Moments of the Fisher Transform: Applications Using Small Samples

This paper presents a theoretical development that extends results on the distribution of the sample correlation coefficient and the Fisher transform (Fisher, 1921). Several well known correlational procedures use the Fisher transform or its square as the basis of hypothesis testing. These procedures assume that the variance of the Fisher transform can be approximated adequately as 1/( 3) n − . We present results that demonstrate that for small sample size this is not necessarily true. We present results that demonstrate that for small sample size this is not necessarily true. Exact moments of the Fisher transform and its square are computed for both null and non-null correlations for small n ( ≤ 20). An extension of the classic series expansion formulae of Hotelling (Hotelling, 1953) for the moments of the Fisher transform and its square are discussed and compared with the exact moments of the Fisher transform and its square. Monte Carlo experiments are used to demonstrate how these results may produce significant improvements in the small sample performance of tests for pattern hypothesis on correlation matrices.