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 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:

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

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:

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

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

Subset stability

The standard odds- or risk ratio of all strata could be calculated, giving risk ratios ${\displaystyle r_{1},r_{2},\dots ,r_{n}}$ , where ${\displaystyle n}$  is the number of strata. If the stratification were removed, there would be one aggregate risk ratio of the collapsed table; let this be ${\displaystyle R}$ .^{[citation needed]}

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 ${\displaystyle R}$  is larger or smaller than all of ${\displaystyle r_{i}}$  for ${\displaystyle i\in 1,\dots ,n}$ . 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 ${\displaystyle R}$  is bounded between ${\displaystyle \min(r_{i})}$  and ${\displaystyle \max(r_{i})}$ , and a well-behaved statistic as being infinitely differentiable and not dependent on the order of the strata. Then the CMH statistic is the unique well-behaved statistic satisfying subset stability.^{[citation needed]}

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.^[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

1. Agresti, Alan (2002). Categorical Data Analysis. Hoboken, 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 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. Ben Klemens (June 2021). "An Analysis of U.S. Domestic Migration via Subset-stable Measures of Administrative Data". Journal of Computational Social Science. 5: 351–382. doi:10.1007/s42001-021-00124-w. S2CID 236308711.
6. Agresti, Alan (2002). Categorical Data Analysis. Hoboken, New Jersey: John Wiley & Sons, Inc. p. 413. ISBN 0-471-36093-7.
7. Day N.E., Byar D.P. (September 1979). "Testing hypotheses in case-control studies-equivalence of Mantel–Haenszel statistics and logit score tests". Biometrics. 35 (3): 623–630. doi:10.2307/2530253. JSTOR 2530253. PMID 497345.

External links

• Introduction to the Cochran-Mantel-Haenszel Test