Exact Tests of Significance in Higher Dimensional Tables

We describe methods used to provide an exact test of significance of the hypothesis that all factors are mutually independent of each other in 2 3 and 2 4 contingency tables. Several numerical examples demonstrate the advantages of exact tests over approximate significance levels. We give bounds on the number of tables needed to perform this exact significance test. In 4 or more dimensions, the number of tables in this enumeration becomes astronomical with even modest sample sizes. Inverting the characteristic function of the exact distribution has proved useful in these situations. 1 Introduction The Fisher exact test is well known in 2 \Theta 2 tables. The central idea is to enumerate all possible outcomes consistent with a given set of marginal totals and add up the probabilities of those tables more extreme than the one observed. Conditional on the margins, a 2 \Theta 2 table is a one-dimensional random variable so there are few conceptual difficulties. In larger tables, ther..