Linear equality constraints: Reformulations of criterion related profile analysis with extensions to moderated regression for multiple groups.

Criterion-related profile analysis (CPA) is a least squares linear regression technique for identifying a criterion-related pattern (CRP) among predictor variables and for quantifying the variance accounted for by the pattern. A CRP is a pattern, described by a vector of contrast coefficients, such that predictor profiles with higher similarity to the pattern have higher expected criterion scores. A review of applications shows that researchers have extended the analysis to meta-analyses, logit regression, canonical regression, and structural equation modeling. It also reveals a need for better methods of comparing CRPs across populations. While the original method for identifying the CRP tends to underestimate the variance accounted for by pattern only, both the pattern identified by the original method and the pattern identified by the new method proposed here have useful and complementary interpretations. Imposing linear equality constraints on regression coefficients yields a more accurate method of estimating the variance accounted for by pattern only, and this constrained approach leads to moderated regression models for investigating whether the CRP is the same in two or more populations. Finally, we show how the elements in Cronbach and Gleser's (1953) classic profile decomposition are related to the linear regression model and the CPA model. Academic ability tests as predictors of college GPA are used to illustrate the analyses. Implications of the profile pattern models for psychological theory and applied decision-making are discussed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).