In contrast to single-indicator models, multiple-indicator models are able to separate measurement error from trait-specific method influences. Moreover, the hypothesis that method effects are trait specific and do not perfectly generalize across traits can be statistically tested. We will only describe three multiple-indicator extensions: a general model that is able to estimate the latent correlations between different trait-method units, a model that is related to the CTCU model, and a model that is an extension of the CTC(M-l) model. Furthermore, we will show how the ideas of the CTUM, the CTCM, and the CT models can be analyzed in the multiple-indicator context. To apply multiple-indicator models, it is necessary to have at least two indicators of each trait-method unit. Hence, an observed variable Yijk has three indices, the first pertaining to the item or test parcel, the second to the trait, and the third to the method.
A multiple-indicator correlation model for our example of three emotional traits and three types of raters is depicted in Figure 20.3a. In this model, a latent variable is defined for the two indicators representing the same trait-method unit. This model allows the estimation of the latent correlations between the trait-method units and the construction of a latent MTMM correlation matrix. The correlations of this matrix represent an error-free variant of the MTMM matrix and, therefore, circumvent one of the major criticisms of the MTMM correlation matrix. Because the MTMM matrix is based on observed correlations, their sizes depend on the reliability of the measures. When measures strongly differ in their reliabilities, the conclusions based on applying the Campbell and Fiske criteria to the MTMM matrix (see Schmitt, chap. 2, this volume) can be misleading (Wothke, 1995). The MTMM model is a very general model without restrictions on the latent correlations. All models depicted in Figure 20.1 for single indicators can be applied to the multiple-indicator case by replacing the observed variables in Figure 20.1 with the latent (trait and method) variables in Figure 20.3a. This means that a second-order factor structure would be defined for the first-order factors in Figure 20.3a, and the fit of these more restricted models could be tested against the fit of the general MTMM correlation model. The residuals of the first-order factors indicate systematic method influences that are specific for a trait-method unit. All other properties of the single-indicator models can be transferred to the multiple-indicator models with a second-order factor structure. In the following discussion, we will present a slightly different way to model these ideas by introducing trait and method factors as firstorder factors. In our view, this approach is more flexible because it allows the testing of hypotheses about the structure of trait and method effects and allows a researcher to relate the latent variables representing method influences to other variables.
An extension of the CTCU model is depicted in Figure 20.3b. In this model there are two indicators for each trait-method unit. All indicators belonging to the same trait are indicators of a common trait factor T.. Hence, there are three trait factors, one for j each emotion considered in our application presented in the last section. Additionally, there is one method factor for each trait-method unit indicating the method influences that are specific (i.e., unique) for one trait. The correlations between the method factors belonging to the same method indicate the generalizability of method effects across traits. In this model, only correlations between
(a) MTMM Correlation Model
(a) MTMM Correlation Model
(b) CTCU Model
(b) CTCU Model
FIGURE 20.3. Multiple-indicator MTMM models, (a) MTMM correlation model; (b) CTCU model: correlated trait/uncorrelated uniqueness model; and (c) CTC(M-l) model: correlated trait/correlated method model with one method factor less than methods considered. Y...: observed variable, i: indicator, i: trait, k: method; T.: latent trait nft ' ' J ' ' J
variable; Mjk: trait-specific method factor; E..fe: Error variable; Au.k: factor loadings. Factor loadings are only depicted for one path for each factor but they are estimated for all variables. The figure shows the general loading pattern. In the applications reported in the text, more-restricted versions are analyzed. In the general version of the CTC(M-l) model, correlations between a trait factor and the method factors that belong to another trait are allowed. However, they are not presented in this figure and are not admitted in the application reported in the text.
method factors belonging to the same method are allowed. Therefore, this model represents the idea of the CTCU model depicted in Figure 20.1b: There are trait-specific method influences (method factors) that are unique to one trait-method unit and that can be correlated across all traits but only if the method factors belong to the same method. In contrast to the model in Figure 20. lb, the model in Figure 20.3b separates measurement error from method-specific influences and represents method-specific influences by latent variables that can be related to other variables. If the three method factors belonging to the same method are identical, the model reduces to a CTUM model. If one allows the method factors of this CTUM model to be corre lated, the model becomes a CTCM model. The multiple-indicator CTUM and CTCM models are very strict variants of the CTCU model implying perfect unidimensionality of the method influences belonging to the same method. A somewhat less-restrictive variant would be to model a general method factor for each method as a second-order factor of all method factors belonging to the same methods. These general method factors can be assumed to be uncorrelated (less-restrictive CTUM model) or correlated (less-restrictive CTCM model). These second-order structures are less restrictive because residuals of the first-order method factors can capture the trait-specificity of a method influence. One would apply these second-order method models if one wants to get a latent variable representing the general trait-unspecific effect of a method. Marsh and Hocevar (1988) have proposed a model that is related to this idea.
The CTCU model depicted in Figure 20.3b allows the decomposition of the variance of the observed indicators into components that represent trait influences, method influences, and influences that are due to measurement error. Because the method factors belonging to different methods are uncorrelated, the model is most appropriate for interchangeable methods (see earlier discussion). This model can also be conceived of as a multiple-trait extension of a so-called latent state-trait model and is related to special models of longitudinal confirmatory factor analysis (Eid, Schneider, & Schwenkmezger, 1999; Marsh & Grayson, 1994a; Steyer, Schmitt, & Eid, 1999; see Khoo, West, Wu, & Kwok, chap. 21, this volume). In this latent state-trait model, the different methods considered are the different occasions on which individuals are measured (e.g., Steyer, Ferring, & Schmitt, 1992; Steyer et al., 1999).
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