The starting point of single-indicator models is the classical MTMM matrix. However, because models of SEM are covariance structure models and the covariance matrix is more informative, SEM of MTMM data is based on the MTMM covariance matrix, not on the correlation matrix (for problems analyzing correlation matrices with SEM, see Cud-eck, 1989). In single-indicator models there is one indicator (observed variable) Yjk for each combination of a trait j and a method k. These observed variables are decomposed in different ways. We will describe five models that are built on different assumptions: (a) the correlated trait model, (b) the correlated trait/correlated uniqueness model, (c) the correlated trait/uncorrelated method model, (d) the correlated trait/correlated method model, and the (e) correlated trait/correlated method (M-l) model.
The correlated trait (CT) model is the simplest model. It assumes that each observed variable Y.k can be decomposed into a common trait variable T. and a residual Ejh. This model is depicted in Figure 20.1a for three trait variables (fear, anger, and sadness) measured each by three methods (self, friend, and acquaintance). The latent trait variable is the common factor of all observed variables that are appropriate for measuring the trait. The correlations between the different trait variables indicate discriminant validity. The variance of an observed variable that is explained by the trait variable indicates convergent validity or consistency, which is the degree of variance that is due to the common trait variable. The variance of the residual is the unexplained variance that is due to measurement error or method-specific influences. The consistency coefficient equals the reliability coefficient of classical test theory if there are no systematic method effects and only measurement error influences. If the residual variable also covers method effects, these method effects do not generalize across traits because the residual variables are assumed to be uncorrelated between traits. This assumption will be violated and the model will not be appropriate if there are systematic method effects, which is the case, for example, when the friend of one target person consistently overestimates the target's fear, anger, and sadness, whereas the friend of another target person consistently underestimates his or her standing on these traits. In this case one would expect a correlation of the residuals belonging to the method friend report. If method effects and error influences are present and the CT model fits the data, the consistency coefficient will be a lower bound for the reliability coefficient, and the unreliability coefficient (the degree of variance that is explained by the residuals) will be the upper bound for method-specific effects. However, it is important to note that even if the model fits the data perfectly, it cannot be determined whether there are method-specific effects in addition to measurement error because the two sources of variance are confounded. The CT model is a rather restrictive model for multimethod research because it assumes that method effects do not generalize across traits. Because correlated method effects can be expected in most applications in psychology, this model is usually too restrictive. The model might be appropriate, however, if only one trait is considered and the different methods are randomly chosen raters from a group of possible raters. For example, when conducting an evaluation of teachers based on the ratings of three students randomly selected from one class of each teacher, the CT model with one trait can be applied. In this case, a one-factor model explains the consistency in the students' ratings. With only one trait, systematic method effects across traits are not of interest.
FIGURE 20.1. Single-indicator MTMM models, (a) CT model: correlated trait model; (b) CTCU model: correlated trait/correlated uniqueness model; (c) CTUM model: correlated trait/uncorrelated method model; and (d) CTC(M-l) model: correlated trait/correlated method model with one method factor less than methods considered. Y' observed variable, j: trait, k: method; T: trait factor, Mk: method factor; E.k.: error variable; XT.fe: factor loadings. Factor loadings are only depicted for one path for each kind of factor, but they are estimated for all variables. SR = self-report; FR = friend report; AR = acquaintance report.
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