The formalization of fairness: issues in testing for measurement invariance using subtest scores

Measurement invariance is an important prerequisite for the adequate comparison of group differences in test scores. In psychology, measurement invariance is typically investigated by means of linear factor analyses of subtest scores. These subtest scores typically result from summing the item scores. In this paper, we discuss 4 possible problems related to this common practice. Specifically, we discuss (a) nonlinearity of the latent variable to subtest relation; (b) suboptimality of the total score as a proxy for the latent variable measured through the item scores; (c) non-normality of the subtest score; and (d) differences in the nature of the latent variable at the item level as compared to the latent variable at the subtest level. Additionally, we give guidelines to overcome these problems and illustrate the issues by analysing data that pertain to a performal IQ data set.