Nonoverlap proportion and the representation of point-biserial variation.

Area of Science:

• Statistics
• Data Analysis
• Psychometrics

Background:

• Point-biserial variation is crucial for understanding relationships between continuous and dichotomous variables.
• Existing effect size measures may not fully capture all degrees of freedom in point-biserial variation.
• Mathematical representations lack a standardized coordinate system for effect size.

Purpose of the Study:

• To develop a complete set of parameters for point-biserial variation.
• To devise an algorithm for generating these parameters using data sorting.
• To introduce a novel nonoverlap proportion (ρpb) measure for effect size.

Main Methods:

• Utilized sorting as an intrinsic property of numbers and labels to generate parameters.
• Represented point-biserial variation algebraically as a Cartesian product of statistical parameters.
• Developed a novel algorithm for estimating the nonoverlap proportion (ρpb) using sorted data and contingency tables.

Main Results:

• Identified [Formula: see text], [Formula: see text], and [Formula: see text] representations for Pearson correlation, Cohen's d, and point-biserial correlation (rpb).
• Demonstrated the importance of sample size proportion invariance in effect size formulation.
• Showcased a case requiring three parameters (rpbd, μ, ρpb) for distinguishing point-biserial variation in CART regression.

Conclusions:

• The novel algorithm provides a comprehensive parameter set for point-biserial variation.
• The nonoverlap proportion (ρpb) offers a valuable new measure for effect size.
• Accurate assessment of effect size requires considering multiple parameters for nuanced interpretation.