Multipleindicator Models With Heterogeneous Indicators

The two multiple-indicator models assume that the indicators are homogeneous indicators of a trait-method unit. This means that they are indicators of the same trait and the same method factors without any unique component of the true score that is not shared with the other indicators of this trait-method unit. This assumption, however, is often too restrictive, particularly when the same indicator is repeatedly measured (e.g., by different raters or on different occasions of measurement). In this case, a unique indicator-specific component can be identified and its nonconsideration would result in the misfit of an MTMM model. There are several ways to consider indicator-specific components. The most prominent is to allow autocorrelations of residuals belonging to the repeatedly measured indicators. This means that all residuals belonging to the same indicator are correlated. Although autocorrelations are admissible representations of indicator-specific influences, they have the disadvantage that they indicate a valid source of variance that is not modeled by latent variables. Consequently, the reliabilities of the indicators will be underestimated. An alternative is to consider a multidimensional trait structure. In these extended models, each indicator measures a different indicator-specific trait factor but a common method factor. Hence, in our example there would be six (correlated) latent trait factors whereas the method factor structure of the model would not change. This way of considering indicator specificity has been adopted in models of latent state-trait theory (e.g., Eid, 1996; Eid & Diener, 1999) and longitudinal confirmatory factor analysis (Marsh & Grayson, 1994a). A second way is to introduce an indicator-specific factor for one of the two indicators (see Eid et al., 1999). The repeated measures of the same indicator are assumed to have substantive loadings on the indicator-specific factor. This indicator-specific factor represents the uniqueness of an indicator that is not shared with the other indicator. Accordingly, there is one indicator-specific factor less than indicators. This approach is similar to the CTC(M-l) approach of modeling method factors. The basic idea of this type of modeling is that if we have two indicators, we need only one indicator-specific factor to contrast the differences between the two indicators. The two different approaches to modeling indicator-specific influences are strongly related and can be transferred to each other under specific conditions (see Eid et al., 1999).

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