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A short note on the maximal point-biserial correlation under non-normality.

Ying Cheng1, Haiyan Liu2

  • 1Department of Psychology, University of Notre Dame, Indiana, USA. ycheng4@nd.edu.

The British Journal of Mathematical and Statistical Psychology
|July 27, 2016
PubMed
Summary

Researchers should be cautious when interpreting point-biserial correlations under non-normal distributions. The maximal correlation may not be a function of probability, can be asymmetric, and still range from -1.0 to 1.0.

Keywords:
maximal point-biserial correlationnon-normalitypoint-biserial correlation

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Area of Science:

  • Statistics
  • Psychometrics

Background:

  • Point-biserial correlation is commonly used to assess the relationship between a continuous and a dichotomous variable.
  • Assumptions of normality are often violated in real-world data, potentially affecting correlation interpretations.

Purpose of the Study:

  • To derive the maximal point-biserial correlation coefficient under various non-normal distributions.
  • To investigate how non-normality influences the behavior and bounds of the point-biserial correlation.

Main Methods:

  • The study mathematically derived the maximal point-biserial correlation.
  • Non-normal continuous distributions examined include uniform, t-distribution, exponential, and mixtures of normal distributions.

Main Results:

  • Maximal point-biserial correlation is not always a function of p (probability of the dichotomous variable equaling 1) under non-normality.
  • The correlation's symmetry around p = 0.5 varies, and its range can still be [-1.0, 1.0].

Conclusions:

  • Existing assumptions about maximal point-biserial correlation being <1 and restricted as p deviates from 0.5 do not universally hold under non-normality.
  • Researchers must exercise caution when interpreting sample point-biserial correlations derived from non-normal data.