Comparison Of The Different Models

The models described so far have different advantages and limitations. Therefore, some guidelines for an appropriate choice of the models are necessary. One major difference between the models is whether they allow correlated methods or not.

Thus, one decision that has to be made is whether it is necessary to allow correlated methods or not. A second difference between the models is whether the researcher wants to define a trait as a common factor from which all methods deviate or whether he or she wants to contrast and compare methods.

One criterion for deciding whether method factors or residuals will be correlated or not is the type of methods considered (Eid, chap. 16, this volume): If methods are interchangeable, it is not likely that there will be correlated method factors or correlated residuals in the models considered. An example of interchangeable methods is the assessment of teacher behavior by randomly selected students. If, for example, three traits of a teacher are measured by three students that are randomly selected for each teacher (i.e., each teacher has different raters), there will be three methods (students). To apply the models, the students must be assigned to one of three groups (method groups). However, it does not make any difference whether a student belongs to Method 1, Method 2, or Method 3. The assignment of students to method groups is totally interchangeable. In this case, it is not reasonable to expect that method factors or residuals of different methods are correlated in the model because of the total inter-changeability of the methods (students). Moreover, one would be interested in a trait measure that reflects a kind of common view of the teacher by his/her students. Because all students have more or less the same access to the teacher's behavior, the average of the ratings or the common factor score might be a good representation of the teacher's behavior. In this case, the CTCU or the CTUM model would be the most appropriate model because the trait factors are defined as common factors, and the models assume that the residuals and method factors are uncorrelated between the three rater groups.

The situation changes when the methods are not interchangeable but differ structurally. Consider, for example, the situation where the well-being of a teenager is assessed by the teenager him- or herself, his or her mother, and his or her father. In this case, the raters are not interchangeable. Moreover, one might assume that the parents have a common view of their child that is not shared with the child. The convergence between the parents' rating might be higher than the convergence between the mother and the child and the father and the child, that is, the two methods father and mother might be more highly correlated. Hence, a model that can capture this stronger method correlation might be most appropriate, which leads us to the CTCM and CTC(M-l) model as the inodels of choice.

In a second step, one has to decide whether the idea of a trait as a common factor from which all methods deviate is meaningful or whether one assumes that it is preferable to contrast the methods. If one is interested in measuring a common factor, one should apply the CTCM model. However, in the case of structurally different methods, the trait loadings could be quite different, which makes it difficult to interpret the common factor. In the case of more than three structurally different methods, it is likely that the assumption of one common factor—which puts constraints on the covariances of the different indicators—might be violated and that the model might have to be rejected. If one would like to contrast the methods, the CTC(M-l) model should be chosen. In this case, the teenager report could serve as the comparison standard, so the trait factors would represent the latent teenager ratings. The method factor mother would represent the deviations of the mother rating from the rating that would have been expected on basis of the self-rating. The method factor values represent over- and underestimations made by the mother. The method factor father has an analogous meaning. A positive correlation of the method factors, for example, would indicate the degree to which mother and father over- vs. underestimate their children in the same direction. To explain parental over- and underestimation, the method factors can be related to other variables.

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