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:

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 under the null hypothesis.[1]

Notes

  1. 1 2 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. JSTOR http://www.jstor.org/stable/3001616. doi:10.2307/3001616.
  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. PMID 13655060. doi:10.1093/jnci/22.4.719.
  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. JSTOR 2282717. doi:10.1080/01621459.1963.10500879.
  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. JSTOR 2530253. doi:10.2307/2530253.
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