testing multivariable covariance structure and means hypotheses via structural equation modeling
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Structural Equation Modeling: AMultidisciplinary JournalPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/hsem20
Testing MultivariableCovariance Structure andMeans Hypotheses viaStructural Equation ModelingTenko RaykovPublished online: 19 Nov 2009.
To cite this article: Tenko Raykov (2001) Testing Multivariable CovarianceStructure and Means Hypotheses via Structural Equation Modeling, StructuralEquation Modeling: A Multidisciplinary Journal, 8:2, 224-256, DOI: 10.1207/S15328007SEM0802_4
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Testing Multivariable CovarianceStructure and Means Hypotheses via
Structural Equation Modeling
Tenko RaykovDepartment of Psychology
Fordham University
A didactic collection of multivariate covariance and mean structure hypotheses is pre-sented, which can be tested using structural equation modeling. The hypotheses re-flect specific structures of the manifest covariance matrix or means, which are oftenof interest in social, behavioral, or educational research or represent assumptions ofwidely applied multivariate analysis methods. This large-sample method (a) is gener-ally applicable with normal and nonnormal data (particularly with very large samplesin the latter case), (b) can be considered complementing corresponding likelihood ra-tio tests in the nonnormality case with very large samples, and (c) is straightforwardlyimplemented in widely circulated structural modeling programs such as LISREL,EQS, AMOS, RAMONA, and SEPATH. The approach is illustrated using data froma two-group cognitive intervention study.
Multivariate analysis methods are presently widely used in the social, behavioral,and educational sciences. They permit the researcher to develop models of complexmultidimensional phenomena, estimate their aspects of interest, and test substan-tive hypotheses reflected in their parameters or their relationships. The methods arewell suited to the nature of social research that typically deals with multifactoriallydetermined phenomena whose study usually yields large bodies of data on multiplevariables. Examination of the variable structure in the resulting data sets representsan important part of multivariate analyses. In addition, many multivariate methodsare based on certain, testable assumptions about manifest relationship patterns thatrequire empirical evaluation, and whose fulfillment ensures that the results ob-tained with them are most trustworthy.
STRUCTURAL EQUATION MODELING, 8(2), 224–256Copyright © 2001, Lawrence Erlbaum Associates, Inc.
Requests for reprints should be sent to Tenko Raykov, Department of Psychology, Fordham Univer-sity, Bronx, NY 10458. E-mail: [email protected]
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Utilizations of a special class of multivariate analysis methods, encompassedby the popular structural equation modeling (SEM) methodology, have seenrapid increase in the social and behavioral sciences over the past three decadesor so. Applications of this methodology for theory testing, construct validation,and the study of latent dimensions and their relationships is becoming widespread, as evidenced by a large body of substantive research literature usingSEM across a number of substantive domains. Apparently less often, however,has this methodology been applied for a related goal—testing special patterns ofrelationships between observed variables. Specifically developed likelihood ra-tio tests usually assuming multinormality seem to have been commonly used inpractice, especially for testing hypotheses about variable relationship structuresthat reflect relevant assumptions of subsequently used multivariate statisticalmethods (e.g., Johnson, 1998).
This article presents a didactic collection of covariance and mean structure hy-potheses that can be tested using a widely applicable and easy to employ SEM ap-proach (cf. Bagozzi & Yi, 1989; Cole, Maxwell, Arvey, & Salas, 1993). Themethod is useful when the goal is to examine the observed multivariable structureor test hypotheses regarding interrelationships in a set of measures and large sam-ples are made available. The approach is generally applicable with normal as wellas nonnormal data and very large samples—with such samples it is also applicablewhen the multinormality assumption usually made in corresponding likelihood ra-tio tests is not fulfilled (e.g., Bentler & Yuan, 1999; Bollen, 1989; Hu, Bentler, &Kano, 1992; Yuan & Bentler, 1997, 1998). An extension of this SEM method per-mits testing group mean differences, without requiring covariance matrix homoge-neity—a main assumption of multivariate analysis of variance—or group sizeequality, as well as time invariance in variable means. In addition, this generalmethod is readily implemented with widely circulated structural modeling pro-grams, such as LISREL (Jöreskog & Sörbom, 1993), EQS (Bentler, 1995), AMOS(Arbuckle, 1997), RAMONA (Browne & Mels, 1994), or SEPATH (Steiger,1995). For applied social research, this ease of implementation is particularly at-tractive because a number of those likelihood ratio tests either require more in-volved matrix computation programming (e.g., using SAS/IML; Johnson, 1998;SAS Institute, 1989) or are less readily accessible, as advanced options in specialsoftware modules (Norusis, 1990; SAS Institute, 1992).
The plan of this article is as follows. The SEM method underlying it is first out-lined. Each of several short subsections then presents an observed multivariablecovariance structure hypothesis of interest and how it can be tested with this ap-proach. The method is subsequently extended to permit testing group mean differ-ences, whether or not covariance matrix homogeneity holds or group sizes areequal, as well as stability over time in variable means. A concluding illustrativesection exemplifies its application on data from a two-group cognitive interven-tion study by Baltes, Dittmann-Kohli, & Kliegl (1986).
TESTING MV HYPOTHESIS VIA SEM 225
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BACKGROUND
Notation
This article utilizes notation commonly employed in the multivariate and SEM lit-erature (e.g., Bollen, 1989; Johnson & Wichern, 1998). In it, x = (x1, x2, …, xp)′ is thevector of p manifest variables x1, x2, …, xp that represent measures of research inter-est (p > 1); θdenotes the vector of all parameters of a considered structural equationModel M with a reproduced covariance Matrix Σ(θ); and Σ is the populationcovariance matrix of x, with unbiased sample estimate S (priming stands for trans-position in this article). As usual in practice, throughout the article all consideredcovariance matrices are assumed positive definite, which implies positive variablevariances (e.g., Rao, 1973).
Underlying Model
This article is based on appropriate modifications of a widely used confirmatoryfactor analysis (CFA) model. It can be considered a special case of a generalcovariance structure analysis model underlying most SEM programs, such asLISREL, EQS, AMOS, RAMONA, and SEPATH (Arbuckle, 1997; Bentler, 1995;Browne & Mels, 1994; Jöreskog & Sörbom, 1993; Steiger, 1995). This CFA modelis defined by the equation
x = Λ ξ + δ (1)
where ξ is a q × 1 vector, typically consisting of latent variables, with covarianceMatrix Φ; Λ is a p × q factor loadings matrix; and δ is a p × 1 error terms vector withzero mean and covariance Matrix Θ (q > 0; cf. Jöreskog & Sörbom, 1993). As usual,the uncorrelatedness of latent variables and error terms is also assumed here, that is,Cov(ξ, δ) = 0q × p where Cov(.,.) denotes matrix of pertinent covariances, and 0q × p isa q × p matrix consisting of zeroes only. Similarly, following widely adopted con-vention, all variable means are assumed vanishing unless otherwise indicated (seelater section).
For the purposes of this article, repeated use is made of a special case of themodel in Equation 1. This special case is obtained with
Λ = Ip and Θ = 0p × p (2)
where Ip is the identity matrix of size p × p and 0p × p is the p × p matrix with zero ele-ments. For simplicity, the model defined by Equations 1 and 2 are referred to asmodel M in the remainder. Model M can be seen as resulting after q = p dummy la-
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tent variables ξ1, ξ2, …, ξp are introduced, which equal each a corresponding ob-served variable, namely, ξi = xi (i = 1, …, p). Equation 1 implies then the identity ofthe observed and “latent” covariance matrices, that is,
Σ = ΛΦΛ′ + Θ = IΦI′ + 0p × p = Φ (3)
(cf. Jöreskog & Sörbom, 1993).Equation 3 is of special benefit when testing manifest multivariable covariance
structure hypotheses. This is because the observed covariance Matrix Σ, beingequal to the latent covariance Matrix Φ, has now each of its elements accessible tothe researcher. In fact, for every of the multivariate covariance structure hypothe-ses considered in the following, the CFA model defined by Equations 1 and 2 isfurther specialized by the introduction of such restrictions on elements of Φ = Σ,which reflect the assertions of the hypothesis.
A path diagram of Model M under consideration, for the special case of p = 4observed variables, is depicted in Figure 1 used in the illustrative section where itis applied to test a number of multivariate hypotheses on data from a cognitive in-tervention study.
TESTING MV HYPOTHESIS VIA SEM 227
FIGURE 1 The basic CFA Model M, for the p = 4 fluid intelligence measures of concern inthis illustration section, as defined in Equations 1 and 2. A corresponding special case of M, or itsextension M′ (Equation 15), with appropriate parameter constraints is used later to test each ofthe observed variable relationship patterns or (group) mean difference hypotheses. (Notationidentical to that of Table 1.)
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Analytic Method
Model M in Equations 1 and 2, considered as parameterizing all elements of thecovariance Matrix Φ, is saturated. This is due to the fact that it has as many parame-ters as there are nonredundant elements of Φ (= Σ; these are the elements along and,say, below its main diagonal). Hence, if fitted to the sample covariance Matrix S, Mwill yield a perfect fit. Thus, once the researcher introduces in M restrictions corre-sponding to a hypothesis of interest (as exemplified in the next section), testing es-sentially any hypothesis about the structure of the population covariance Matrix Σof the observed variables becomes possible using this method (with large samples).This is achieved by the comparison of the chi-square fit indexes of the so-con-strained model and the saturated Model M. More specifically, the difference be-tween the chi-square values of the restricted model, χ2, and of M, namely 0, repre-sents a large-sample test of the corresponding multivariable structure hypothesisspecified by the restrictions (see next section). That is, its large-sample test statisticis χ2 – 0 = χ2 (e.g., Jöreskog & Sörbom, 1993). This statistic can be judged for sig-nificance by comparing its value to the difference in degrees of freedom of bothmodels, namely, d – 0 = d, with d being the degrees of freedom of the restrictedmodel.
When the analyzed variables are multivariate normally distributed, the re-stricted model is fitted using the maximum likelihood (ML) method minimizingthe fit function (e.g., Bollen, 1989):
FML(S,Σ(θ)) = –ln|SΣ–1| + tr(SΣ–1) – p (4)
where | . | denotes determinant and tr(.) matrix trace. The chi-square value men-tioned previously is obtained as
χ2 = (N – 1)FML,min (5)
where FML,min denotes the attained minimum of the fit function in Equation 4 and Nis the sample size. This chi-square value is a realization of a random variable thatasymptotically (i.e., with large samples) follows a chi-square distribution, with de-grees of freedom being those of the model, under the hypothesis that the fittedmodel is true (e.g., Bentler, 1995).
When the multinormality assumption is up to mildly violated, the chi-square inEquation 5 could still be used with some caution as an overall fit index (e.g.,Jöreskog & Sörbom, 1993). In particular, Satorra-Bentler’s scaled test statistic andstandard errors (Bentler, 1995) have much to recommend then (e.g., Chou,Bentler, & Satorra, 1991), as have recent robust SEM methods (Yuan & Bentler,1997, 1998; see also Bentler & Yuan, 1999). With more serious violations of nor-
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mality these robust approaches and, with very large samples, the asymptoticallydistribution-free method (ADF; e.g., Browne & Arminger, 1995; Hu et al., 1992)can be used, for example, after specific corrections of the latter (Yuan & Bentler,1997; Yung & Bentler, 1994). The ADF method minimizes the fit function
FADF(S,Σ(θ)) = (s – σ(θ))′ W–1 (s – σ(θ)) (6)
where s and σ(θ) are the strung-out vectors containing all nonredundant elementsof S and Σ(θ), respectively, and W–1 is a special weight matrix taking into accountfourth-order moments of the observed variables (e.g., Browne & Arminger, 1995).The fit function in Equation 6 is more general than that in Equation 5, with the latterfit function being asymptotically equivalent to a special case of Equation 6 (e.g.,Bentler, 1995). The test statistic with the ADF method for the correspondingmultivariable covariance structure hypothesis in the case of nonnormality equalssimilarly the pertinent chi-square value of the restricted model,
χ2ADF = (N – 1)FADF,min (7)
where FADF,min is the attained minimum of the fit function in Equation 6 (e.g.,Bentler, 1995). With very large samples, the statistic in Equation 7 presents a possi-bility of testing patterns of relationships in observed variable sets beyond themultinormality case typically addressed by conventional likelihood ratio tests ofsuch hypotheses (e.g., Bock, 1975; Johnson, 1998; Timm, 1975). Hu et al. (1992)provided an extensive and informative study of its performance across a variety ofdesign conditions, and for many of them justify a requirement for a very large sam-ple size. As mentioned before, recent correction and robustness studies by Yungand Bentler (1994), Yuan and Bentler (1997, 1998), and Bentler and Yuan (1999)presented modifications of the ADF and related methods with improved perfor-mance at less than (very) large samples. We emphasize, however, that applicationof the ADF method typically requires very large samples, which may present apractical limitation to the social science researcher in particular empirical settings.
As an informative adjunct to the test statistics in Equation 6, and similarlyEquation 7, the root mean square error of approximation (RMSEA) can be used(Steiger & Lind, 1980; see also Browne & Cudeck, 1993; and Steiger, 1998, for itsmultipopulation extension). Because it is based on an estimate of the minimalvalue of the fit function if the restricted model were fitted to the populationcovariance matrix, the RMSEA also informs about the extent of violation of thetested hypothesis in the data whereby account is simultaneously taken of modelcomplexity. It is therefore especially useful when assessing the degree of departureof the empirical covariance matrix from the hypothesized or tested multivariablestructure.
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With multiple studied populations, this SEM approach is directly applicable byextending the test statistics in Equations 6 and 7 to a weighted sum of the fit func-tions in each group (e.g., Bentler, 1995). In case of multinormality within each ofsay r > 1 groups, the extended fit function for the ML method is
(r)F = n1FML(1)(S1,Σ(θ1)) + n2 FML
(2)(S2,Σ(θ2)) + … + nr FML(r)(Sr,Σ r(θr)) (7r)
where FML(s)(Ss,Σs(θs)) is the ML fit function in Equation 4 for the sth group with
sample size Ns = ns + 1, s = 1, …, r. Thereby, cross-group equality constraints maybe tested in the same way with nested models, whose restrictions reflect group-in-variant aspects of the multivariable hypothesis (see next section; Bentler, 1995).The multigroup extension in the nonnormality case is completely analogous to thatin Equation 7r, the only change being the use of the corresponding fit function 6 ineach group (i.e., with FADF
(s)(Ss,Σs(θs)) instead of FML(s)(Ss,Σs(θs)) in the right-hand
side of 7, s = 1, …, r).1
TESTING COVARIANCE MATRIX HYPOTHESES WITHSTRUCTURAL MODELS
In this section, a sequence of hypotheses about the population covariance Matrix Σof a set of observed variables, x1, x2, …, xp, is discussed, which are tested with thepresent large-sample SEM method.
Equality to a Prespecifed Covariance Matrix
Here the hypothesis specifies the value of each element of the populationcovariance matrix. This hypothesis is of interest when previous extensive researchin a substantive domain has suggested (relatively precise) values of all interrela-tionship indexes of a set of observed variables, and the researcher wishes to exam-ine the empirical evidence in favor or against the assertion that the relationshipspattern of the measures is representable by this set of numbers. Thus, the null hy-pothesis is
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1As a general comment to all multivariable hypotheses tested in this article (and others that could betested with this or related SEM methods), if the restricted model is rejected—and thus its correspondingnull hypothesis disconfirmed—modifications indexes pertinent to the restrained elements of thecovariance Matrix Φ = Σ (or means, see later) might suggest which parts of the model contribute mark-edly to its rejection (e.g., Jöreskog & Sörbom, 1993). Specifically, the elements of Φ there might indi-cate restrictions on variable relationships that do not follow the restraints imposed on them by the testedhypothesis. Such suggestions need to be tested by corresponding restraint relaxations and comparison ofthe resulting chi-square difference of the two resulting nested models, in the way indicated previously.
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1Ho: Σ = Σo
where Σo is a prespecified covariance matrix (of the same size as Σ) with typical ele-ment σo,ij (i, j = 1, …, p).
Using the present SEM method, this hypothesis is tested against the alternative1H1: Σ ≠ Σo by testing the model obtained through adding the following restrictionsto the saturated Model M defined in Equations 1 and 2:
φij = σo,ij, i, j = 1, …, p (8)
where φij is the typical element of Φ(i, j = 1, …, p). The straightforward implemen-tation of this test in the widely circulated program LISREL (Jöreskog & Sörbom,1993), in form of input file, is provided in the Appendix, utilizing data from the il-lustrative section.
As mentioned before, this large-sample SEM test is generally applicable inthe normality as well as nonnormality case for observed variables (particularlywith very large samples in the latter case; see earlier section). Withmultinormality, the likelihood ratio test of 1Ho: Σ = Σo is stated, for example, inJohnson (1998).
Sphericity
This hypothesis asserts that there are no (linear) relationships between the studiedmeasures and that they are equivariable. It is of interest when a preliminary aim is toreduce a given data set on a large number of variables, for example, in principalcomponent analysis or factor analysis. If this assumption is correct, there is no pointin pursuing such data reduction because this hypothesis implies that there are nocommon sources of variability in the data and the coordinate axes are already inmaximum variability directions (and in addition exhibit equal variability alongthem). Furthermore, it is important to have an easily applicable test for sphericitybecause its holding for appropriate orthonormally transformed repeated measurelevels is necessary and sufficient for the validity of corresponding F tests inwithin-subject designs (e.g., Timm, 1975; see next subsection).
Thus, the null hypothesis to test here is
2Ho: Σ = σ2Ip
where σ2 is the common variance of the measures. With the present SEM method,2Ho is tested against the alternative 2H1:Σ ≠ σ2Ip by evaluating the fit of the modelobtained from M in Equations 1 and 2 after adding the equality restrictions on thediagonal elements and fixing to 0 the off-diagonal elements of the Matrix Φ:
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φ11 = φ22 = … = φpp , and φij = 0, i ≠ j = 1, …, p (9)
The implementation of this test is provided by the second LISREL input file inthe Appendix. We note that in the mulinormality case this is a test of variable inde-pendence (and variance homogeneity), because then independence is implied fromuncorrelatedness (e.g., Johnson & Wichern, 1998).
A useful relaxation of 2Ho is also testable with this method. The relaxationstates that, although there are no (linear) relationships, the variables may have dif-ferent variances. That is, the null hypothesis is that Σ is a diagonal matrix with pos-sibly differing diagonal elements:
21Ho : Σ = diag(σ11, σ22, …, σpp)
where diag(. , . , …, .) denotes diagonal matrix with the stated elements along themain diagonal. The Hypothesis 21Ho is tested with the model resulting from M inEquations 1 and 2 after imposing only the zero-fixing restrictions on the off-diago-nal elements of Φ:
φij = 0, i ≠ j = 1, …, p (10)
The LISREL input file for testing 21Ho is obtained from that of 2Ho by merely delet-ing the line imposing the variance equality restriction (the second to last line; seeAppendix). Like the preceding test, in the mulinormality case, that of 21Ho, is in facta test of variable independence (without necessarily variance homogeneity).
As indicated earlier, either of the two large-sample SEM tests in this subsectionis in general applicable in the normality as well as nonnormality case (in the latterwith very large samples; see earlier section). With multinormality, the likelihoodratio tests for the hypotheses 2Ho and 21Ho are given, for example, in Johnson andWichern (1998).2
Compound Symmetry
A covariance Matrix Σ exhibits compound symmetry if it has the same elementsalong its main diagonal and equal (possibly to another number) elements off it. Thisfeature becomes of special relevance in repeated measure designs. The reason isthat here compound symmetry of the covariance matrix for the levels of the re-
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2The result of the test of 21Hois also provided by LISREL, EQS, AMOS, RAMONA, and SEPATH, aspertinent to the so-called “independence model.” The present SEM test is included here only for pur-poses of completeness of the article and exemplification of generality of this SEM method.
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peated factor (within-subject factor) is (a) sufficient for the corresponding F ratiosto follow an F distribution and therefore be evaluated with reference to the latter;and at the same time (b) easier to evaluate than sphericity of the covariance matrixpertaining to orthonormally transformed levels that would necessitate first workingout the needed contrast matrix, then applying it on the levels, and finally testing theresulting covariance matrix for sphericity via a test for 2Ho of the preceding subsec-tion. Even though when sphericity (or compound symmetry, for the same matter, incorresponding cases) does not hold corrections of the degrees of freedom of these Fratios can be performed and the F test still conducted, the latter usually becomesconservative (to an extent related to which correction is used; e.g., Timm, 1975; seealso Maxwell & Delaney, 1990).
Because compound symmetry necessitates that the correlations between anytwo levels of the repeated factor be identical (e.g., Maxwell & Delaney, 1990), in-dependently of its sufficiency for the validity of the F tests this pattern of the corre-lation matrix is also of interest in its own right when it is of relevance for asubstantive question to examine the evidence in the sample for identity of the inter-relationship indexes among all repeated measure levels. An informal and incon-clusive check could obviously involve a direct comparison of these correlations.However, a formal test of this covariance structure is possible using the SEM ap-proach in this article. The null hypothesis to test is
30
2
1
1H :
. .Σ =
σ
ρ ρρ ρ
. .ρ ρ 1
……
…
for some σ2 and ρ. The alternative Hypothesis 3H1 asserts that there are no two num-bers σ2 > 0 and ρ with the property that Σ has all its main diagonal elements equal toσ2 and all off-diagonal entries equal to ρσ2. Using the present SEM approach, 3Ho istested with the model obtained from M defined in Equations 1 and 2 by adding thefollowing restrictions:
φ11 = φ22 = … = φpp , andφij = φi′j′ , i ≠ j, i ′ ≠ j′ (i, j = 1, …, p, i′, j′ = 1, …, p)
(11)
The LISREL input file for this test is provided in the Appendix. Like precedingutilizations of the present method, this large-sample SEM test is generally applica-ble with normal as well as nonnormal data (with very large samples). The likeli-hood ratio test of 3Ho with normal data is presented, for example, in Johnson(1998). We note that the test of 3Ho can be considered a generalization of that of2Ho, whereby the former hypothesis is obtained from the latter by relaxing the re-
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quirement of all off-diagonal elements being equal to zero yet still keeping themconstrained for equality.
Uncorrelatedness of Subsets of Measures
This covariance matrix pattern is of importance when the manifest variables canbe partitioned into subsets of variables, for example, based on some substantivecriterion or previous research, and the analyst wishes to examine whether thesesubsets are uncorrelated (without any restrictions on their relationships withinthe subsets). For example, with two variable subsets this test is relevant whencontemplating carrying out canonical correlation analysis on a large set of mea-sures of two substantively distinct types. Should this hypothesis or assumptionbe retained, there is no point in conducting the canonical analysis as then thetwo variable subsets are unrelated. This test is a generalization of the earlier testof variable uncorrelatedness (see previous subsection on testing for sphericity),with the latter hypothesis being obtained from the present one when each vari-able subset contains only one measure.
For this test, first assume that the partition of the population Matrix Σ, whichcorresponds to the variable subsets, is as follows:
Σ
Σ Σ ΣΣ Σ Σ
Σ Σ Σ
=
11 12 1
21 22 2
1 2
……
…
k
k
k k kk
. . . .
where Σhg is a ph × qg matrix (ph < p, q; < p, h, g = 1, …, k; p1+p2+…+pk = p = q1 +q2+…+qk). Under the null hypothesis of interest, Σ is a block-diagonal matrix, thatis,
40
11
22H
kk
:. . . .
Σ
ΣΣ
Σ
=
0 0
0 0
0 0
……
…
where the symbol 0 denotes for simplicity correspondingly sized matrices consist-ing of zeros only.
With the present SEM approach, 4Ho can be tested against the alternative that atleast one element of Σ, which lies outside of Σhh (h = 1, …, k), is nonzero. This isaccomplished by testing the model obtained from M in Equations 1 and 2 throughadding the following restrictions on Φ:
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φhg,ij = 0 (i = 1, …, ph, j = 1, …, qg, h, g = 1, …, k) (12)
for all elements of the corresponding (to Σ’s partitioning) submatrices Φhg of Φ, h >g; h, g = 1, …, k. Thus, the restricted model (a) does not constrain any element ofΦhh along the main diagonal of Φ (i.e., the main diagonal blocks); and (b) fixes to 0all elements of Φ outside Φhh, h = 1, …, k. Its LISREL input file is similarly pro-vided in the Appendix.
An extension of this test is possible when interest lies in the modified null Hy-pothesis 41Ho resulting from 4Ho by (a) relaxing some elements of (some or all)off-diagonal matrix blocks Σhg from their restriction to 0 in Equation 12 (h > g; h, g =1,…,k);or (b) imposingon the lattermatricesother restrictionsof interest, forexam-ple, Σhg = oΣhg being prespecified matrices (h > g; h, g = 1, …, k). The test of 41Ho isaccomplished with Model M in Equations 1 and 2 after imposing Equation 12 onlyfor the elements that are not of concern for (a), or introducing the corresponding al-ternative restrictions on the elements of Σhg for (b), (h > g, h, g = 1, …, k).
As before, the discussed large-sample tests of 4Ho or 41Ho are generally applica-ble whether or not normality is violated (and with very large samples in the lattercase). With multinormality, the test is one of variable subset independence, be-cause then the latter is implied from subset uncorrelatedness (e.g., Bock, 1975). Inthis normality case, the likelihood ratio test of 4Ho is given, for example, in Timm(1975) and the one for 41Ho can be similarly obtained.
Group Invariance of Variable Interrelationship Indexes
In multivariate analysis of covariance (MANCOVA), group identity of thecovariance matrix of the dependent variables and the covariates is a sufficient con-dition for identity of the pertinent regression hyperplanes, whose identity repre-sents an important MANCOVA assumption (e.g., Timm, 1975). This assumptiongeneralizes the regression homogeneity requirement (equality of regression slopesacross groups) in univariate analysis of covariance. Violation of it leads to poten-tially misleading statistical and substantive conclusions, because the MANCOVAresults could then be affected by a substantial under- or overadjustment for groupdifferences on the covariates (e.g., Huitema, 1980; Tabachnick & Fidell, 1996). It istherefore desirable to have an easily applicable means in practice for testing thissufficient condition. The SEM method in this article offers such a means.
If the dependent variables are collected in the vector y = (y1, y2, …, yu)′ and thecovariates in z = (z1, z2, …, zv)′, where u and v are integers, for r > 0 studied groupsthe null hypothesis corresponding to this homogeneity assumption is
5H o : Σyz(1) = Σyz
(2) = … = Σyz(r)
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where Σyz(s) = Cov(y, z) is the covariance matrix of dependent variables and
covariates in the sth group, s = 1, …, r.To test 5Ho, the following procedure is used. First, postulate in each of the r
groups the saturated Model M defined in Equations 1 and 2 with altogether p = u +v manifest variables, namely, x1 = y1 (= ξ1), …, xu = yu (= ξu), xu+1 = z1 (= ξu+1), …,and xp = zv (= ξp). Then impose the following group-invariance restraints on theoverall covariance Matrix Φ(= Σ = Cov(x,x)):
φyz,ij(1) = φyz,ij
(2) = … = φyz,ij(r) (13)
where φyz,ij(s) is the typical element of the dependent variables and covariates
covariance Matrix Φyz(s) = Σyz
(s) = Cov (y, z) in the sth group, i = 1, …, u, j = 1, …, v, s= 1, …, r.
As seen from Equation 13, this restricted model constrains for equality acrossall groups each element of the covariance matrix of dependent variables andcovariates, but leaves free all other elements of Φ = Σ. (Note that unlike allcovariance matrices considered thus far, Σyz(s) = Φ yz(s) need not be symmetric oreven square, s = 1, …, r.) Like the preceding tests, this large-sample SEM-basedone is generally applicable with normal as well as nonnormal data (and very largesamples in the latter case). With multinormality, a likelihood ratio test of 5Ho ispresented, for example, in Timm (1975).3
A special case: Testing homogeneity of a dependent variables’covariance matrix. An important special case of this method permits testingthe null hypothesis of homogeneity of the covariance matrix of all dependent vari-ables. This test is obtained from the immediately preceding one for y = x = z, andevaluates the null hypothesis
6H o: Σ(1) = Σ(2) = … = Σ(r)
where Σ(1), Σ(2), …, Σ(r) are the r group covariance matrices of the observed variables inx. (This special case of the previous test of 5Ho is identical to that by Jöreskog &Sörbom, 1993.) The null Hypothesis 6Ho is of major relevance for a number ofmultivariate analysis methods, for example, the multivariate approach to repeatedmeasure analysis, multivariate analysis of variance, (linear) discriminant analysis.The popular Box’s M test (e.g., Johnson, 1998) is oftentimes used in practice to test6Ho, but like Bartlett’s test of variance homogeneity (of which the former is amultivariate generalization), it is quite sensitive to nonnormality and thus cannot dis-
236 RAYKOV
3This is not the only assumption of MANCOVA (e.g., Huitema, 1980), and this discussion is notmeant to imply that it is the only important, or most important, assumption.
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entangle it from lack of homogeneity (e.g., Morrison, 1976). Unlike these two tests, asimplied from the immediately preceding subsection, its large-sample SEM test with y= x = z—and thus postulating Equation 13 for all i≠ j (i, j = 1, …, p)—offers an alterna-tivewayof testingcovariancematrixhomogeneityacrossgroups in thegeneralcaseofnonnormality with very large samples (see earlier section).
Time Invariance of a Variable Relationship Pattern
The tests discussed so far in this section assumed that the analyzed groups were ran-dom samples from different populations. Sometimes it can be of interest to test ifthe pattern of relationships between several observed variables remains stable overtime, for example, in repeated measure designs where it may be of concern to testidentity of the covariance matrix of a variable set across two or more assessments.For example, it is frequently desirable to find out if a specific training program hasproduced no changes in the variable interrelationship pattern (e.g., from pretest toposttest). Similarly, examining stability of certain behavioral characteristics (the“flip side” of change over time) can be of main interest in an empirical study, for ex-ample, in a context of late life maintenance of cognitive functioning. The featurethat sets this type of concern apart from the ones dealt with up to this point is the factthat the same individuals have been observed at the repeated assessment occasions,and, unlike the previous tests, one needs to account for the so-induced dependen-cies between the measurements.
To show how the present large-sample method of testing multivariablecovariance structures can be used to examine time-invariance of interrelationshippatterns, denote the observed variables at lth assessment as x(l) = (xl1, xl2, …, xlp)′, l= 1, …, m, and enclose all variables across all measurement occasions into theoverall vector x = (x(1)′, x(2)′,…, x(m)′)′. The hypothesis of time invariance of the ob-served variable pattern can be represented by the equality of the manifestcovariance matrices across measurement points:
7Ho: Σ(1) = Σ(2) = … = Σ(r)
where Σ(l) = Cov(x(l), x(l)) are the covariance matrices of x(l), l = 1, …, m. The matricesΣ(l) (l = 1, …, m) represent the submatrices of the covariance matrix of x, Σ, whichcorrespond to the partitioning x′ = (x(1)′| x(2)′|…| x(m)′). Alternatively, the null hypoth-esis can include only a subset of the repeated assessment points, implying obviousmodifications in the following Equation 14 and subsequent model fitting.
To test 7Ho with this SEM method, one fits to the sample covariance Matrix S ofx the Model M defined in Equations 1 and 2, with the added constraints
φij(1) = φij(2) = … = φij(m) (14)
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on all elements φij(l)of the covariance matrices Φ(l) = Σ(l)at lth measurement occasion(i ≠ j, i, j = 1, …, p, l = 1, …, m). As before, this large-sample SEM test is generallyapplicable in the multinormality case as well as in the case of nonnormality withvery large samples.4
TESTING MEAN GROUP DIFFERENCES WITHUNCONSTRAINED RELATIONSHIP BETWEEN
COVARIANCE MATRICES
Motivation
A major assumption of a number of multivariate analysis methods, such as (linear)discriminant analysis, some repeated measures designs, and especially MANOVA,is the group identity in the covariance matrix of analyzed variables. When this as-sumption is violated, the results of statistical tests may be misleading as demon-strated repeatedly in the literature (e.g., Johnson & Wichern, 1998; Morrison,1976). However, relatively limited research seems to be available on the robustnessof MANOVA then, which provides clear practical guidance and in particular meth-odological alternatives (e.g., Algina & Coombs, 1996; Coombs, Algina, & Oltman,1996; Keselman & Lix, 1997; Maxwell & Delaney, 1990; Olson, 1976; see alsoCole et al., 1993). Although it is well known that MANOVA will likely be affectedto a limited extent with different covariance matrices in the groups as long as sam-ple sizes are equal (and large; e.g., Tabachnick & Fidell, 1996), it seems that manyempirical cases when covariance matrix homogeneity is untenable deal with con-siderably differing sample sizes. Whereas it is possible to drop randomly chosen in-dividuals from the larger group(s) to achieve equality in group size and rely on thepreviously mentioned robustness property, in many cases—particularly with largediscrepancies in the sample sizes—such an approach may be at least questionableas it is wasteful of empirical information from the original samples (see alsoKeselman & Lix, 1997).
238 RAYKOV
4To test any of these covariance structure hypotheses for the correlation Matrix P rather thancovariance Matrix Σ (with single samples, and correspondingly modified hypotheses as indicated in thefollowing), only a minor modification of the approach used so far is needed. The same basic CFA ModelM in Equations 1 and 2 is then employed, but any model is fitted to the correlation, rather thancovariance, matrix using RAMONA (Browne & Mels, 1994) or another SEM program that implementsthe correct correlation distribution theory (e.g., EQS6, Bentler, personal communication; SEPATH,Steiger, 1995). The effect is that all latent variances are automatically fixed to unity by the constraints inEquation 2, and thus Φ (= Σ) becomes the correlation matrix. For each hypothesis to be tested then, thecorresponding restrictions on the elements of Φ are introduced (keeping in mind that Φ = Σ is now thecorrelation matrix and thus has the constant 1 along its diagonal). Alternatively, a modelreparameterization can be used that allows one to use the covariance distribution theory and is outlinedin Bentler, 1995.
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This suggests the desirability of such a means for testing group mean differ-ences, which does not assume particular covariance matrix relationships (e.g., ho-mogeneity) or equality of sample sizes. Such an approach that is readily applied inpractice is provided by the large-sample SEM method in this article. To use it inthis particular context, one needs to analyze the mean structure of the variables foreach group (e.g., Bentler, 1995). This proceeds on the basis of an extension ofModel M in Equations 1 and 2, which is defined next. As indicated in the earliersection, M made the conventional assumption that all variable means were zero.This assumption is commonly adopted when covariance structures are examined(e.g., Jöreskog & Sörbom, 1993), because it is convenient and does not limit thegenerality of the obtained results, and because the covariance matrix of x is identi-cal to the covariance matrix of the mean-corrected observed variables (e.g., John-son & Wichern, 1998; cf. Cole et al., 1993).
Basic Model’s Extension
When group means are of interest and do not vanish in the population, the assump-tion of zero variable means obviously needs to be abandoned. To respond to this ne-cessity, this section is based on the following modification of Model M (E(.) de-notes mean below):
x = Λξ + δ, Λ = Ip ; Θ = 0q × xp, E(x) (= E(ξ)) = µ, E(δ) = 0 (15)
where in addition to the notation used with the earlier Equations 1 and 2 the symbol0 denotes for simplicity the p × 1 vector consisting of zeroes only. For ease of refer-ence, we denote the model defined in Equation 15 by M′. Model M′ differs fromModel M only in allowing the mean of x = ξ to be possibly distinct from zero (com-pare Equations 1 and 2 with 15).
Procedure
When testing for mean differences across say r > 1 groups under consideration, thenull hypothesis is
8Ho: µ(1) = µ(2) = … = µ(r)
where µ(s) = E(x(s)) designates the observed variable means in the sth group, s = 1,…, r.
With the present SEM method, to test 8Ho, (a) postulate Equation 15 in each ofthe r groups (i.e., introduce the latent means µ from Equation 15 in all groups asfree parameters); and (b) set all group means correspondingly equal to one another
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(cf. an alternative procedure in Jöreskog & Sörbom, 1993, chapter 10). That is, theonly added constraint to M′ defined in Equation 15, after postulating it in the rgroups, is identical to the statement of 8Ho:
µ(1) = µ(2) = … = µ(r) (16)
We note that Model M′, when postulated in each of the r groups without Equa-tion 16, is saturated. Hence, imposing in this multisample version of M′ the restric-tion of concern here, namely Equation 16, makes it overidentified. Thus, as in thepreceding section, the difference in the chi-square values of the so-restrictedmodel, χ2, and of the multisample version of M′, 0, represents a large-sample teststatistic for 8Ho, namely, χ2 – 0 = χ2. If the null Hypothesis 8Ho is rejected, inspec-tion of the modification indexes pertaining to the latent mean constraints may sug-gest components of them that if freed from the group identity restriction maysubstantially improve the fit (see Footnote 1). Releasing them and testing the re-sulting difference in chi-squares against the difference in degrees of freedom pro-vides a test of separate variable mean differences across groups. This examinationcan also be achieved by comparing these means’ confidence intervals obtainedfrom the saturated multisample Model M′ (or using the sample means and theirstandard errors) at appropriately protected significance levels with, for example,Bonferroni’s method, for overlapping over groups (e.g., Raykov, 1995, and illus-trative section; for alternative multiple hypotheses testing methods, see Shaffer,1995).
We emphasize that the present large-sample SEM method of examining groupmean differences only assumes Equation 15, and additionally 16 when testing thenull Hypothesis 8Ho. That is, the approach is applicable whether or not (a) thecovariance matrix of the observed variables x is the same across the groups that asmentioned is an important assumption of MANOVA, and (b) the group sizes areequal (cf. Cole et al., 1993). Similarly to the previous tests, this one is generally ap-plicable with multinormality as well as nonnormality—particularly with verylarge samples in the latter case (e.g., Hu et al., 1992). Furthermore, this method isapplicable with orthogonal or nonorthogonal, one- or multiway designs wherebythe r groups represent their cells pertaining to a specific null hypothesis of interest(not necessarily involving all design cells). In case of multinormality, covariancematrix homogeneity and small samples, however, this method is not to be pre-ferred to a corresponding MANOVA, because the latter is a finite sample ratherthan a large-sample approach as the present one.5 Similarly, with small samples
240 RAYKOV
5An alternative method of testing the null Hypothesis 8Ho, as well as 6Ho, using (an earlier version of)LISREL is provided by Kuhnel (1987). His approach leads to the same results as this section’s methodfor testing 8Ho or 6Ho, but the parameterization underlying his (a) is distinct from the present one, (b) ismore involved to utilize in practice, and (c) uses an alternative mode of analyzing means (viz. fitting the
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(e.g., under, say, 100), MANOVA approaches deserve closer consideration andgeneral preference, because a main assumption of the SEM method discussed inthis article is not fulfilled, namely that of large samples. For small to moderatelylarge samples, and particularly with covariance heterogeneity and considerablydisparate group sizes, further research is needed that could provide guidance be-tween the two methods and shed light on specific conditions under which one ofthem could be preferable.
A Test of Time Invariance in Variable Means
A modification of the method outlined in this section is also applicable in amultivariate repeated measure design with a single group, when the concern is withtesting for mean stability (i.e., no change) over time in a set of longitudinally as-sessed variables. Suppose the set of p measures (x1, x2, …, xp) is repeatedly observedat t assessment occasions, and the recorded variables are denoted as x(l) = (xl1, xl2, …,xlp)′, l = 1, …, t. As before, enclose all variables across all repeated measurementpoints into the overall vector x = (x(1)′, x(2)′,…, x(t)′)′. Testing time-invariance in themeans of x(l), l = 1, …, t, amounts to testing the null hypothesis
9Ho: µ (1) = µ (2) = … = µ(t)
where µ (l) is the mean of x(l), l = 1, …, t.To test 9Ho with the method in this section, use a modification of the procedure
outlined in the preceding subsection. To this end, postulate Model M′ in Equation15 for the overall vector x and constrain for equality the means of the vectors x(l), l= 1, …, t, for example, introduce the following restrictions reflecting 9Ho:
µ(1) = µ(2) = … = µ(t) (17)
Similarly to the preceding developments, the difference in chi-square values ofModel M defined in Equation 15, with and without the additional constraints of
TESTING MV HYPOTHESIS VIA SEM 241
model to the cross-products-around-zero matrix, as opposed to the covariance/mean matrix of concernhere; his fitting approach is unnecessary since the release of LISREL7 some 10 years ago). Hisparameterization represents each individual case data as group mean plus deviation from it (i.e., “er-ror”), and thus the covariance Matrix Σ becomes the error covariance Matrix Θ (rather than the latentcovariance Matrix Φas in this article; see Equations 1 and 2). To obtain the variable means then, in effectone has to regress each observed variable on a dummy unit constant added to the original data set. Wetherefore find Model M′ of this section (Equation 15), in contrast to Kuhnel’s, to be (a) direct, (b) using amore general and natural parameterization, (c) readily portable across SEM programs, and (d) easier andquicker to use in practice (compare the short LISREL input files 6 and 8 for testing these hypotheses 8Ho
and 8Ho in this article’s Appendix with the ones in Kuhnel’s article).
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Equation 17 (both times with the same other restrictions if any imposed before onparameters of the latent covariance Matrix Φ), represents a large-sample test statis-tic of 9Ho. We note that this test requires only 15, and 17 when 9Ho is evaluated, andlike all SEM tests in this article is generally applicable with normal data as well as inthe nonnormality case—particularly with large samples in the latter (e.g., Hu et al.,1995). As an alternative approach in the multinormality case, Hotelling’s T2 testcan be used with a correspondingly constructed contrast matrix (e.g., Morrison,1976); with lack of normality and large samples, a counterpart χ2 test asymptoti-cally obtained from the T2 test can be used (e.g., Johnson & Wichern, 1998). Wenote that this T2 test has a finite-sample rather than asymptotic nature and is there-fore to be preferred to the current large-sample one in the case of small samples.
ILLUSTRATION ON DATA FROM A TWO-GROUPCOGNITIVE INTERVENTION STUDY
Here data from Baltes et al.’s (1986) two-group cognitive intervention study of re-serve capacity in fluid intelligence, with four consecutive assessments, is used to il-lustrate the present SEM approach to testing multivariable structure hypotheses.For the specific purposes of this section, we will focus on data from four fluid intel-ligence tests measuring the fluid intelligence subabilities induction and figural rela-tions and two perceptual speed tests (e.g., Horn, 1982). Baltes et al. (1986) de-scribed in detail the contents of the tests, study design, and training programadministered to the experimental group, which targeted induction and figural rela-tions. For further information regarding preliminary data analyses that did not findserious violations of multinormality, see also Raykov (1995). The samplecovariance matrices of concern to the majority of the analyses in this section arepresented in Table 1. Because we are going to carry out 10 tests altogether on thisdata set, using Bonferroni’s method (e.g., Bock, 1975) we first determine as a con-servative significance level for each of them α* = .05/10 = .005 (see Shaffer, 1995,for alternative approaches).
Testing Covariance Structure Hypotheses
First we are interested in testing whether the covariance matrix of the four fluid in-telligence measures—three measuring induction and one figural relations—equalsat first posttest in the experimental group the following covariance Matrix Σo ob-tained in another study. (Only the lower half of that symmetric matrix is reproducedin Table 2.)
The path diagram of Model M defined by Equations 1 and 2 for this case of p =4 observed variables is depicted in Figure 1.
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To test this null Hypothesis 1Ho, we employ the first listed LISREL input file inthe Appendix (utilizing the ML fit function in the earlier Equation 4, given the previ-ously mentioned findings of the preliminary analyses). We obtain a chi-square value(χ2) = 40.16, degrees of freedom (df) = 10, associated probability (p) < .001, andRMSEA = .14. (We note that there are no free parameters in this fitted model, as di-rectly deduced from Equations 1, 2, and 8.) The reported fit indexes are unaccept-able, and we reject the null Hypothesis 1Ho: Σ= Σo in favor of the alternative 1H1: Σ ≠Σo. We thus conclude that there is sample evidence against the particular covariance
TESTING MV HYPOTHESIS VIA SEM 243
TABLE 1Covariance Matrices of Used Variables From Baltes, Dittmann-Kohli, & Kliegl’s
(1986) Study
Variable
AI FR TI CF PS1 PS2
Experimental groupa
328.11200.05 199.24324.97 220.12 377.21151.73 117.50 167.97 114.6091.42 57.82 92.06 40.88 73.41
130.63 91.37 125.42 71.02 62.18 194.77Control groupb
217.99149.86 262.79219.87 171.01 319.22117.89 148.86 149.82 161.9864.28 82.22 80.29 58.10 109.4472.40 106.58 73.68 89.85 62.03 156.58
Note. AI, FR = Induction and Figural Relations tests, respectively, specifically developed by Balteset al. (1986) for the purposes of this empirical study; TI = Thurstone’s Standard Induction test; CF =Culture Fair test; PS1, PS2 = Perceptual Speed tests.
an = 161. bn = 87.
TABLE 2Tested Covariance Matrix Σo of x = (x1, x2, x3, x4)′
Variable
x1 x2 x3 x4
313.35198.34 205.54312.64 221.75 352.54142.45 111.35 160.54 135.59
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Matrix Σo as a hypothetical one for the four fluid measures of concern. Thereby, thehigh RMSEA is well in excess of the proposed critical value of .05 for reasonablygood fit (e.g., Browne & Cudeck, 1993) and therefore indicates marked departure ofthesamplecovariancematrix fromthishypothesizedstructure inΣo (ascouldalsobeanticipated comparing the entries in Table 2 to their counterparts in Table 1).
Next, it is of interest to test whether there are no (linear) relationships betweenthese four fluid measures and if they exhibit equal variability (the latter assump-tion will be relaxed later). Should this hypothesis of sphericity, 2Ho, be consideredacceptable, we could interpret the four fluid tests as measuring independent abili-ties (in general, unrelated), and, for instance, there will be no point in attempts toextract sources of common variance. With the second LISREL input file in the Ap-pendix, we obtain χ2= 735.83, df = 9, p < .001, RMSEA = .71. These are unaccept-able fit indexes, and we reject the null Hypothesis 2Ho : Σ = σ2Ip in favor of thealternative 2H1: Σ ≠ σ2Ip stating that these fluid measures either are not independ-ent of one another, are not equivariable, or both. The relaxed null Hypothesis 21Ho,assuming only variable independence (in general uncorrelatedness), is similarlyrejected after looking at the unacceptable fit indexes of its model: χ2 = 671.89, df =6, p < .001, RMSEA = .83. (To fit this model, only the second to last line of the sec-ond input file in the Appendix is deleted.) These and immediately preceding re-sults have been in fact expected, given that the fluid measures assess what is foundto be markedly related fluid intelligence subabilities (e.g., Horn, 1982). In either ofthese two SEM tests the salience of the RMSEA is again notable, suggesting sub-stantial departure of the sample covariance matrix from the hypothesizedcovariance structure of independence, with or without equivariance.
We are next concerned with examining whether the results of Thurstone’s Stan-dard Induction test (Baltes et al., 1986) exhibit compound symmetry and are thusequicorrelated over the four repeated assessment points in the experimental group.The pertinent covariance matrix for this fluid measure is presented in Table 3.
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TABLE 3Covariance Matrix of the Four Repeated Assessments WithThurstone’s Standard Induction Test (Experimental Group)
Variable
TI1 TI2 TI3 TI4
307.46296.52 377.21295.02 365.10 392.47291.02 355.88 358.25 376.84
Note. TIi = Thustone’s Standard Induction test at the ith repeatedassessment (i = 1, …, 4).
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To test this hypothesis of compound symmetry, 3Ho, we employ the thirdLISREL input file in the Appendix and find unacceptable fit indexes: χ2 = 88.09, df= 8, p < .001, RMSEA = .25. They present evidence against the assumption ofcompound symmetry. Like in many repeated measure investigations in the socialsciences, this assumption is excessively strong in the intervention study under con-sideration. The rejection of 3Ho is not surprising given that one would usually ex-pect that assessment points further apart in time would tend to have lowercorrelations than those closer in time. The large RMSEA indicates considerablediscrepancy between the sample covariance matrix and the rather stringent com-pound symmetry requirements.
Next, the question is addressed if the four experimental fluid intelligence tests atfirst posttest, as a group, are independent from the pair of corresponding perceptualspeed tests that are generally thought to tap to a lesser degree into intellectual abili-ties related to figural relations and induction (e.g., Baltes et al., 1986). The fourthLISREL input file in the Appendix yields unacceptable fit indexes also for this nullHypothesis 4Ho:χ2= 91.39, df = 8, p < .001, RMSEA = .26. We thus reject 4Ho claim-ing that these twosetsofvariablesare independent (uncorrelated ingeneral).This re-sult is also not unexpected, given that perceptual speed tests tap to some extent intothe “mechanics” of our brains, which is considered a major component of fluid intel-ligence (Horn, 1982). The salient RMSEA again indicates marked violation of thenull hypothesis of variable set independence, which might be interpreted also as asuggestionofpossiblyconsiderablemeasureset intercorrelations. (Thisstatement isnot meant to imply conclusive evidence for substantial correlations across the twovariable sets.)
If one wishes to consider applying MANCOVA on the four fluid measures ofconcern across the two groups using the perceptual tests as covariates, as indicatedbefore it is helpful to examine group homogeneity of the covariance matrix of de-pendent variables and covariates. To this end, use is made of the fifth LISREL in-put file in the Appendix. The results are unacceptable fit indexes: χ2 = 31.99, df =8, p < .001, RMSEA = .137 (Steiger, 1998), leading to a rejection of the pertinentnull Hypothesis 5Ho of group identity in the corresponding covariance matrix.Thus, an important assumption of MANCOVA, homogeneity of regressionhyperplanes, may be violated in this data set (see also Footnote 3). We note that an-other relevant assumption of MANCOVA is not fulfilled here, namely that thecovariates are measured without error (e.g., Huitema, 1980). The two observationsimply that if one were nonetheless to apply MANCOVA on this data set, the re-sults may be misleading.6
TESTING MV HYPOTHESIS VIA SEM 245
6Although this point is not the concern of the article, SEM is an alternative, large-sample methodol-ogy that does not need either of these two assumptions (e.g., Arbuckle, 1997; see also Raykov, 1995, foran alternative modeling of this study data).
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Next, we are interested to see if the four fluid measures exhibit group invariantpatterns of relationship. To test this null hypothesis, 6Ho, of covariance homogene-ity, the sixth LISREL input file in the Appendix is used and yields χ2 = 51.36, df =10, p < .001, RMSEA = .18 (Steiger, 1998). These fit indexes cannot be consideredacceptable, which suggests that the fluid measures of concern are not exhibitingthe same relationship patterns in the experimental and control groups.
Then the question is examined whether the pattern of interrelationships of thefour fluid measures is invariant across first and second posttest. To evaluate thisnull hypothesis of time invariance of their covariance matrix, 7Ho, use is made ofthe seventh input file in the Appendix. The covariance matrix needed is given inTable 4.
The obtained fit indexes are χ2 = 6.01, df = 10, p = .81, and RMSEA = .06, witha 90% confidence interval (.0; .05), indicating an acceptable fit. (LISREL did notprovide such intervals for most analyses reported in this section, due to the small pvalue associated with their chi-squares; e.g., Jöreskog & Sörbom, 1993.) We con-clude that these fluid measures have a time-invariant pattern of interrelationships.The minimal RMSEA suggests a high degree of similarity of two fluid covariancematrices at the assessments under consideration, as can also be anticipated by in-specting the upper and lower 4 × 4 submatrices (tested for identity here) along themain diagonal of the covariance matrix in Table 4. This finding could be explainedby considering particular effects of the training program and subjects followingthe corresponding tutor’s instructions across the relatively short time of approxi-mately 1 month between the two assessments (Baltes et al., 1986; the test of timeinvariance of relationships pattern from second to last posttests yields a similar re-
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TABLE 4Covariance Matrix and Means (Bottom Row) of the Four Fluid Measures at First and
Second Posttests (Experimental Group)
Variable
AI1 FR1 TI1 CF1 AI2 FR2 TI2 CF2
328.11200.05 199.24324.97 220.12 377.21151.73 117.50 167.97 114.60310.49 200.81 330.12 154.36 338.01202.09 173.22 224.54 120.96 202.29 213.32322.90 225.76 365.10 168.14 333.19 228.49 392.47156.45 123.75 170.60 99.90 161.13 128.33 174.70 125.53
Means 49.08 71.42 53.30 57.73 51.66 73.37 54.82 59.65
Note. AI, FR = Induction and Figural Relations tests; TI = Thurstone’s Standard Induction test; CF= Culture Fair test.
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sult). We note that time invariance in the covariance matrix—as found in this sam-ple—does not imply, and is not implied by, such in the variable means. Thisempirical study in fact offers an example of this lack of relation between their timepatterns, as shown in the following subsection. (With multinormality, for example,the means are independent from variances and covariances and hence can havecompletely different patterns of change over time; e.g., Bock, 1975.)
Testing Mean Difference Hypotheses
It is next of interest to explore the evidence, if any, for group mean differences in thefluid intelligence tests at first posttest. To test this null hypothesis, 8Ho, use is madeof the second to last LISREL input file in the Appendix. To this end, we need addi-tionally to data from Table 4 the control group means of the fluid measures, whichare presented in Table 5.
The restricted Model M′ with the sole constraint that the variable means areidentical across groups (i.e., with restrictions in Equation 16) yields χ2 = 29.08, df= 4, p < .001, and RMSEA = .23 (Steiger, 1998) that represent unacceptable fit in-dexes. It is therefore concluded that the means of the four fluid intelligence testsdiffer across groups. To find which particular tests may have different means in thegroups, we also fit the saturated Model M′ (with no mean restrictions). The latterhas an identical LISREL input file, apart from the changed last keyword on themodel definition line in the control group from KA=IN to KA=FR. We obtain per-fect fit because this model is saturated but focus in it on the estimates of the latentmeans (which are therefore identical to the observed means) and in particular theirstandard errors. In the order of presentation of these measures in Table 5, theirmean estimates are as follows (standard errors given in parentheses): 49.08(1.43),71.42(1.12), 53.30(1.54), and 57.73(.85) in the experimental group; and41.51(1.59), 62.04(1.75), 44.96(1.93), and 55.34(1.37) in the control group. Con-structing their (approximate) 95% confidence intervals by adding and subtractingfrom the estimates twice their standard errors and comparing these intervals forsame measure across groups (e.g., Raykov, 1995), we see that only the induction
TESTING MV HYPOTHESIS VIA SEM 247
TABLE 5Control Group Means of the Four Fluid Intelligence Tests at First Posttest
Variable
AI FR TI CF
Mean 41.51 62.04 44.96 55.34
Note. AI, FR = Induction and Figural Relations tests; TI = Thurstone’s Standard Induction test; CF= Culture Fair test.
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and figural relation tests exhibit higher means in the experimental group. Thisfinding may be attributed to a training program effect targeting these fluidsubabilities, which however has not shown marked transfer at first posttest to theculture-reduced measure. (The same conclusions are reached by using the standarderrors of the observed means, as mentioned before, instead of this model’s asymp-totic standard errors.) In general, mean differences can also be examined withinthis modeling approach by testing the corresponding group equality restrictions(null hypothesis) in the means via nested models; this procedure can also be usedin a one-tailed fashion (e.g., Hays, 1994) as long as a one-tailed alternative hypoth-esis is specified before the data is inspected. In this particular example, we decidedto use a more exploration-oriented approach, rather than one based on such a sta-tistical test of pertinent restrictions. The reported application dealt with a compari-son of corresponding confidence intervals in order to be readily in a position tosense the direction of the experimental or control group differences in means ifany, because we were not willing to assume a prespecified alternative with regardto these group differences but instead wished to be opened to either finding.
Finally, we want to see if there is time invariance also in the experimentalmeans of these fluid measures from first to second posttests, in addition to thisinvariance in their relationships pattern found earlier. The last LISREL input filein the Appendix yields χ2 = 49.81, df = 14, p < .001, and RMSEA = .13. These in-dexes indicate lack of acceptability of the null Hypothesis 9Ho. As with the preced-ing model, we find by releasing the mean identity constraints that at secondposttest all fluid means appear somewhat lower, although none is (individually)significantly so. Although participants’ performance on individual tests (ignoringthe other measures) remains at approximately the same level, when taking into ac-count these tests’ intercorrelations a considerable drop emerges overall in the per-formance on the four measures (for similar multivariate significances, see, e.g.,Tabachnick & Fidell, 1996).
Summary
The reported tests of the above 10 null hypotheses using the SEM method outlinedin this article have shed much light on the interrelationships between the focusedfluid intelligence and perceptual speed measures. Rejecting 9 of these hypothesesimplies that the corresponding multivariate covariance structures of these measuresdo not have the patterns stipulated in the hypotheses and that there are mean differ-ences of fluid measures across groups and assessments. In particular, the findingssuggest that at first posttest (a) the fluid tests and the perceptual measures are not in-dependent as groups; (b) the fluid measures are not independent (and with the samevariance) and do not exhibit relationships representable by the covariance Matrix
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Σo in Table 2; (c) the repeated Thurstone’s induction test results do not comply tothe stringent requirements of compound symmetry; (d) a MANCOVA may notnecessarily be an appropriate method of studying mean differences in the fluidmeasures controlling for possible group differences in perceptual speed tests (seealso Footnotes 3 and 5); (e) the covariance matrices of the fluid measures differacross groups as do their means (apart from the culture-fair test); and (f) the rela-tionships pattern of the fluid measures is invariant across first and second posttest,unlike their four means (when accounting for measure interrelationships).
CONCLUSION
This article presented a didactic collection of tests of manifest multivariable struc-tures using the popular SEM methodology. Specifically, the concern was with test-ing hypotheses about variable relationship patterns and mean differences. The hy-potheses focused on multivariable covariance structures of special interest,represented assumptions of multivariate analysis methods—for example, analysisof variance and covariance, repeated measure analysis, discriminant analysis—orwere concerned with group mean differences or time invariance.
The method underlying this article requires availability of large samples be-cause it makes intensive use of the SEM methodology that is based an asymptotic,rather than finite-sample, statistical theory. Given the increasing popularity of thismethodology in the social, behavioral, and educational sciences, the intention wasto discuss such a general approach to testing these (and other similar) hypotheses,which would be easily applicable by social researchers in practice. In addition, themethod is readily implemented in widely circulated SEM programs, such asLISREL, EQS, AMOS, and RAMONA. Because SEM can be used with normal aswell as nonnormal data—especially with very large samples (see earlier sec-tion)—in the latter case this approach can be considered complementing corre-sponding likelihood ratio tests (LRT), typically assuming multinormality, ofmultivariable structure hypotheses (e.g., Bock, 1975; Johnson, 1998; Johnson &Wichern, 1998; Srivastava & Carter, 1989; Timm, 1975). A direct extension ofthis method provided a large-sample means of testing group mean difference with-out requiring covariance matrix homogeneity—a main assumption of multivariateanalysis of variance—and regardless of group size ratio, as well as time invariancein means of longitudinally assessed multiple measures. Moreover, even in themultinormality case where this SEM method can be considered an alternative tothose LRTs, it seems at times to offer greater ease of handling particularly with re-gard to its software implementation (see the short LISREL input files in the Ap-pendix), as long as the sample size is large in order for the results of this method tobe trustworthy.
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In this context, limitations of the discussed general SEM method lie in this re-quirement for large samples and for very large samples with nonnormality (e.g.,Hu et al., 1992; see earlier section). This requirement may be hard to meet in prac-tice to a satisfactory extent, even in cases where the multinormality assumption ap-pears plausible, and this is a clear limitation to a wider applicability of theapproach. We stress that this same principal limitation is shared with theabove-mentioned LRTs in case of multinormality. This is due to the fact that alsothey are of asymptotic nature, being by construction based on the large-sample dis-tribution of –2lnΛ, where Λ is the corresponding likelihood ratio (e.g., Rao, 1973).Further research, which goes beyond the confines of this article, is thereforeneeded to clarify if the asymptotic theory on which the present SEM models arebased may obtain practical relevance at smaller sample sizes than the ones neededfor such relevance of the asymptotic theory behind the corresponding LRTs, or re-versely. Future comparative research also needs to address the issue of test perfor-mance with the present SEM models in the light of possible specific robustnessfeatures of the corresponding LRTs, of which currently relatively little is known,across a number of nonnormality cases. It is worth emphasizing in this regard thatrecent developments of SEM robust estimation and testing methods—particularlyof the Satorra-Bentler’s scaled type statistics and standard errors (Bentler & Yuan,1999; Chou et al., 1991; Yuan & Bentler, 1998) and corrections of the ADFmethod (Yuan & Bentler, 1997, 1998; Yung & Bentler, 1994)—lead to improvedperformance in general of SEM methods and therefore seem to hold promise forwider applicability of the SEM approach in this article, particularly withnonnormal data, yet demonstration of this awaits further research.
Although the relevance of availability of large samples cannot be overstated forthis SEM method to be strictly applicable, it is conversely emphasized that withsmall samples it cannot be recommended. This is because, as repeatedly men-tioned before, its nature is asymptotic, and therefore with small samples it mayyield misleading results (e.g., Raykov & Widaman, 1995; cf. Bentler & Yuan,1999). Thus, with a limited number of studied participants the social researcher isurged to consider use of general linear model related techniques and correspond-ing tests of specific multivariable structure hypotheses mentioned previously.With such samples, one could also rely for example on some robustness features ofMANOVA with nonnormality or covariance matrix heterogeneity, particularlywith equal group sizes (Algina & Coombs, 1996; Cole et al., 1993; Coombs et al.,1996; Tabachnick & Fidell, 1996). With some samples, the possibility of randomdropping of participants from the larger group might also be considered for usewith MANOVA in the heterogeneity case, as well as some normalizing transfor-mations if needed to bring variable distribution(s) closer to normal and thus im-prove the performance of that analysis of variance method.
We conclude by noting that the CFA model on which this article is based, as de-fined in Equations 1 and 2 or 15, is a very general one (cf. Jöreskog & Sörbom,
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1993, chapter 9). It covers very many, if not essentially all, situations currently ofconcern to social, behavioral, and educational research where an examination ofthe relationship patterns of multiple measures may be of interest. This articleshowed how one can use the present large-sample SEM method to test a few spe-cific hypotheses about multivariable structures, but as indicated earlier essentiallyany hypothesis about the population covariance matrix, Σ, of a set of observedvariables can be tested with appropriate restrictions on the elements of the latentcovariance Matrix Φ (= Σ). We therefore hope that the general SEM approach inthis article has great potential as a useful method of testing a variety of empiricallyand theoretically important hypotheses about variable relationship patterns, groupmean differences, or both in social, behavioral, and educational research.
ACKNOWLEDGMENTS
This research was supported by a grant from the Max Planck Society for Advance-ment of Science.
I thank P. B. Baltes, F. Dittmann-Kohli, and R. Kliegl (Max Planck Institute forHuman Development, Berlin) for the permission to use data from their project“Aging and Plasticity in Fluid Intelligence.” I am grateful to G. A. Marcoulidesand three anonymous referees for a number of useful and suggestive commentsand criticism on earlier versions of the paper, which contributed considerably to itsimprovement. I also thank R. Wilcox and J. Algina for valuable discussions on ro-bustness issues.
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APPENDIX
LISREL Input Files for the Tested Multivariate Hypotheses
(1) LISREL8 INPUT FILE FOR TESTING Ho: “SIGMA=SIGMA_o”DA NI = 4 NO=161CM FI = <see Table 1>MO NX=4 NK=4 PH=SY,FI LX=ID TD=ZEMA PH313.35198.34 205.54312.64 221.75 352.54142.45 111.35 160.54 135.59OU
(2) LISREL8 INPUT FILE FOR TESTING Ho: “SPHERICITY”DA NI = 4 NO=161CM FI = <see Table 1>MO NX=4 NK=4 PH=DI,FR LX=ID TD=ZEEQ PH 1 1 PH 2 2 PH 3 3 PH 4 4OU
(3) LISREL8 INPUT FILE FOR TESTING Ho: “COMPOUND SYMMETRY”DA NI = 4 NO=161CM FI = <see Table 3>MO NX=4 NK=4 PH=SY,FR LX=ID TD=ZEEQ PH 1 1 PH 2 2 PH 3 3 PH 4 4EQ PH 2 1 PH 3 1 PH 4 1 PH 3 2 PH 4 2 PH 4 3OU
(4) LISREL8 INPUT FILE FOR TESTING Ho: “VARIABLE SUBSETINDEPENDENCE”DA NI = 4 NO=161CM FI = <see Table 1>MO NX=6 NK=6 PH=SY,FR LX=ID TD=ZE
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FI PH 5 1 PH 5 2 PH 5 3 PH 5 4FI PH 6 1 PH 6 2 PH 6 3 PH 6 4OU
(5) LISREL8 INPUT FILE FOR TESTING MANCOVA HOMOGENEITYASSUMPTION * EXPERIMENTAL GROUPDA NI = 6 NO=161 NG=2CM FI = <see Table 1>MO NX=6 NK=6 PH=SY,FR LX=ID TD=ZEST .1 ALLST 300 PH 1 1 PH 2 2 PH 3 3 PH 4 4 PH 5 5 PH 6 6OU NSCONTROL GROUPDA NO=87 NI=6CM FI = <see Table 1>MO PH=SP LX=IN TD=INEQ PH 5 1 PH 1 5 1EQ PH 5 2 PH 1 5 2EQ PH 5 3 PH 1 5 3EQ PH 5 4 PH 1 5 4EQ PH 6 1 PH 1 6 1EQ PH 6 2 PH 1 6 2EQ PH 6 3 PH 1 6 3EQ PH 6 4 PH 1 6 4ST .1 ALLST 300 PH 1 1 PH 2 2 PH 3 3 PH 4 4 PH 5 5 PH 6 6OU NSNote. The start values used in this example are specific to the analyzed data set andmay not be appropriate for others.
(6) LISREL8 INPUT FILE FOR TESTING COVARIANCE HOMOGENEITY *EXPERIMENTAL GROUP (Jöreskog & Sörbom, 1993, ch. 9)DA NI = 4 NO=161 NG=2CM FI = <see Table 1>MO NX=4 NK=4 PH=SY,FR LX=ID TD=ZEOUCONTROL GROUPDA NO=87 NI=4CM FI = <see Table 1>MO PH=IN LX=IN TD=INOU
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(7) LISREL8 INPUT FILE FOR TESTING TIME INVARIANCE OF VARI-ABLE RELATIONSHIP PATTERNSDA NI = 8 NO=161CM FI = <see Table 4>MO NX=8 NK=8 PH=SY,FR LX=ID TD=ZEEQ PH 1 1 PH 5 5EQ PH 2 2 PH 6 6EQ PH 3 3 PH 7 7EQ PH 4 4 PH 8 8EQ PH 2 1 PH 6 5EQ PH 3 1 PH 7 5EQ PH 4 1 PH 8 5EQ PH 3 2 PH 7 6EQ PH 4 2 PH 8 6EQ PH 4 3 PH 8 7OU
(8) LISREL8 INPUT FILE FOR TESTING GROUP MEAN DIFFERENCES *EXPERIMENTAL GROUPDA NI=4 NO=161 NG=2CM FI = <see Table 1>ME FI = <see Table 4>MO NX=4 NK=4 PH=SY,FR LX=ID TD=ZE KA=FROUCONTROL GROUPDA NO=87 NI=4CM FI = <see Table 1>ME FI = <see Table 5>MO PH=SP LX=IN TD=IN KA=INOU
(9) LISREL8 INPUT FILE FOR TESTING TIME INVARIANCE IN MEANSDA NI = 8 NO=161CM FI = <see Table 4>ME FI = <see Table 4>MO NX=8 NK=8 PH=SY,FR LX=ID TD=ZE KA=FREQ PH 1 1 PH 5 5EQ PH 2 2 PH 6 6EQ PH 3 3 PH 7 7EQ PH 4 4 PH 8 8EQ PH 2 1 PH 6 5EQ PH 3 1 PH 7 5
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EQ PH 4 1 PH 8 5EQ PH 3 2 PH 7 6EQ PH 4 2 PH 8 6EQ PH 4 3 PH 8 7EQ KA(1) KA(5)EQ KA(2) KA(6)EQ KA(3) KA(7)EQ KA(4) KA(8)OU
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