Proportional Reduction in Error Interpretations for the Squared Generalized Multiple, Partial, and Multiple-Partial Correlation Coefficients and Their Special Cases

Abstract

Proportional Reduction in Error (PRE) interpretations for three generalized product-moment statistics are provided. The PRE interpretation for M2, the generalized multiple correlation coefficient, theoretically consolidates and extends those developed for R2, Rs2, Moron's multiple τ2, T2, multiple ϕ2 and various other hybrid special cases of M2. That for partial Γ2, provides PRE interpretations for partial r2, partial rs2, Kendall's partial τ2, partial tau-b2, partial ϕ2, and various other hybrid special cases of partial Γ2. The PRE interpretation for the square of the generalized multiple-partial correlation coefficient in effect defines multiple-partial correlation coefficients for rank, ordinal, dichotomous and hybrid systems of variables. This research has several implications. First, it documents that generalized summary statistics such as M2 have valid interpretations even when the generalized regression equation has a constant term of zero due to being based on the comparison of pairs of observations. Second, it provides PRE interpretations for the ordinal special cases based on Kendall's tau-b: the square of multiple tau-b, the square of partial tau-b, and a new multiple- partial tau-b statistic. Finally, it provides PRE interpretations for summary statistics for systems of variables that combine interval, rank, ordinal, and dichotomous information.