Exact Stratified Linear Rank Tests for Ordered Categorical and Binary Data

Abstract

We present an efficient algorithm for generating exact permutational distributions for linear rank statistics defined on stratified 2 × c contingency tables. The algorithm can compute exact p values and confidence intervals for a rich class of nonparametric problems. These include exact p values for stratified two-population Wilcoxon, Logrank, and Van der Waerden tests, exact p values for stratified tests of trend across several binomial populations, exact p values for stratified permutation tests with arbitrary scores, and exact confidence intervals for odds ratios embedded in stratified 2 × c tables. The algorithm uses network-based recursions to generate stratum-specific distributions and then combines them into an overall permutation distribution by convolution. Where only the tail area of a permutation distribution is desired, additional efficiency gains are achieved by backward induction and branch-and-bound processing of the network. The algorithm is especially efficient for highly imbalanced categorical data, a situation where the asymptotic theory is unreliable. The backward induction component of the algorithm can also be used to evaluate the conditional maximum likelihood, and its higher order derivatives, for the logistic regression model with grouped data. We illustrate the techniques with an analysis of two data sets: The leukemia data on survivors of the Hiroshima atomic bomb and data from an animal toxicology experiment provided by the U.S. Food and Drug Administration.