title: A short note on the maximal point-biserial correlation under non-normality.
abstract: The aim of this paper is to derive the maximal point-biserial correlation under non-normality. Several widely used non-normal distributions are considered, namely the uniform distribution, t-distribution, exponential distribution, and a mixture of two normal distributions. Results show that the maximal point-biserial correlation, depending on the non-normal continuous variable underlying the binary manifest variable, may not be a function of p (the probability that the dichotomous variable takes the value 1), can be symmetric or non-symmetric around p = .5, and may still lie in the range from -1.0 to 1.0. Therefore researchers should exercise caution when they interpret their sample point-biserial correlation coefficients based on popular beliefs that the maximal point-biserial correlation is always smaller than 1, and that the size of the correlation is always further restricted as p deviates from .5.
authors.affiliation: Department of Psychology, University of Notre Dame, Indiana, USA. ycheng4@nd.edu.
authors.forename: Ying
authors.initials: Y
authors.lastname: Cheng
authors.affiliation: Department of Psychology, University of Notre Dame, Indiana, USA.
authors.forename: Haiyan
authors.initials: H
authors.lastname: Liu
mesh_terms: D003198:Computer Simulation; D003627:Data Interpretation, Statistical; D015233:Models, Statistical; D012044:Regression Analysis; D018401:Sample Size; D016008:Statistical Distributions; D013223:Statistics as Topic
keywords: maximal point-biserial correlation; non-normality; point-biserial correlation