It allows an investigator to test the association between a binary predictor or treatment and a binary outcome such as case or control status while taking into account the stratification.
[1] Unlike the McNemar test, which can only handle pairs, the CMH test handles arbitrary strata sizes.
[2][3] Extensions of this test to a categorical response and/or to several groups are commonly called Cochran–Mantel–Haenszel statistics.
[4] It is often used in observational studies in which random assignment of subjects to different treatments cannot be controlled but confounding covariates can be measured.
We consider a binary outcome variable such as case status (e.g. lung cancer) and a binary predictor such as treatment status (e.g. smoking).
The stratified data are summarized in a series of 2 × 2 contingency tables, one for each stratum.
The i-th such contingency table is: The common odds-ratio of the K contingency tables is defined as: The null hypothesis is that there is no association between the treatment and the outcome.
The test statistic is: It follows a chi-squared distribution asymptotically with 1 degree of freedom under the null hypothesis.
If the stratification were removed, there would be one aggregate risk ratio of the collapsed table; let this be
[citation needed] One generally expects the risk of an event unconditional on the stratification to be bounded between the highest and lowest risk within the strata (or identically with odds ratios).
This is comparable but not identical to Simpson's paradox, and as with Simpson's paradox, it is difficult to interpret the statistic and decide policy based upon it.
Klemens[5] defines a statistic to be subset stable iff
, and a well-behaved statistic as being infinitely differentiable and not dependent on the order of the strata.