In statistics, the Cochran–Mantel–Haenszel test is a test used in the analysis of stratified or matchedcategorical data. 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. Unlike the McNemar test which can only handle pairs, the CMH test handles arbitrary strata size. It is named after William G. Cochran, Nathan Mantel and William Haenszel. Extensions of this test to a categorical responseand/or to several groups are commonly called Cochran–Mantel–Haenszel statistics. It is often used in observational studies where random assignment of subjects to different treatments cannot be controlled, but confounding covariates can be measured.
Definition
We consider a binary outcome variable such as case status and a binary predictor such as treatment status. The observations are grouped in strata. The stratified data are summarized in a series of 2 × 2 contingency tables, one for each stratum. The i-th such contingency table is:
Treatment
No treatment
Row total
Case
Ai
Bi
N1i
Controls
Ci
Di
N2i
Column total
M1i
M2i
Ti
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. More precisely, the null hypothesis is and the alternative hypothesis is. The test statistic is: It follows a distribution asymptotically with 1 df under the null hypothesis.
Related tests
The McNemar test can only handle pairs. The CMH test is a generalization of the McNemar test as their test statistics are identical when each stratum shows a pair.
Breslow-Day test for homogeneous association. The CMH test supposes that the effect of the treatment is homogeneous in all strata. The Breslow-Day test allows to test this assumption. This is not a concern if the strata are small e.g. pairs.
