The Power of the Mantel—Haenszel Test

The Mantel-Haenszel (MH) test (Mantel and Haenszel 1959) is a widely used method for analyzing sets of two-by-two tables that compare the outcome of two treatments in several strata (Bailar and Anthony 1977). The procedure tests the null hypothesis of no partial association between treatment and outcome in any of the strata, or tables, against the alternative that the partial association, as measured by the common odds ratio, differs from 1. Birch (1964) showed that for a fixed number of tables, both the MH test and an earlier version due to Cochran (1954) are asymptotically identical to the uniformly most powerful unbiased test of this hypothesis. This article is concerned with calculating the power of the MH test as a prelude to embarking upon a study. We consider versions of the test with and without corrections for continuity. The spirit behind the approximations is that, although the odds ratio is the appropriate parameter for computing asymptotic power under local alternatives, in practice studies are designed for nonlocal alternatives. Our approach is based on evaluating a weighted difference of proportions of success in each table, rather than a weighted average of log-odds ratios, and is valid even if the odds ratios differ from table to table. The adequacy of our approximations to power is evaluated for planning three types of studies: the comparative binomial (CB), the double dichotomy (DD), and the survey (SY). In the CB trial, the proportion λ i , of patients in the ith strata, as well as the proportion pi of patients in the ith strata who are assigned to the first treatment, are constants (i = 1, 2, …, N). For the DD only the pi 's are constant, whereas for the SY both vectors are random variables. We first approximate power under the condition that for each stratum NE(λ i ) is large. This approximation cannot distinguish between corrected and uncorrected statistics nor among the different sampling schemes. In some sampling schemes, the number of patients per table is fixed but the number of tables increases. The more accurate approximation presented in this article is based on the expected value and variance of the numerator and denominator of the MH statistic for each design and yields different estimates of power for each design. Simulation studies show that for the CB the new approximations for power are closer to the exact values than previously obtained formulas. We also present designs in which the CB power differs markedly from that of the DD, and we show that our approximations do an excellent job of tracking these differences. This comparison also indicates that the strategy of poststratification may in some situations lead to an appreciable loss of power.