Although already Raju et al. (2002) demonstrated major similarities of item response theory (IRT) and confirmatory factor analysis (CFA), Skrondal and Rabe-Hesekth (2007:723) recently noted that still “factor analysts and item-response theorists rarely cite each other, although their work is closely related and often published in the same journal, Psychometrika“.
In the paper at hand, we strive to overcome this shortcoming by cross-validating results from a polytomous IRT approach, the partial credit model (PCM; Masters 1982, 1988) by means of ordinal confirmatory factor analysis (CFA; Bollen 1989). Hitherto, in a couple of German large-scale educational surveys, the PCM was used in order to classify school classes along a set of parental socio-economic status variables that were deduced from Bourdieu’s (1986) cultural capital theory in order to assign crucial resources to comparably disadvantaged schools (Bonsen et al. 2007). However, these analyses typically assume the one-dimensionality of the underlying latent social composition variable on the student level. In CFA terminology, this is equivalent to a group-factor model whose group-factor correlations are all set to one (Rindskopf & Rose 1988:55f.) – which might impose an untenable restriction on the data.
Hence, in a first step, we fit a conventional PCM and test for the one-dimensionality of our latent social composition variable. In a second step, we only use those social composition items that achieved an acceptable fit in the PCM model to build up an ordinal CFA within subsequently both reflective and formative specifications (Diamantopoulos & Winkelhofer 2001). Tentative results based on a sample of highest-track secondary school students (N=3310) and their parents (N=2729) in the German federal state of North-Rhine Westphalia suggest that while in the PCM, a satisfactory-fitting one-dimensional social composition index could be obtained, this does not equally hold for the (reflective) CFA wherein a four-factor structure with an additional error correlation suited best. Next steps involve i) further cross-validation of the student-level results obtained by both PCM and CFA by means of a latent class analysis (LCA); and ii) comparing results from multilevel item response models (Raudenbush & Sampson 1999), multilevel CFA (Bauer 2003), and multilevel LCA (Vermunt 2003) that will have been fitted directly on the school-class level. We aim to conclude with a critical discussion of whether rigorous factor structure tests such as PCM and CFA are suitable for assessing school-classes’ social composition within the more ‘lax’ framework of cultural capital theory.

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