In educational research usually a polar distinction is made to characterize phenomena: categorical versus continuous. In contrast, Ercikan and Roth (2006) demonstrate that the nature of many phenomena is categorical and continuous simultaneously. They argue that a polarization is not productive for educational research and propose to integrate both approaches. In order to meet this inquiry, recent developments in latent variable modeling aim to combine categorical and continuous approaches in a common modeling framework. We present an application of a so-called hybrid latent variable model (Muthén, 2008, Masyn, Henderson, & Greenbaum, 2010). For this purpose, we use data from a pilot study of the German National Educational Panel Study (NEPS; Blossfeld, Rossbach & von Maurice, 2011) and discuss our findings and their implications for other studies.
In our analysis, we investigate the latent structure of Rosenberg’s global self-esteem (GSE) scale. The widely used instrument comprises 10 items on self-esteem and it is typically treated as a unidimensional scale. Half of the items are positive (e.g. I take a positive attitude toward myself) and half are reverse phrased (e.g. I wish I could have more respect for myself). Several researchers employed factor analyses and found not only one, but also a second oblique or nested factor that is related to the negatively worded items. Some researchers interpret this second factor as a meaningful construct whereas others refer to a meaningless method factor (e.g. Marsh, 1996; DiStefano & Motl, 2006; Quilty, Oakman & Risko, 2006). The studies, however, are limited as they are bound in a variable-centered latent factor analytic framework and thus only consider latent factors. Therefore, we extent this framework in such a way that we also take qualitative differences between individuals into account by introducing a categorical latent variable. Such differences are traditionally modeled within a person-centered latent class framework. We integrate variable-centered and person-centered analyses and employ factor mixture analysis (FMA) to estimate a model with 2 classes and 1 continuous factor (see Muthén & Asparouhov, 2006 for applications in tobacco dependence). This model strikes the assumption of a unidimensional scale that holds for all students. But it differs from the variable-centered factor-analytic perspective that assumes that the GSE is multidimensional in nature in a crucial point. We rather argue that it is possible to disclose two classes of persons, and in each class the GSE scale is unidimensional. The classes may have different factor loading patterns. To sum up, we estimate a model with two latent classes and one GSE factor. Within the classes, the continuous factors represent quantitative differences in GSE between class members. We then investigate differences in the factorial structure between the two classes. If the factor loadings are equal in both classes strict factorial invariance applies and the same trait is observed in the two classes. Without factorial invariance the two classes cannot be compared with respect to the factor. Between these two extreme poles weak invariances of single items can be used to evaluate the properties of the instrument on item level.

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