Cochran–Mantel–Haenszel statistics

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:

${\displaystyle R={{\sum _{i=1}^{K}{{A_{i}D_{i}} \over T_{i}}} \over {\sum _{i=1}^{K}{{B_{i}C_{i}} \over T_{i}}}},}$

The null hypothesis is that there is no association between the treatment and the outcome. More precisely, the null hypothesis is ${\displaystyle H_{0}:R=1}$ and the alternative hypothesis is ${\displaystyle H_{1}:R\neq 1}$. The test statistic is:

${\displaystyle \xi _{CMH}={[{\sum _{i=1}^{K}(A_{i}-{N_{1i}M_{1i} \over T_{i}})]^{2}} \over {\sum _{i=1}^{K}{N_{1i}N_{2i}M_{1i}M_{2i} \over T_{i}^{2}(T_{i}-1)}}}.}$

It follows a ${\displaystyle \chi ^{2}}$ distribution asymptotically with 1 df under the null hypothesis.[1]

• 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.[5]
• Conditional logistic regression is more general than the CMH test as it can handle continuous variable and perform multivariate analysis. When the CMH test can be applied, the CMH test statistic and the score test statistic of the conditional logistic regression are identical.[6]
• 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.