Cochran–Mantel–Haenszel statistics
In
Definition
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:
Treatment | No treatment | Row total | |
Case | Ai | Bi | N1i |
Controls | Ci | Di | N2i |
Column total | M1i | M2i | Ti |
The common
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.[1]
Subset stability
The standard odds- or
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.
Klemens[5] defines a statistic to be subset stable iff is bounded between and , and a well-behaved statistic as being
Related tests
- The McNemar test as their test statistics are identical when each stratum shows a pair.[6]
- 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.[7]
- 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.
Notes
- ^ ISBN 0-471-36093-7.
- JSTOR 3001616.
- PMID 13655060.
- JSTOR 2282717.
- S2CID 236308711.
- ISBN 0-471-36093-7.
- PMID 497345.