Robustness of Estimators of the Squared Multiple Correlation and Squared Cross-Validity Coefficient to Violations of Multivariate Normality

A monte carlo experiment was conducted to evaluate the robustness of two estimators of the population squared multiple correlation (R²_p) and one estimator of the population squared cross-validity coefficient (R²_cv) to a common violation of multivariate normality. Previous research has shown that these estimators are approximately unbiased when independent and dependent variables follow a joint multivariate normal distribution. The particular violation of multivariate normality studied here consisted of a dependent variable that may assume only a few discrete values. The discrete dependent variable was simulated by categorizing an underlying continuous variable that did satisfy the multivariate normality condition. Results illustrate the attenuating effects of categorization upon R²_p and R²_cv. In addition, the distributions of sample squared multiple correlations and sample squared cross-validity coefficients are affected by categorization mainly through the attenuations of R²_P and R²_cv. Consequently, the formula estimators of R²_p and R² _cv were found to be as accurate and unbiased with discrete dependent variables as they were with continuous dependent variables. Substantive researchers who use categorical dependent variables, perhaps obtained by rating scale judgments, can justifiably employ any of the three estimators examined here.