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A short note on the maximal point‐biserial correlation under non‐normality
Author(s) -
Cheng Ying,
Liu Haiyan
Publication year - 2016
Publication title -
british journal of mathematical and statistical psychology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.157
H-Index - 51
eISSN - 2044-8317
pISSN - 0007-1102
DOI - 10.1111/bmsp.12075
Subject(s) - mathematics , normality , correlation , variable (mathematics) , statistics , exponential function , range (aeronautics) , binary number , normal distribution , point (geometry) , distribution (mathematics) , function (biology) , mathematical analysis , geometry , materials science , arithmetic , evolutionary biology , composite material , biology
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.