CFA Approaches to MTMM Data

Using CFA approaches to MTMM data, researchers can define models that posit a priori trait and method factors and test the ability of such models to fit the data. In the general MTMM model (Marsh, 1989; Marsh & Grayson, 1995; Widaman, 1985); (a) there are at least three traits (T = 3) and three methods (M = 3); (b) T X M measured variables are used to infer T + Ma priori factors; (c) each measured variable loads on one trait factor and one method factor but is constrained so as not to load on any other factors; (d) correlations among trait factors and among method factors are freely estimated, but correlations between trait and method factors are fixed to be zero; and (e) the uniqueness of each scale is freely estimated but assumed to be uncorrelated with the uniquenesses of other scales. This general model with correlated traits and correlated methods (CFA-CTCM), provides apparently unambiguous interpretation of convergent validity, discriminant validity, and method effects: large trait factor loadings indicate support for convergent validity, large method factor loadings indicate the existence of method effects, and large trait correlations—particularly those approaching 1.0—indicate a lack of discriminant validity.

A taxonomy of models (Marsh, 1989, 1993b; Widaman, 1985) was proposed to evaluate MTMM data that systematically varied the way that traits and methods were represented. Particularly important was the correlated uniqueness model (CFA-CTCU) in which method effects are inferred from correlated uniquenesses among measured variables based on the same method instead of method factors. Correlated uniquenesses reflect the covariation between two measured variables that are measured with the same method after taking into account the effects of the trait factors. The rationale is that correlations among all measures should be explained in terms of the correlated traits so that any residual covariation between two variables measured with the same method reflects method effects. To the extent that these correlated uniquenesses are consistently large, statistically significant, and interpretable, there is support for method effects in addition to the effects of the traits. From a practical perspective, the CFA-CTCU model almost always results in proper solutions, whereas the traditional CFA-CTCM model typically results in improper solutions. For example, Marsh and Bailey (1991), using 435 MTMM matrices based on real and simulated data, showed that the CFA-CTCM model typically resulted in improper solutions (77% of the time), whereas the CFA-CTCU model nearly always (98% of the time) resulted in proper solutions. Improper solutions for particularly the CFA-CTCM models were more likely when the MTMM design was small (i.e., 3 Trait X 3 Method vs. 5 Trait X 5 Method), when the sample size was small, and when the assumption of unidimensional method effects was violated. From this practical perspective, the complications in comparing the different MTMM models may be of limited relevance because in many applications only the CFA-CTCU model results in a proper solution. Because of the inherent instability of CFA-MTMM models, Marsh and Grayson (1995) recommended that studies should contain at least four traits, at least three methods, and a sample size of at least 250.

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