Exact Inference for Kendall'sSand Spearman's ρ with Extension to Fisher's Exact Test inr×cContingency Tables

Abstract

Abstract We formulate the problem of exact inference for Kendall's S and Spearman's D algebraically, using a general recursion formula developed by Smid for the score S with ties in both rankings. Analogous recursion formulas are shown to hold for the score D as well as for a log transform, F, of the score used in Fisher's exact test of independence in contingency tables. A new implementation of Mehta and Patel's network algorithm is then applied to obtain exact significance levels of either S or D for observations from both continuous and discrete distributions. A simple extension is made to obtain Fisher's exact test in r x c contingency tables. Observed CPU times for contingency table problems four to six of Mehta and Patel and problems four and five of Clarkson, Fan, and Joe are roughly 2/3 of those obtained using Clarkson's et al. implementation of the network algorithm. It is shown that a hierarchy, with F > S > D, holds regarding the rate of aggregation. An algorithm for rapid lexicographic enumera...