Nonoverlap proportion and the representation of point-biserial variation

We consider the problem of constructing a complete set of parameters that account for all of the degrees of freedom for point-biserial variation. We devise an algorithm where sort as an intrinsic property of both numbers and labels, is used to generate the parameters. Algebraically, point-biserial variation is represented by a Cartesian product of statistical parameters for two sets ofR 1data, and the difference between mean values ( δ ) corresponds to the representation of variation in the center of mass coordinates, ( δ , μ ). The existence of alternative effect size measures is explained by the fact that mathematical considerations alone do not specify a preferred coordinate system for the representation of point-biserial variation. We develop a novel algorithm for estimating the nonoverlap proportion ( ρ pb ) of two sets ofR 1data. ρ pb is obtained by sorting the labeledR 1data and analyzing the induced order in the categorical data using a diagonally symmetric 2 × 2 contingency table. We examine the correspondence between ρ pb and point-biserial correlation ( r pb ) for uniform and normal distributions. We identify theR 2,P 1, andS + 1representations for Pearson product-moment correlation, Cohen’s d , and r pb . We compare the performance of r pb versus ρ pb and the sample size proportion corrected correlation ( r pbd ), confirm that invariance with respect to the sample size proportion is important in the formulation of the effect size, and give an example where three parameters ( r pbd , μ , ρ pb ) are needed to distinguish different forms of point-biserial variation in CART regression tree analysis. We discuss the importance of providing an assessment of cost-benefit trade-offs between relevant system parameters because ‘substantive significance’ is specified by mapping functional or engineering requirements into the effect size coordinates. Distributions and confidence intervals for the statistical parameters are obtained using Monte Carlo methods.