Can There Be Infinitely Many Models Equivalent to a Given Covariance Structure Model?

Transcription

1 STRUCTURAL EQUATION MODELING, 8(1), Copyright 2001, Lawrence Erlbaum Associates, Inc. TEACHER S CORNER Can There Be Infinitely Many Models Equivalent to a Given Covariance Structure Model? Tenko Raykov Department of Psychology Fordham University George A. Marcoulides Management Science California State University, Fullerton The problem of equivalent models has plagued researchers since the earliest developmental stages of structural equation modeling (SEM; Marcoulides & Schumacker, 1996). Equivalent models are those that provide the same sets of statistical fit indexes (e.g., chi-square and p values) as a hypothesized model but may imply very different substantive interpretations of the data (Stelzl, 1986). As a consequence, equivalent models represent a very serious threat to behavioral and social theory development and construct validation using SEM. In fact, unless the model equivalence problem is addressed, any originally considered model regardless of how well it fits the data remains only one possible means of its explanation (e.g., MacCallum, Wegener, Uchino, & Fabrigar, 1993). Although the problem of equivalent models has received considerable methodological attention for more than a decade (e.g., Bollen, 1989; Breckler, 1990; Hayduk, 1996; Hershberger, 1994; Jöreskog & Sörbom, 1993; Lee & Requests for reprints should be sent to Tenko Raykov, Department of Psychology, Fordham University, Bronx, NY raykov@murray.fordham.edu

2 INFINITELY MANY EQUIVALENT MODELS 143 Hershberger, 1991; Luijben, 1991; MacCallum et al., 1993; Raykov, 1997; Raykov & Penev, 1999; Stelzl, 1986; Williams, Bozdogan, & Aiman-Smith, 1996), distinguishing between equivalent models cannot be achieved using currently available fit indexes. Model equivalence can only be managed via substantive considerations or considerations pertaining to design and data collection features (apart from the case of multiple-population versions of single-group equivalent models, where statistical distinction becomes possible with appropriate group constraints, if substantively correct; Raykov, 1997). MacCallum et al. (1993) indicated that equivalent models exist routinely and for essentially any structural equation model there are potentially many equivalent models that represent equally plausible means of describing the analyzed data. Once a hypothesized model is proposed, some of its equivalent models can be obtained using specifically developed rules (e.g., Hershberger, 1994; Lee & Hershberger, 1990; Stelzl, 1986), whereas others can be obtained utilizing special parameter restrictions (e.g., Raykov & Penev, 1999). From this perspective, it is important to know whether for any hypothesized model there exist only a limited (although possibly large) number of equivalent models or if there can be infinitely many such equivalent models. In the former case, the desire is to be in a position to assess the substantive meaningfulness and likelihood of each of a finite number of models that may have generated the data and possibly rule out most of them as means of data description and explanation. In the latter case, however, another level of difficulty with potentially hard to anticipate implications is added to efforts of resolving the equivalence problem. The purpose of this article is to address the question of whether there may be infinitely many models equivalent to a hypothesized one. Specifically, an example of a set of infinitely many models equivalent to an a priori hypothesized covariance structure model is presented. The intention is to demonstrate and highlight the difficulty and seriousness of the problem of equivalent models by providing what represents an unlimited host of alternatives to an understanding of how a studied phenomenon might operate as reflected in an initially considered model of it. Subsequently, issues pertaining to managing the equivalence problem are discussed. NOTATION AND BACKGROUND This article makes extensive use of notation commonly employed in the SEM literature (e.g., Bollen, 1989). The vector y =(y 1, y 2,,y p ) t denotes observed data on p manifest variables of interest (p 2), θ denotes the vector of parameters of a structural equation model M under consideration, Ω is its parameter space containing all possible values of θ, and Σ(θ) is the covariance matrix implied by M at θ.

3 144 RAYKOV AND MARCOULIDES The concern of this article is with special cases of the general linear structural relationship model (Jöreskog & Sörbom, 1993). They are contained in the so-called Submodel 3B of the comprehensive software program LISREL. Accordingly, the model is defined by the following equations: η = Bη + ζ, and y = Λη + ε (1) In Equation 1, η is the vector of latent variables (factors, constructs, latent dimensions), B is the matrix of structural regression coefficients relating the latent variables between themselves, Λ is the factor loading matrix, ζ is the structural regression residuals vector (with a covariance matrix Ψ), and ε the error terms vector (with a covariance matrix Θ). It is emphasized that Equation 1 defines the most general LISREL model that encompasses the so-called general LISREL model (Jöreskog & Sörbom, 1993). 1 Currently, the model in Equation 1 covers the vast majority of models used in routine applications of the SEM methodology in behavioral and social research. From Equation 1, with the usual assumption of invertibility of the matrix I q B (where I q is the q q identity matrix and q is the number of latent variables in the model), it follows that the covariance matrix implied by a LISREL model M has the form Σ(θ) = ΛΣ ηη Λ t + Θ = Λ(I q B) 1 Ψ(I q B t ) 1 Λ t + Θ (2) where Σ ηη is the covariance matrix of the latent variables in η. Equation 2 implies that different matrices appearing in its right-hand side may lead to identical reproduced covariance matrices in its left-hand side. Obviously, this is because due to Equation 2 the latter-reproduced matrices equal sums and products of matrices, yet from these sums and products, one cannot deduce uniquely the individual matrices on the right-hand side of Equation 2. In broad terms, this represents the essence of the phenomenon of equivalent models. For the remaining sections, the following statement that defines two models M and M as equivalent is important (e.g., Raykov & Penev, 1999; Stelzl, 1986). Definition: The structural equation models M and M are equivalent if they reproduce the same sets of corresponding covariance matrices, Σ(θ) and Σ (θ ), when their parameters θ and θ vary across their parameter spaces Ω and Ω, respectively. 1 That general LISREL model utilizes a different, but not necessary, notation for the latent independent variables and their indicators, viz. ξ and x respectively, as well as δ for the error terms associated with x (Jöreskog & Sörbom, 1993, p. 190).

4 INFINITELY MANY EQUIVALENT MODELS 145 AN EXAMPLE OF INFINITELY MANY EQUIVALENT MODELS This section considers a set of infinitely many models equivalent to an hypothesized covariance structure model, called M o.m o is the basic model used in this article and is depicted in Figure 1. As seen in Figure 1, M o is a simple, two-factors with two-indicators model, with unitary factor loadings. Although this simple model was selected for illustrative purposes, it is important to point out that the ideas presented in this article are equally applicable to the case of any number of indicators per factor. M o assumes that there is a direct positive effect of the latent variable η 1 on the latent variable η 2, each measured by indicators that act as τ-equivalent tests (i.e., measuring the same true score, in the same units of measurement, yet with potentially different error variances), and is identified. Because the replacement rule by Lee and Hershberger (1990) is applicable to model M o, it follows that M o is equivalent to model M1 depicted in Figure 2. Model M1 is identical to M o in all aspects except the one-way arrow from η 1 into η 2 (and pertinent structural residual), which is exchanged in M1 by a two-way (positive) covariance of the two constructs. M1 is also identified and, although not of main interest in this aricle, plays an auxiliary role to enable a direct demonstration of model equivalence later. The next model to be considered is equivalent to M1 and identified, as formally shown by Raykov & Penev (1999, Appendix 2), and for consistency is referred to as model M2 and depicted in Figure 3. From the equivalence of M o and M1, and the transitivity property of the model equivalence relation (i.e., from M being equivalent to M and the latter to M, it follows that M and M are equivalent; Stelzl, 1986), it is deduced that M o and M2 are equivalent. FIGURE 1 Model M o.

5 146 RAYKOV AND MARCOULIDES FIGURE 2 Model M1 (auxiliary model). FIGURE 3 Model M2 (the first member of the infinite series ϒ of models equivalent to M o ). We note that M2 has different substantive implications than those that follow from M o. Whereas M o states that η 1 causally affects η 2, no relation at all is assumed between the corresponding two constructs in M2. 2 Instead, in M2 the common source of the relation between the four observed variables y 1 to y 4 is not a relation between η 11 and η 22, but a third construct η 3 that is unmatched by a 2 The constructs η 11 and η 22 in M2 need not be identical to η 1 and η 2, respectively, in M1, as seen when writing out the definition equations of both models in terms of relations of manifest to latent variables (e.g., Raykov & Penev, in press, Appendix 2).

6 INFINITELY MANY EQUIVALENT MODELS 147 separate model term in M o. We stress that all three constructs in M2 are orthogonal to each other, whereas in M o the two underlying latent variables are correlated, with the specific earlier cause effect assumption incorporated at the latent level (as indicated in Raykov & Penev, 1999, in M2 the variance of η 3 is assumed to be less than the smaller of the variances of η 1 and η 2 in model M1). The fact that M o and M2 are equivalent means that M o and M2 reproduce the same set of implied covariance matrices (see definition of equivalence). Thus, any model that has the property of reproducing the same set of implied covariance matrices as that by M2 will by necessity be equivalent to M o as well (due to the aforementioned transitivity property of the equivalence relation). This feature will hold for any of the infinitely many models constructed next that are each equivalent to M2, and thus equivalent to M o too. This infinite series of equivalent models, denoted ϒ, is defined as follows. The first member of ϒ is model M2, which has one added construct (viz. η 3 ) that has unitary loadings on all four observed variables and is orthogonal to the remaining two unrelated constructs (η 11 and η 22 ). The kth member of the series, for k 2 (that is, k becoming larger than any finite number), has k such added constructs with the following four properties: (a) unit loadings on all manifest variables, (b) are orthogonal to the same unrelated constructs η 11 and η 22, (c) are unrelated among themselves, and (d) have the additional feature that their sum equals the construct η 3 in model M2. The kth member of ϒ is depicted in Figure 4, whereby the triple of vertically located dots stands for the intermediate added constructs η 5,,η k +2 in the general case (k 2). Given property (d) of the added constructs, it follows that the covariance matrix implied by any member of ϒ is identical to that reproduced by model M2. Thus, FIGURE 4 The kth member of the infinite series ϒ of models equivalent to M o (k 2; η η k +3 = η 3 in Model M2).

7 148 RAYKOV AND MARCOULIDES any member of ϒ is equivalent to M2 and is therefore equivalent to the basic model M o. Furthermore, because M2 is identified, the variance of η 3, σ 32 is identified. Then the variances of the added constructs in the kth member of ϒ will be identified (and thus so will be also that model, as all remaining parameters in it are obviously identified) if assumed equal to one another, that is, each having variance σ 32 /k (k 2). It is important to note that, with this final assumption, the degrees of freedom of each model in ϒ are identical. We also stress that each member of ϒ has different substantive implications from one another and from M o, because that member model has a different number of added unrelated constructs. Each of them implies an additional characteristic (i.e., unrelated to the others) source of variability in the observed variables y 1 to y 4. Thus, ϒ represents an infinite series of identified models equivalent to the basic covariance structure model M o. CONCLUSION This article addressed the problem of equivalence in structural equation modeling. In particular, it addressed the issue of whether researchers can safely assume that there are no more than a finite number of models equivalent to an initially hypothesized covariance structure model. The article provided a counterexample that highlights the difficulty and extent to which model equivalence is a very serious problem for SEM researchers. The article also illustrates the need and importance of further research into model equivalence, particularly with respect to their differentiation. Similarly, it highlights the relevance of thorough studies into the nature of latent variables so as to provide as much as possible additional information about modeled latent dimensions widely used in the behavioral, social, and educational sciences. With this additional information that could largely be of nonstatistical type, perhaps a more successful distinction between equivalent accounts of studied phenomena may be achieved. ACKNOWLEDGMENTS This research was supported in part by a faculty fellowship grant from Fordham University. We are indebted to D. Ozer for valuable discussions on model equivalence. REFERENCES Bentler, P. M. (1995). EQS: Structural equations program manual. Encino, CA: Multivariate Software. Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.

8 INFINITELY MANY EQUIVALENT MODELS 149 Breckler, S. (1990). Applications of covariance structure modeling in psychology: Cause for concern? Psychological Bulletin, 107, Hayduk, L. A. (1996). LISREL issues, debates, and strategies. Baltimore, MD: Johns Hopkins University Press. Hershberger, S. L. (1994). The specification of equivalent models before the collection of data. In A. von Eye and C. C. Clogg (Eds.), Latent variables analysis (pp ). Thousand Oaks, CA: Sage. Jöreskog, K. G., & Sörbom, D. (1993). LISREL 8: User s guide. Chicago: Scientific Software. Lee, S., & Hershberger, S. (1990). A simple rule for generating equivalent models in covariance structure modeling. Multivariate Behavioral Research, 25, Luijben, T. C. W. (1991). Equivalent models in covariance structure analysis. Psychometrika, 56, MacCallum, R. C., Wegener, D. T., Uchino, B., N., & Fabrigar, L. R. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114, Marcoulides, G. A., & Schumacker, R. E. (Eds.). (1996). Advanced structural equation modeling: Issues and techniques. Mahwah, NJ: Lawrence Erlbaum Associates, Inc. Raykov, T. (1997). Equivalent structural equation models and group equality constraints. Multivariate Behavioral Research, 32, Raykov, T., & Penev, S. (1999). On structural equation model equivalence. Multivariate Behavioral Research, 34, Stelzl, I. (1986). Changing a causal hypothesis without changing the fit: Some rules for generating equivalent path models. Multivariate Behavioral Research, 21, Williams, L. J., Bozdogan, H., & Aiman-Smith, L. (1996). Inference problems with equivalent models. In G. A. Marcoulides and R. E. Schumacker (Eds.), Advanced structural equation modeling. Issues and techniques (pp ). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.