Michael Eid, Tanja Lischetzke, and Fridtjof W Nussbeck

Models of confirmatory factor analysis (CFA) or structural equation modeling (SEM) have generally become the most often applied methodological approaches besides Campbell and Fiske's traditional approach of inspecting correlation matrices (e.g., Eid, 2000; Eid, Lischetzke, Nussbeck, & Trierweiler, 2003; Kenny, 1976, 1979; Marsh, 1989; Marsh & Grayson, 1995; Saris & van Meurs, 1991; Widaman, 1985). This is mainly due to the fact that SEM is an approach that tries to explain the correlations and covariances of variables by a set of underlying latent variables (factors). Hence, the multitrait-multimethod (MTMM) matrix proposed by Campbell and Fiske (1959) can be taken as input for more complex analyses by SEM. In contrast to Campbell and Fiske's approach, SEM has several advantages. First, SEM makes it possible to separate unsystematic measurement error from systematic individual differences that are due to trait and method effects. Second, measurement models for trait as well as method factors can be defined. This makes it possible to relate the latent trait and method variables to other latent variables. This is particularly important if one wants to explain trait and method effects by other variables. Third, SEM allows an empirical testing of the assumptions on which a model is based. Consequently, many hypotheses about the structure of trait and method effects can be tested.

The aim of this chapter is to illustrate these advantages by presenting several MTMM models that have been defined in the framework of SEM. In the first part of the chapter, we will discuss models that have been developed to analyze an MTMM matrix with the typical structure described by Campbell and Fiske (1959) and Schmitt (chap. 2, this volume). An important characteristic of this first type of MTMM models is that there is only one indicator for each trait-method unit. The major limitation of these single-indicator MTMM models is that unsystematic measurement error and systematic method-specific influences can be separated only if strong assumptions are fulfilled. The second part of the chapter shows how this limitation can be circumvented by selecting several indicators for each trait-method unit (multiple-indicator models).

Over the last years, many structural equation models for MTMM data have been proposed. Widaman (1985), for example, developed a taxonomy of 16 models of CFA for analyzing MTMM data with t traits and m methods by crossing four different types of trait structures (no trait factor, general trait factor, t orthogonal trait factors, t oblique trait factors) with four different types of method structures (no method factor, general method factor, m orthogonal method factors, m oblique method factors). In addition to the models covered by Widaman, several other CFA-based approaches have been developed (Eid, 2000; Eid et al„ 2003; Kenny & Kashy, 1992; Marsh, 1993b; Marsh & Hocevar, 1988). Moreover, models for analyzing MTMM data have been defined in the frameworks of other methodological traditions such as variance component models (e.g., Millsap, 1995b; Wothke, 1995, 1996) or multiplicative correlation models (Browne, 1984; Dudgeon, 1994;

Wothke & Browne, 1990), which imply special CFA models (Dumenci, 2000). In this chapter we will not present all MTMM models that have been developed in the CFA framework. We will concentrate on those models that are most often applied and discuss their strengths and weaknesses. Moreover, we assume that all variables are centered (deviations from the mean), which means that we focus on covariance structures and do not deal with mean structures.

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