The correlated trait/uncorrelated method (CTUM) model (e.g., Marsh & Grayson, 1995; Widaman, 1985) is a restricted version of the CTCU model. In the CTUM model, the correlated residuals are explained by the existence of method factors (see Figure 20.1c). An observed variable Y.fe is decomposed into a trait component T., a method component Mk, and a residual component E.k that is not explained by trait and method factors: Y.k = XTjk T + hyk Mk + Ejk> where Kjk are trait loadings and lMjk are method loadings. Whereas all trait factors can correlate, the method factors are assumed to be uncorrelated. In contrast to the CTCU model, the CTUM model allows the identification of a consistent method effect. Consequently, the variance of an observed variable can be decomposed into the variance that is due to the trait factor (consistency or convergent validity), the variance that is due to the method factor (method specificity), and the variance that is due to the residual (influences due to measurement error and method-specific effects that are specific for a trait). Hence, this model is in a better position to separate true method-specific effects from measurement error. However, the residual might not only comprise measurement error but also trait-specific method effects. We will come back to this issue when we present multiple-indicator models. Because method factors are specified in the CTUM model, method-specific effects can be related to other criterion variables to explain method effects. Hence, the CTUM model solves the first two restrictions of the CTCU model by introducing method factors, however, not without costs. The CTUM model is a restrictive variant of the CTCU model because the introduction of method factors puts restrictions on the covariances of the residuals in the CTCU model—they have to follow the assumptions of a congeneric factor model. This means that in the case of more than three traits
(i.e., more than three indicators for a method factor), the CTUM model is more restrictive than the CTCU model and can be statistically tested against the CTCU model. From a substantive point of view, the method factors represent method effects that generalize across traits in a specific manner. This assumption might be violated when method effects are trait-specific (see following section). With respect to the correlation of different methods, the CTUM is restricted in the same way as the CTCU model because associations between different methods are not allowed.
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