The hypothesis-based investigation of patterns of relatedness by means of confirmatory factor models: The treatment levels of the Exchange Test as example

Confirmatory factor analysis was originally proposed as a method for the confirmatory investigation of the structure of covariance matrices (Jöreskog, 1970). In such investigations the focus was on the correct assignment of manifest to latent variables, and it has proven to be a valuable research tool in a large number of applications. The theoretical importance of this focus results from the association of the latent variable with a specific concept that is defined in terms of a psychological construct (MacCorquodale & Meehl, 1948). Great advances in psychological science have been achieved by means of this research tool. However, confirmatory factor analysis is not really restricted to the standard approach that means investigating the correct assignment of manifest to latent variables. Starting with the work by McArdle (1986, 1988; McArdle & Epstein, 1987) and Meredith and Tisak (1984, 1990) confirmatory factor analysis has been refined by fixing loadings to specific sizes for investigating the presence of specific relationships mostly described as curves in repeated-measures data. This refinement has led to so-called growth curve and latent curve models (Duncan & Duncan, 2004; McArdle, 2009). These models proved to be very useful in the area of developmental and educational research. Furthermore, there are fixed-links models (Schweizer, 2006a, 2006b, 2007, 2008, 2009) that aim at the hypothesis-guided decomposition of variances and covariances. Within this approach hypotheses can give rise to various types of specific patterns of the relatedness of the manifest and latent variables that provide the outset for such an investigation. Growth curve and latent curve models as well as fixed-links models share the same mathematical foundations although they have been developed in different directions. An example that is especially appropriate for highlighting the special characteristics of such patterns shall provide the outset for the presentation of the rationale. The example is a measure of working memory capacity (Schweizer, 1996) that includes several treatment levels, which give rise to precise relational expectations based on cognitive theory, which can easily be summarized by a formal hypothesis. This measure requires the mental exchange of simple figures to bring two series of simple figures to a match. Figure 1 gives an example of such a pair of two series. The hypothesis-based investigation of patterns of relatedness by means of confirmatory factor models: The treatment levels of the Exchange Test as example