Cochran–Mantel–Haenszel statistics

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.^[1] 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.^[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 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 (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 strata. 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:

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

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 $\chi ^{2}$ distribution asymptotically under the null hypothesis.^[1]

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 the strata are pairs.^[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. It should be noted that this is not a concern if the strata are small e.g. pairs.

Notes

1. Agresti, Alan (2002). Categorical Data Analysis (PDF). Hooken, New Jersey: John Wiley & Sons, Inc. pp. 231–232. ISBN 0-471-36093-7.
2. William G. Cochran (December 1954). "Some Methods for Strengthening the Common χ2 Tests". Biometrics. 10 (4): 417–451. doi:10.2307/3001616. JSTOR http://www.jstor.org/stable/3001616.
3. Nathan Mantel and William Haenszel (April 1959). "Statistical aspects of the analysis of data from retrospective studies of disease". Journal of the National Cancer Institute. 22 (4): 719–748. doi:10.1093/jnci/22.4.719. PMID 13655060.
4. Nathan Mantel (September 1963). "Chi-Square Tests with One Degree of Freedom, Extensions of the Mantel–Haenszel Procedure". Journal of the American Statistical Association. 58 (303): 690–700. doi:10.1080/01621459.1963.10500879. JSTOR 2282717.
5. Agresti, Alan (2002). Categorical Data Analysis (PDF). Hooken, New Jersey: John Wiley & Sons, Inc. p. 413. ISBN 0-471-36093-7.
6. Day N.E., Byar D.P. (September 1979). "Testing hypotheses in case-control studies-equivalence of Mantel–Haenszel statistics and logit score tests.". Biometrics: 623–630. doi:10.2307/2530253. JSTOR 2530253.