Nonoverlap proportion and the representation of point-biserial variation

PLoS One. 2020 Dec 28;15(12):e0244517. doi: 10.1371/journal.pone.0244517. eCollection 2020.


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 of [Formula: see text] data, 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 (ρpb) of two sets of [Formula: see text] data. ρpb is obtained by sorting the labeled [Formula: see text] data and analyzing the induced order in the categorical data using a diagonally symmetric 2 × 2 contingency table. We examine the correspondence between ρpb and point-biserial correlation (rpb) for uniform and normal distributions. We identify the [Formula: see text], [Formula: see text], and [Formula: see text] representations for Pearson product-moment correlation, Cohen's d, and rpb. We compare the performance of rpb versus ρpb and the sample size proportion corrected correlation (rpbd), 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 (rpbd, μ, ρpb) 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.

MeSH terms

  • Algorithms*
  • Data Analysis*
  • Monte Carlo Method
  • Regression Analysis*
  • Sample Size

Associated data

  • figshare/10.6084/m9.figshare.11591334.v2

Grants and funding

The author(s) received no specific funding for this work. The author, Stanley Luck, is a member of Vector Analytics, LLC, which is a science consulting company. Since there is no salary, Vector Analytics did not provide any funds for this work. Vector Analytics, LLC did not have any additional role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. The specific roles of the author are articulated in the ‘author contributions’ section.