Conditional and unconditional tests for the extended Stuart-Maxwell design

A design is given that includes the designs for the Stuart-Maxwell and Cochran tests. Two models, called the conditional and unconditional, are given for this extended design and the corresponding Wald-type test statistics derived. These test statistics differ by a simple constant but purportedly have the same asymptotic χ2 null distribution. Since the unconditional test statistic is always larger than the conditional, if inference is based on their asymptotic χ2 null distributions, the unconditional test will always be more critical of the null hypothesis. The test of Stuart is based on a model that is neither conditional nor unconditional from this perspective, but coincides with the conditional test statistic. The discussion here addresses an apparent contradiction in the literature and indicates that for the unconditional tests inference should be based on resampling p-values that reflect the model. The design for the extended Stuart-Maxwell tests is consistent with that for Cochran–Mantel–Haenszel general association test.