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Statistical Power for a Simultaneous Test of Factorial and Predictive Invariance

A common finding in studies of differential prediction across groups is that although regression slopes are the same or similar across groups, group differences exist in regression intercepts. Building on earlier work by Birnbaum (1979), Millsap (1998) presented an invariant factor model that would explain such intercept differences as arising due to unique variance or measurement error in the predictors and criterion. The model can be tested even when only a single predictor (p = 1) and criterion are present, but Millsap (1998) only illustrated the use of the test in very large samples. The present study investigates the Type I error and power performance of the test of fit across a range of sample sizes and effect sizes in simulated data, using an alternative model that includes group differences in measurement intercepts. Surprising results showed little power for the chi-square test in the p = 1 case but good power for a diagnostic indicator based on negative unique variance estimates. The p = 2 case showed better power in modest samples. A real example from an application on reading disabilities in Share and Silva (2003) is given. The implications of the results for studies of differential prediction involving group differences in regression intercepts are discussed.