Derivation of Analytic Formulas for the Sample Moments of the Sample Correlation over Permutations of Data

Pearson's correlation is among the mostly widely reported measures of association. The strength of the statistical evidence for linear association is determined by the p-value of a hypothesis test. If the true distribution of a dataset is bivariate normal, then under specific data transformations a t-statistic returns the exact p-value, otherwise it is an approximation. Alternatively, the p-value can be estimated by analyzing the distribution of the sample correlation under permutations of the data. Moment approximations of this distribution are not as widely used since estimation of the moments themselves are numerically intensive with greater uncertainties. In this paper we derive an inductive formula allowing for the analytic expression of the sample moments of the sample correlation under permutations of the data in terms of the central moments of the data. These formulas placed in a proper statistical framework could open up the possibility of new estimation methods for computing the p-value.