Bias in Estimation and Hypothesis Testing of Correlation

Abstract

This study examined bias in the sample correlation coefficient, r, and its correction by unbiased estimators. Computer simulations revealed that the expected value of correlation coefficients in samples from a normal population is slightly less than the population correlation, ρ, and that the bias is almost eliminated by an estimator suggested by R.A. Fisher and is more completely eliminated by a related estimator recommended by Olkin and Pratt. Transformation of initial scores to ranks and calculation of the Spearman rank correlation, rS, produces somewhat greater bias. Type I error probabilities of significance tests of zero correlation based on the Student t statistic and exact tests based on critical values of rS obtained from permutations remain fairly close to the significance level for normal and several non-normal distributions. However, significance tests of non-zero values of correlation based on the r to Z transformation are grossly distorted for distributions that violate bivariate normality. Also, significance tests of non-zero values of rS based on the r to Z transformation are distorted even for normal distributions.