The Cochran–Mantel–Haenszel (CMH) methodology is a suite of tests applicable to particular tables of count data. Inference is conditional on the treatment and outcome totals on each stratum being known before sighting the data. The CMH tests are important for analyzing randomised blocks data when the responses are categorical rather than continuous. This overview will first review the traditional CMH tests and then explore new alternative presentations of the ordinal CMH tests. Next the ordinal CMH tests will be extended so they can be used to test for higher moment effects. Finally unconditional analogues of the extended CMH tests will be developed.