A L Tests for Cross-classifications: Approximation of Attained Significance Levels

Several recent articles have outlined methodologies for exact conditional analyses of data which are summarized in an r × c matrix of counts. Depending upon the purposes of the experimenter, several types of null and alternative hypotheses are appropriate. Some of the procedures [Agresti & Wackerly, 1977; Freeman & Halton, 1951] test for independence between two variables, while others [Klotz & Teng, 1977] are designed to compare several treatments with respect to observed responses on an ordinal categorical variable. Extentions of Fisher's [1971] now famous experiments involving the tea-tasting lady have also been considered [Wackerly, McClave & Rao, 1978]. When testing the null hypothesis of independence of two variables in a crossclassification table, various types of alternative hypotheses may be appropriate. For example, if the two variables are measured on a strictly nominal scale, we might be interested in the broad alternative of "statistical dependence". A natural test statistic in that case is the standard chi-square statistic or a nominal measure of association such as Goodman and Kruskal's lambda or tau. Alternately, if both variables are ordinal categorical, we might wish to detect whether there is a monotonic relationship between the variables. In that case we might use the alternative hypothesis that the proportion of concordant pairs of observations is unequal to the proportion of discordant pairs, and employ Kendall's tau-b as the test statistic. In the case of comparing several treatments on an ordinal categorical variable, the null hypothesis of independence corresponds to homogeneity of the treatments. We might then wish to use a Kruskal-Watlis type statistic for detecting response shifts among those treatments. In practice, cross-classification tables often occur in which the overall sample size or the cell frequencies are too small to employ asymptotically derived sampling distributions for these test statistics. In such cases, as an alternative procedure we can conduct an exact test of independence, conditional on the observed marginal frequencies. Probably the best known test of this nature is Fisher's exact test of independence for 2 × 2 tables. However,