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The Exact Noncentral Distributions of Spearman's r and other Related Correlation Coefficients

Abstract This article presents a general construction principle for the non-central distribution of Spearman's r without the usual reference to the underlying distributions of the two continuous random variables. The only distribution used is the bivariate distribution on the rank pairs in an n × n table. The noncentral distribution is generated by the calculation of the unequal probabilities of (n!)2 bipermutations. This distribution is approximated also by a skew distribution derived from the binomial distribution. Partial and multiple correlation coefficients and tricorrelation coefficients are also studied.