODELING, 10(4), 534–543Copyright © 2003, Lawrence Erlbaum Associates, Inc.

Requests for reprints should be sent to Tenko Raykov, Department of Psychology, Fordham Univer-sity, Bronx, NY 10458. E-mail: raykov@fordham.edu

tion of focal latent dependent constructs or ensure that they are comparable withregard to explained variance in studied populations (e.g., Illustration on Data sec-tion, which follows). The proposed procedure is based on testing nonlinear con-straints appropriately imposed on model parameters. The method is readily em-ployed with the widely available SEM program, LISREL (Jöreskog & Sörbom,1996), and is illustrated on data from a cognitive intervention study.

BACKGROUND AND NOTATION

The procedure discussed in this article is applicable with covariance structuremodels to test for equality of proportions explained variance in latent constructs,referred to as focused variables in the remainder, which are regressed on generallydifferent sets of latent independent variables only (possibly after obtaining the re-duced form equations for involved predictors of the focused variables). We are notconcerned with models in which focused variables are involved in loops and recip-rocal relations with some of their predictors, or are associated with disturbanceterms correlated with some of those predictors.

The models of concern in this article are special cases of the general linearstructural relations (LISREL) model defined by the following pair of equations us-ing the widely adopted LISREL notation (e.g., Jöreskog & Sörbom, 1996):

y = Λη + ε (1)

η = Bη + ζ (2)

where y denotes the vector of observed variables; η that of latent variables; Λ is thematrix of factor loadings; ε the vector of pertinent manifest error terms; B = (βjk) thematrix of latent regression coefficients; ζ the associated vector of latent disturbanceswith covariance matrix Ψ = (ψjk); as usual, ε and ζ are assumed with zero means anduncorrelated with one another and with η (j, k = 1, … , q, where q > 1 is the number oflatent variables; and no latent or disturbance variance is assumed zero. This model isfrequently referred to as Submodel 3 of the general LISREL model, where variancesand covariances of latent independent variables are corresponding elements of Ψ;Jöreskog & Sörbom, 1996).

The idea underlying the testing procedure in this article is the same whether oneis dealing with two or more focused variables and irrespective of whether the latterare explained in terms of the same or different predictor sets. For simplicity of ex-position, in the case of identical predictor sets and two focused variables (num-bered pth and p + 1st say, after their predictors), their structural equations are

TESTING EQUALITY IN LATENT R-SQUARES 535

1

1

(3)p

p pj j pj

η β η ζ�

�

� ��

and

where η1, …, ηp – 1 are independent predictors uncorrelated with ζp and ζp + 1,whereas other latent variables in the model not playing explanatory roles for ηp

and/or ηp + 1 may have any relations among themselves (including reciprocal rela-tions, loops, or disturbance terms correlated with some of their predictors oramong themselves) and may be explained in terms of ηp, ηp + 1, or any of their pre-dictors in Equations 3 and 4. The class of covariance structure models defined byEquations 1, 3, and 4, including their analogs in the case of different predictor setsand multiple focused variables (see next section), covers a wide range of models inbehavioral research. In particular, a subclass of it contains models of interest inlongitudinal studies where constructs measured at later occasions are related toearlier assessed constructs (e.g., Illustration on Data section).

Following general principles in regression analysis and their applicationswithin the SEM framework (e.g., Bollen, 1989; Pedhazur, 1997) in the generalLISREL model of Equations 1 and 2, the ith latent variable, ηi (assumed depend-ent), is associated with proportion explained variance, R2i, defined as

R2i = 1 – Var(ζi)/Var(ηi) = 1 – ψii/Var(ηi) (5)

where Var(.) denotes variance in the participant population (1 < i < q; assumingnonzero latent variance). In the case of Equations 1, 3, and 4 describing models ofinterest in this article, we are concerned with testing the following null hypothesis:

Ho: R2p = R2p+1 (6)

with regard to the latent dependent variables ηp and ηp+1 in Equations 3 and 4 (p < q).Equation 6 remains the same in models where two focused variables are explained interms of different predictor sets (see the following), and is straightforwardly ex-tendedtocoverallpertinentR2 coefficients incaseofat least threefocusedvariables.

A Procedure for Examining Equality in Proportions ofExplained Latent Variance

We initially describe the method for the case of an identical predictor set for twofocused variables and then extend it to that of distinct predictors. In the formercase, according to Equation 5 the null hypothesis in Equation 6 will be fulfilledif and only if the corresponding ratios subtracted from 1 are equal. That is,

ψpp/Var(ηp) = ψp+1, p+1/Var(ηp+1) (7)

536 RAYKOV

1

1 1, 11

(1 ) (4)p

p p j j pj

p qη β η ζ�

� � �

�

� � � ��

Using simple algebra and well-known formulas for variance of sums of randomvariables on Equations 3 and 4 (e.g., Hays, 1994), Equation 7 holds if and only if

To be in a position to test Equation 8—and thus the null hypothesis of Equation6—within currently available implementations of the SEM modeling framework,we introduce an additional parameter, π1, which is defined as the denominator ofthe left-hand side of Equation 8 and used as an auxiliary quantity next. This is ac-complished with the following implicit equation (cf. Jöreskog & Sörbom, 1996)that results with direct rearrangements from the one setting π1 equal to that denom-inator expression:1

Equation 8 is now rewritten with Equation 9 as

Equation 10 is equivalent to the null hypothesis of Equation 6 and represents anonlinear constraint on parameters of the model defined by Equations 1, 2, and 4.Therefore, testing Equation 6 is tantamount to testing the validity of the restrictionin Equation 10, whereby we note that the preceding developments in this sectionrendered Equation 10 in a form readily implemented with the latest versions of thepopular SEM program, LISREL (Jöreskog & Sörbom, 1996). Hence, testing thenull hypothesis of concern in Equation 6 can be conducted via the difference inchi-square values associated with the two nested models with and without restric-

TESTING EQUALITY IN LATENT R-SQUARES 537

1, 11 1

2 21, 1, 1,

1 1 1 1 1 1

(8)

2 2

pp p p

p p

pj jj pj pk jk p j jj p j p k jkj j k p j j k p

ψ ψ

β ψ β β ψ β ψ β β ψ

� �

� �

� � �

� � � � � � � � � �

�

� �� � � �

12 2 2 211 1 1 1 1

2 1 1

2 (9)p

p pj jj p pj pk jk pj j k p

ψ π β β ψ β β β ψ β�

� � �

� � � � �

� � �� �

1If p = 2 (i.e., two focused constructs, η2 and η3) are regressed on a single latent predictor η1, thenthe path from ζ2 into η2 can be set equal to β21 and that from ζ3 into η3 equal to β31. (The effect of thismodel reparameterization is division of the units of measurement underlying ψ22 and ψ33 by β21

2 andβ31

2, respectively.) This renders Equation 8 equivalent to the constraint ψ*22 = ψ*

33 for theso-reparameterized latent disturbances (with resulting variances ψ*

22 = ψ22/β212 and ψ*

33 = ψ33/β312).

This linear restriction, ψ*22 = ψ*

33, and hence Equation 6 equivalent to it, is readily tested with mostpopular CSA programs (e.g., also EQS, Bentler, 2003; and Mplus, Muthén & Muthén, 2002). (Thesame approach is applicable in the case of distinct single predictors of focused variables; see the follow-ing. Therefore, the following developments in the main text can be carried out under the assumptionthat the predictor set contains at least 2 variables, i.e., p > 2.)

12 1 11, 1 1, 1 1, 1, 1

1 1 1

2 (10)p

p p p j jj pp p j p k jk ppj j k p

ψ β ψ ψ π β β ψ ψ π�

� �� � � � �

� � � � �

� �� �

tion (Equation 10). The software code for this purpose is presented in the Appen-dix where it is applied on data used in the next section.2

In the case of distinct predictor sets, for the sake of completion of this article,with appropriate numbering of the latent variables, the model of concern is definedby Equation 1 and the following pair of equations:

and

where ζp and ζp+1 are uncorrelated with any of the ηs appearing on the right-handsides of Equations 11 and 12; all ηs there are independent variables, and not all vari-ables ηp+2, ηp+3, … , ηp+r are identical to variables from the set η1, η2, … , ηp-1 (incase of identity, pertinent sub-indexes are identical as well; see also Footnote 1). Asmentioned earlier, the null hypothesis (Equation 6) is found to be equivalent to

In analogy to the aforementioned developments, we first introduce an auxiliaryparameter, π1, defined as equal to the denominator of the left-hand side of Equation13. That is, similarly

As mentioned earlier, one now finds, via algebraic rearrangements, that the nullhypothesis (Equation 6) is equivalent to the nonlinear parameter constraint:

538 RAYKOV

1

1

(11)p

p pj j pj

η β η ζ�

�

� ��

1 1, 1 1 11

(1 , 0) (12)r

p p p m p m pm

p r q rη β η ζ� � � � � � �

�

� � � � � ��

2An anonymous referee of this article proposed a two-step alternative to the described testingmethod: (a) using nonlinear constraints, restrict to 1 the variances of the latent dependent variableswhose proportion explained variance are to be compared; and (b) test if the variances of their distur-bance terms are equal; the latter test is tantamount to that of equal proportions explained latent vari-ances (see Equation 7). We do not follow this approach here because it does not preserve the metric(unit of measurement) of the dependent variables, a feature of special relevance in longitudinal studiesas well as in others where loss of that metric may be incompatible with research questions of interest.

12

1 1 1

1, 1

2 1, 1 1 , 1 1, 1 1, 1 1 , 11 1

(13)

2

2

ppp

pj jj pj pk jkj j k p

p pr

p p m p m p m p p m p p n p m p nm m n r

ψ

β ψ β β ψ

ψ

β ψ β β ψ

�

� � � � �

� �

� � � � � � � � � � � � � � � � �

� � � �

�

�

�

� �

� �

12 2 2 211 1 1 1 1

2 1 1

2 (14)p

p pj jj p pj pk jk pj j k p

ψ π β β ψ β β β ψ β�

� � �

� � � � �

� � �� �

which is also readily implemented with the latest versions of the popular SEM pro-gram, LISREL (Jöreskog & Sörbom, 1996).

ILLUSTRATION ON DATA

We use in this section data from a cognitive intervention study by Baltes,Dittmann-Kohli, and Kliegl (1986). Their concern was with examining older adults’reserve capacity in age-sensitive fluid intelligence tests after tutor-guided traininginto test-relevant skills. For our purpose, we employ data from two consecutive as-sessments using two pairs of tests of induction (inductive reasoning) and figural re-lations that can be considered subabilities of the broad cluster of fluid intelligence(e.g., Horn, 1982). As induction markers, we utilize the so-called ADEPT Inductionand Thurstone’s Standard Induction tests (Baltes et al., 1986); as figural relationsmeasures, we use the ADEPT Figural Relations and Culture-Fair tests (Baltes et al.,1986). Our next aim is to examine whether subsequent induction and figural rela-tions can be predicted equally well in terms of initial induction and figural relationsabilities. The covariance matrix of the eight observed variables is presented in Table1. Initial data exploration suggested up to mild deviations from normality, and thus,the maximum likelihood method is applied for model fitting and testing (Jöreskog &Sörbom, 1996).

Given our objective of testing equality in latent predictive power, we utilize amodel relating second assessment to first assessment abilities. The unconstrainedmodel is depicted in Figure 1. In this model, due to the longitudinal nature of theoriginal study, as customary (e.g., Jöreskog & Sörbom, 1996) we allowcovariances between measurement errors pertaining to the same repeatedly pre-sented fluid measures; we similarly constrain for time invariance the correspond-ing factor loadings of the latter, as the same latent constructs—induction andfigural relations—have been assessed at both measurement points with the sametests (e.g., Tisak & Meredith, 1988; Tisak & Tisak, 2000).

When fitted to data, the model in Figure 1 is found to be associated with accept-ablegoodness-of-fit indexes:χ2(13,N=161)=21.16,p= .07, rootmeansquareerrorof approximation (RMSEA) = .061, with a 90% confidence interval (CI) (0; .11). Inthis model, the explained proportion of variance in induction at the second assess-ment is estimated at .91, whereas in figural relations it is .83. To accomplish our ob-jective, we need to ascertain whether the observed difference of 8% in their percent-ages explained variance may be due to chance fluctuations in the available sample or

TESTING EQUALITY IN LATENT R-SQUARES 539

1, 1

2 11, 1 1 , 1

11

1, 1 1, 1 1 , 11

(15)

2

p pr

p p m p m p m ppm

p p m p p n p m p n ppm n r

ψ

β ψ ψ π

β β ψ ψ π

� �

�� � � � � � �

�

�� � � � � � � � � �

� � �

�

�

�

�

perhaps reflecting a genuine population difference in explanatory power of thesetwo latent predictors of subsequent fluid abilities.

To this end, as indicated earlier, we first introduce the auxiliary parameter, π1,defined as the denominator of the left-hand side of Equation 8. In this empiricalcase, constraint (Equation 8) rendering equal proportions of explained variance insecond assessment induction and figural relations has the following form (ob-tained by substituting p = 3 in Equation 8):

Setting π1 equal to the denominator of the left-hand side of Equation 16, one ob-tains the implicit equation for this additional parameter as

With π1 defined in Equation 17, Equation 16 is equivalent to

which is a nonlinear constraint equivalent to the null hypothesis (Equation 6) of in-terest here with respect to subsequent induction and figural relations as focusedvariables. (The software implementation of the critical restrictions, Equations 17and 18, is provided in the Appendix.)

Imposing the next constraint (Equation 18) in the initially fitted model of Figure 1brings about a minor increase in the chi-square value: χ2(14, N = 161) = 23.00, p =.06, RMSEA = .061, with a 90% CI (0; .11). The resulting decrement in fit is not sig-nificant: increase inchi-squarevalues is1.84for thedifference indegreesof freedombeing 1, with associated p > .05. This indicates that Equation 18 represents a plausi-

540 RAYKOV

TABLE 1Fluid Intelligence Tests Covariance Matrix

IR11 IR12 FR11 FR12 IR21 IR22 FR21 FR22

217.99219.87 319.22149.86 171.01 262.79117.89 149.82 148.86 161.98221.82 275.13 167.85 155.80 343.09245.32 326.42 209.41 192.75 342.58 450.25146.69 173.84 195.64 158.04 188.83 232.85 282.79111.40 142.09 139.89 134.69 146.20 185.60 183.89 196.31

Note. N = 161; IRij = jth induction (inductive reasoning) measure at ith assessment occasion, FRij= jth figural relations measure at ith assessment (i = 1, 2; ADEPT induction and Thurstone’s Inductiontests used as 1st and 2nd Induction measures, and ADEPT figural relations and Culture-Fair tests as 1stand 2nd Figural relations measures).

33 44

2 2 2 231 11 32 22 31 32 21 41 11 42 22 41 42 21(16)

2 2ψ ψ

β ψ β ψ β β ψ β ψ β ψ β β ψ�

� � � �

2 2 2 111 1 31 32 22 31 31 32 212 (17)ψ π β β ψ β β β ψ� � �� � �

2 1 2 1 144 41 11 33 1 42 22 33 1 41 42 21 33 12 (18)ψ β ψ ψ π β ψ ψ π β β ψ ψ π� � �� � �

ble constraint, and hence, the null hypothesis (Equation 6) equivalent to it may be re-tained. The last finding suggests that initial induction and figural relations have thesame explanatory power, as a pair of predictors, for subsequent induction and figuralrelations abilities. Their common latent R2 is estimated at 89% in the last fitted, tena-ble model. This result shows a fairly high proportion explained subsequent abilityvariance in terms of earlier assessment induction and figural relations.

CONCLUSION

This article outlined a covariance structure analysis method for testing equality inproportions explained latent variance. The approach can be used with structuralequation models where the focused latent constructs are explained in terms of la-tent independent variables only, but future developments in SEM software maymake it possible to relax this requirement. The proposed procedure does not neces-sitate that the same set of predictors be used for the latent dependent variables of

TESTING EQUALITY IN LATENT R-SQUARES 541

FIGURE 1 Covariance structure model of relations between earlier and subsequent fluid in-telligence abilities, inductive reasoning and figural relations. Factor loadings of the same mark-ers are set equal across time. For simplicity, error terms and latent disturbance terms are not ex-plicitly represented. Covariances between errors of the same markers are represented withcurved double-headed arrows. IR1, IR2 = induction (inductive reasoning) ability at 1st and 2ndassessment point; FR1, FR2 = figural relations ability at 1st and 2nd assessment occasion; AIi =ADEPT Induction test at ith measurement occasion; TIi = Thurstone’s Standard Induction testat ith assessment point; AFi = ADEPT Figural Relations test at ith measurement; CFi = Cul-ture-Fair test at ith assessment (i = 1, 2).

interest, and in fact, these sets do not even need to have any predictor in common.The method is directly extended to the case of k > 2 focused dependent variables,which is accomplished, for example, by introducing k – 1 additional parameters,π1, π2, … , πk-1, defined in the same manner as π1 mentioned earlier (see Equations9 and 14) and fitting a model imposing the corresponding versions of Equations 10or 15 for each of the k – 1 consecutive pairs of focused variables (i.e., introducing k– 1 constraints of the type of restriction [Equations 10 or 15], depending onwhether predictor set is the same or not for the focused variables). The outlined ap-proach is applicable with software allowing the introduction of nonlinear con-straints (e.g., LISREL), and can also be used with models having only observedvariables after introducing associated dummy latent variables equal to them (e.g.,Jöreskog & Sörbom, 1996). The testing procedure does not depend on variable dis-tribution, as long as the appropriate fit function for the manifest variable distribu-tion is used when fitting the model and sample size is large. The proposed methodcan be utilized either in a one tailed or two tailed fashion to examine differences inlatent R2 indexes in a prespecified direction or regardless of such (e.g., Hays,1994). If the approach is also used to generate a hypothesis about a latent explainedvariance relation that is to be tested subsequently, the samples its two applicationsare carried on should be independent of one another to avoid the possibility of cap-italization on chance. The method presented in this article adds a useful instrumentto the armory of behavioral, social, and educational researchers concerned withpredictive power of possibly distinct sets of independent variables for latent con-structs or dependent variables of interest within given covariance structure models.

ACKNOWLEDGMENTS

This research was conducted during a visiting scientist fellowship at the ResearchInstitute on Addictions, SUNY Buffalo.

I am grateful to P. M. Bentler, G. A. Marcoulides, R. E. Millsap, and three anon-ymous referees for valuable discussions on proportion explained latent variance,and to P. B. Baltes, F. Dittmann-Kohli, and R. Kliegl for permission to use datafrom their project “Aging and Fluid Intelligence.”

REFERENCES

Baltes, P. B., Dittmann-Kohli, F., & Kliegl, R. (1986). Reserve capacity of the elderly in aging-sensitivetasks of fluid intelligence: Replication and extension. Psychology and Aging, 1, 172–177.

Bentler, P. M. (2003). EQS structural equations program manual. Encino, CA: Multivariate Software.Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.Hays, L. W. (1994). Statistics. Orlando, FL: Harcourt Brace.Horn, J. L. (1982). The aging of human abilities. In B. B. Wolman (Ed.), Handbook of developmental

psychology (pp. 847–870). New York: McGraw-Hill.

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Jöreskog, K. G., & Sörbom, D. (1996). LISREL8 user’s guide. Chicago: Scientific Software.Muthén, B. O., & Muthén, L. K. (2002). MPLUS user’s guide. Los Angeles: Author.Neale, M. (1997). Mx user’s guide. Richmond, VA: Virginia Commonwealth University, Department of

Psychiatry.Pedhazur, E. J. (1997). Multiple regression in behavioral research. Orlando, FL: Harcourt Brace.Tisak, J., & Meredith, W. (1988). Exploratory longitudinal factor analysis in multiple populations.

Psychometrika, 54, 261–281.Tisak, J., & Tisak, M. S. (2000). Permanency and ephemerality of psychological measures with appli-

cation to organizational committment. Psychological Methods, 5, 175–198.

APPENDIXLISREL Source Code for Testing Equality in Proportions

Explained Latent Variance

TESTINGEQUALITYOFPROPORTIONSEXPLAINEDLATENTDEPENDENTVARIANCE IN TERMS OF LATENT PREDICTORSDA NO=161 NI=8CM=<SEE TABLE 1>LAIR11 IR12 FR11 FR12 IR21 IR22 FR21 FR22MO NY=8 NE=4 PS=SY,FI BE=FU,FI TE=SY,FI AP=1 ! AP = PA(1) = π1

LEIR1 FR1 IR2 FR2VA 1 LY 1 1 LY 3 2 LY 5 3 LY 7 4FR LY 2 1 LY 4 2 LY 6 3 LY 8 4FR PS 1 1 PS 2 2 PS 2 1FR PS 3 3 PS 4 4FR BE 3 1 BE 3 2 BE 4 1 BE 4 2EQ LY 2 1 LY 6 3EQ LY 4 2 LY 8 4FR TE 1 1 TE 2 2 TE 3 3 TE 4 4 TE 5 5 TE 6 6 TE 7 7 TE 8 8FR TE 5 1 TE 6 2 TE 7 3 TE 8 4CO PS(1,1)=PA(1)*BE(3,1)**-2-BE(3,2)**2*PS(2,2)*BE(3,1)**-2- C2*BE(3,1)**-1*BE(3,2)*PS(2,1)CO PS(4,4)=PS(3,3)*BE(4,1)**2*PS(1,1)*PA(1)**-1+ CBE(4,2)**2*PS(2,2)*PS(3,3)*PA(1)**-1+ C2*BE(4,1)*BE(4,2)*PS(2,1)*PS(3,3)*PA(1)**-1ST 1 ALLOU NSNote. The used start values may not be appropriate for other data sets. (The sym-bol “C” denotes continuation of current command on next line.)

TESTING EQUALITY IN LATENT R-SQUARES 543