CFA – SEM: Modeling Causal Processes

1. CFA – SEM: Modeling Causal Processes BUSI6280

2. The term structural equation modeling conveys two key aspects of the procedure: • That the causal processes under study are represented by a series of structural (i.e., regression) equations, • That these structural relations can be modeled pictorially to enable a clearer conceptualization of the theory under study.

3. Why Use SEM ? • SEM lends itself well to the analysis of data for inferential purposes. • Whereas, traditional multivariate procedures are incapable of either assessing or correcting for measurement error, SEM provides explicit estimates of these parameters. • SEM procedures can incorporate both unobserved (i.e. latent) and observed variables.

4. Purpose of Factor Analysis • The factor analytic model (EFA or CFA) focuses solely on how the observed variables are linked to their underlying latent factors. • Factor analysis is concerned with the extent to which the observed variables are generated by the underlying latent constructs and thus strength of the regression pathsfrom the factors to the observed variables (the factor loadings) are of primary interest. • Although inter-factor relations are also of interest, any regression structure among them is not considered in the factor analytic model.

5. Type of Models • Measurement model: latent variables and their observed measures (i.e., the CFA model) • Structural model: Model withlinks among the latent variables. • Full (Complete) Model: a measurement model and a structural model • Recursive model: Direction of cause is from one direction only • Non-recursive model: reciprocal or feedback effects (often different from one another).

6. Three different scenarios or models (Jöreskog 1993) • Strictly Confirmatory (SC) • Alternative Models (AM) • Model Generating (MG)

7. The SC Scenario • The researcher postulates a single model based on theory, collects the appropriate data, and then tests the fit of the hypothesized model to the sample data. • From the results of this test, the researcher either rejects or fails to reject the model. No further modifications to the model are made.

8. The AM Scenario • The researcher proposes several alternative (i.e., competing) models, all of which are grounded in theory. • Following analysis of a single set of empirical data, the researcher selects one model as most appropriate in representing the sample data.

9. The MG Scenario • The researcher, having postulated and rejected a theoretically derived model on the basis of its poor fit to the sample data, proceeds in an exploratory(rather than confirmatory) fashion to modify and re-estimate the model. • The primary focus here is to locate the source of misfit in the model. Jöreskog noted that, although respecification may be either theory- or data-driven, the ultimate objective is to find a model that is both substantively meaningful and statistically well fitting.

10. SEM procedures – alternative computer programs • AMOS-Arbuckle, 1995 • EQS-Bentler, 1995 • LISCOMP-Muthén, 1998 • CALIS-SAS Institute, 1992 • RAMONA-Browne, Mels, & Coward, 1994 • SEPATH-Steiger, 1994 • LISREL program is the most widely used, 1970s

11. SEM - Language Exogenous latent variables are synonymous with independent variables; they “cause” fluctuations in the values of other latent variables in the model. Changes in the values of exogenous variables are not explained by the model. Rather, they are considered to be influenced by other factors external to the model. Endogenous latent variables are synonymous with dependent variables and, as such, are influenced by the exogenous variables in the model, either directly, or indirectly.

12. SEM - Language By convention, observed measures are represented by Roman letters and latent constructs by Greek letters: • Those that are exogenous are termed X-variables. • Those that are endogenous are termed Y-variables. • The measurement model may be specified either in terms of LISREL exogenous notation (i.e., X-variables), or in terms of its endogenous notation (i.e., Y-variables). • The exogenous latent constructs are termed as ξ (xi). • The endogenous latent constructs are termed as η (eta).

13. x is a q x 1 vector of observed exogenous variables y is a p x 1 vector of observed endogenous variables. ξis an n x 1 vector of latent exogenous variables η is an m x 1 vector of latent endogenous variables. Λ is a q x n matrix of coefficients (λij) linking x and ξ. δ is a q x 1 vector of random disturbance term (errors of measurement) associated with x vector. ε is a p x 1 vector of random disturbance term (errors of measurement) associated with y vector. SEM – LanguageThe Measurement Model

14. SEM – LanguageThe Structural Model • Г(gamma) is an m x n matrix of coefficients (γij) that relates the n exogenous factors to the m endogenous factors. • B(beta) is an m x m matrix of coefficients (βij) that relates the m endogenous factors to one another. • ζ(zeta) is an m x 1 vector of residuals (ζi) representing errors in the equation relating η and ξ. • Ф (phi) is an n x n matrix of coefficients (φij) that captures the variance/covariance between ξs. • Ψ (psi) is the m x m matrix of covariance between ζs.

15. The Structural Model Measurement Model for the X-variables (1):x=xξ+δ Measurement Model for the Y-variables (2): y= yη+ε Structural Equation Model (3): η=Bη + Γξ + ζ

16. The following minimal assumptions are presumed to hold for the system of equations • ε is uncorrelated with η (construct) • δ is uncorrelated with ξ (construct) • ζ is uncorrelated with ξ and η (construct) • ζ, ε, and δ are mutually uncorrelated. • E(η) = 0 • E(ξ) = 0 • E(δ) = 0 • E(ε) = 0 • E(ζ) = 0 • (I-B) is nonsingular so that (I-B)־¹ exists. This makes the equation 3 to be written in the reduced form.

17. Symbol Representation Unobserved (latent) Factors Observed Variable Path coefficient for regression of observed variable on unobserved factors Path coefficient for regression of one factor on another. Residual error (disturbance) in prediction of unobservedfactors Measurement error associated with observed variable.

18. Summary of Matrices, Greek Notation, and Programs Codes Matrix Program Matrix Greek Letter Matrix Element Code Type • Measurement Model • Lambda-X x x LX Regression • Lambda-Y y y LY Regression • Theta delta Qδ θδ TD Var/cov  • Theta epsilon Qe θe TE Var/cov  • Structural Model • Gamma Γ γ GA Regression  • Beta в ß BE Regression • Phi Φ φ PH Var/cov  • Psi Ψψ PS Var/cov  • Xi (or Ksi) --- ξ --- Vector • Eta --- η --- Vector • Zeta --- ζ --- Vector • Var/cov = variance-covariance

19. δ1 ζ1 X1 λx11 ε1 Y1 λx21 λy11 ξ1 η1 δ2 X2 λy21 ε2 Y2 δ3 X3 λx31 The Structural Model - η1 predicted by ξ1

20. An important corollary of SEM is that the variances and covariance of dependent (or endogenous) variables, whether they be observed or unobserved, are never parameters of the model; these are explained by the exogenous variables. • In contrast, the variance and covariance of independent variable are important parameters that need to be estimated.

21. δ1 X1 λ11 δ2 X2 λ21 ξ1 δ3 X3 λ31 CFA Part ζ1 η1 λ11 Y1 ε1 λ21 Y2 ε2 CFA Part MEASUREMENT (CFA) MODELS

22. CFA Model error ReadSC ASC error WriteSC error TalkSC SSC error InteractSC

23. CFA with Greek Notation δ1 x1 λ11 ξ1 δ2 x2 λ11 ф21 δ3 x3 λ32 ξ2 δ4 x4 λ42

24. Regression Equations (X’s) • x1 = λ11ξ1 + δ1 • x2 = λ21ξ1 + δ2 • x3 = λ32ξ2 + δ3 • x4 = λ42ξ2 + δ4 Or in matrix form X =Λxξ + δ

25. The parameters of this model are Λx, Φ,and Θδ Where: • Λx represents the matrix of regression coefficients related to the ξs (described earlier). • Φ (phi) is an η x η symmetrical variance-covariance matrix among the η exogenous factors. • Θδ (theta-delta) is a symmetrical q x q variance-covariance matrix among the error of measurement for the q exogenous observed variables

26. The general factor analytic model can be expanded as: X = Λx ξ + δ x1 Λ11 0 δ1 x2 Λ210 ξ1 δ2 = + x3 0 Λ32ξ2 δ3 x4 0 Λ42 δ4

27. Λ is the Loadings Matrix • The Λ matrix is often termed the factor-loading matrix because it portrays the pattern by which each observed variable is linked to its respective factor.

28. The Y’s • y1 = λ11η1 +ε1 • y2 = λ21η1 +ε2 • y3 = λ32η2 +ε3 • y4 = λ42η2 +ε4 Or in matrix form Y = Λyη +є

29. Matrix Notation for Loadings with Regression Model Y = Λyη + ε y1 λ 11 0 ε1 y2 = λ 21 0 η1 + ε2 y3 0 λ32η2 ε3 y4 0 λ32 ε4

30. A just-identified model is one in which there is a one-to-one correspondence between the data and the structural parameters. • Number of data variances and covariances equal number of parameters to be estimated. • However, despite the capability of the model to yield a unique solution for all parameters, the just-identified model is not scientifically interesting because it has no degrees of freedom and therefore can never be rejected.

31. Overidentified Model • An overidentified model is one in which the number of estimable parameters is less than the number of data points (i.e., variance, covariance of the observed variable). • This situation results in positive degrees of freedom that allows for rejection of the model, thereby rendering it scientific use. The aim in SEM, then, is to specify a model such that it meets the criterion of overidentification.

32. Underidentified Model An underidentified model is one in which the number of parameters to be estimated exceeds the number of variances and covariances. As such, the model contains insufficient information (from the input data) for the purpose of attaining a determinate solution of parameter estimation; that is, an infinite number of solutions are possible for an underidentified model.

33. Suppose there are 12 observed variable, this means that we have 12(12+ 1)/2=78data points. Suppose that there are 30unknown parameters. Thus, with 78 data points and 30 parameters to be estimated, we have an overidentified model with 48 degrees of freedom. It is important to point out, however, that the specification of an overidentified model is a necessary, but not sufficient condition to resolve the identification problem. Indeed, the imposition of constraints on particular parameters can sometime be beneficial in helping the researcher to attain an overidentified model

34. No Scale Set for Constructs Linked to the issue of identification is the requirement that every latent variable have its scale determined. This requirement arises because these variable are unobserved and therefore have no definite metric scale;

35. Assume CFA Model with 12 variables (items) and 4 factors (3 items per factor). • We can assume that there are 12 regression coefficient (λs) • There are 12 error variance (δs). • There are 4 factors variances (which may be standardized and therefore set to 1). • There are 6 covariances between factors. • If the factor variances are not set to 1 then then one of the λ parameters for each factor can be fixed to a value of 1.00 (they are therefore not to be estimated). The rationale underlying this constraint is tied to the issue of statistical identification. In total, then, there are 30 parameters to be estimated for this CFA model.