A comparison of exact tests for trend with binary endpoints using bartholomew's statistic

Tests for trend are important in a number of scientific fields when trends associated with binary variables are of interest. Implementing the standard Cochran-Armitage trend test requires an arbitrary choice of scores assigned to represent the grouping variable. Bartholomew proposed a test for qualitatively ordered samples using asymptotic critical values, but type I error control can be problematic in finite samples. To our knowledge, use of the exact probability distribution has not been explored, and we study its use in the present paper. Specifically we consider an approach based on conditioning on both sets of marginal totals and three unconditional approaches where only the marginal totals corresponding to the group sample sizes are treated as fixed. While slightly conservative, all four tests are guaranteed to have actual type I error rates below the nominal level. The unconditional tests are found to exhibit far less conservatism than the conditional test and thereby gain a power advantage.