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Concept
Cochran–Mantel–Haenszel statistics
Summary
In statistics, the Cochran–Mantel–Haenszel test (CMH) is a test used in the analysis of stratified or matched categorical 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 response and/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.
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 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:
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.
The standard odds- or risk ratio of all strata could be calculated, giving risk ratios , where is the number of strata. If the stratification were removed, there would be one aggregate risk ratio of the collapsed table; let this be .
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).
It is easy to construct examples where this is not the case, and is larger or smaller than all of for .
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.
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