Investigating sources of within & between-group differences and measurement invariance (MI) across groups is fundamental to any meaningful group comparison based on observed test scores. It is shown that by placing certain restrictions on the multigroup confirmatory factor model, it is possible to investigate the hypothesis that within & between-group differences are due to the same factors.
Moreover, the modeling approach clarifies that absence of measurement bias implies common sources of within & between-group variation. It is shown how the influence of background variables can be incorporated in the model.
The advantages of the modeling approach as compared with other commonly used methods for group comparisons is discussed and illustrated by means of an analysis of empirical data.
[Keywords: common factor model, within-group differences, between-group differences, measurement invariance, confirmatory factor analysis]
…The paper is organized as follows. First, the multigroup CFA model is presented. We show that observed scores are decomposed into common factor scores and a regression residual, which comprises measurement error and item specific error. This decomposition has the advantage that groups can be compared with respect to the means and covariances of the factors. Second, we explain the concept of MI on a theoretical level and on a more practical level in the context of the multigroup common factor model. The multigroup common factor model corresponding to MI is characterized by a set of invariance restrictions across groups. Third, we show that MI implies that between-group differences are unlikely to be due to other factors than those capturing systematic within-group differences. We discuss how this result can be used in practice. By comparing a model with the invariance restrictions across groups to a less restricted model in a likelihood ratio test, one can examine not only whether MI holds but also whether between-group differences are due to differences in the same factors as the within-group differences. Fourth, we discuss how the multigroup model can be extended to include background variables. The way in which background variables are integrated can be guided by the outcome of tests of MI (1992, 1998). Finally, we briefly discuss the advantages of multigroup CFA as compared with other commonly used methods and present, for the purpose of illustration, an analysis of scores of African and Caucasian Americans on an IQ test (1980).