Is the Pearson r² Biased, and if So, What Is the Best Correction Formula?

Abstract

In this study the authors investigated the use of 5 (i.e., Claudy, Ezekiel, Olkin-Pratt, Pratt, and Smith) R ² correction formulas with the Pearson r ². The authors estimated adjustment bias and precision under 6 X 3 X 6 conditions (i.e., population ρ values of .0, .1, .3, .5, .7, and .9; population shapes normal, skewness = kurtosis = 1, and skewness = -1.5 with kurtosis = 3.5; ns = 10, 20, 40, 60, 100, and 200 respectively). Results indicate that the sample Pearson r ² is marginally biased at small sample sizes and small population effect sizes, and that the Ezekiel and the Smith R ² corrections work well in controlling this bias.

Keywords:

• bivariate relationship
• correlation
• effect sizes
• Pearson r
• statistical bias