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The Correlated Trait/Correlated Uniqueness Model

The correlated trait/correlated uniqueness (CTCU) model is an extension of the correlated trait model that allows generalization of methods effects across traits by correlated residual variables (Kenny, 1979; Marsh, 1989; Marsh & Grayson, 1995). The residual variables (uniqueness) are correlated in a method-specific manner (see Figure 20.1b). Whereas the basic decomposition is the same as in the CT model (Y.fc = Xrj/i T. + Ejk), all residuals Ejk with the same method index k can be correlated in the CTCU model. The CTCU model is a reasonable model for MTMM data and widely applied (e.g.,

Marsh, 1989; Marsh & Grayson, 1995). However, it is restricted in three ways (Bagozzi, 1993; Eid, 2000; Lance, Noble, & Scullen, 2002). First, as in the CT model, measurement error is confounded with method specificity because the residuals comprise both aspects. Hence, it is not possible to separate unreliability from method specificity. Consequently, the consistency coefficient is a lower bound of reliability, and the variance explained by the residuals is the upper bound of the method specificity. Second, "true" (error-free) method effects cannot be related to other external variables because pure method effects are not represented in the model. Third, correlations between different methods are not permitted. This assumption might be too restrictive for applications in which some methods resemble one another more than other methods, for example, if the two other-rater groups in Figure 20.1 hold a common view of the target person that is not shared with the target's view.

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