# Cochran–Mantel–Haenszel statistics

In statistics, the Cochran–Mantel–Haenszel statistics are a collection of test statistics used in the analysis of stratified categorical data.[1] They are named after William G. Cochran, Nathan Mantel and William Haenszel.[2][3][4][5] One of these test statistics is the Cochran–Mantel–Haenszel (CMH) test, which allows the comparison of two groups on a dichotomous/categorical response. It is used when the effect of the explanatory variable on the response variable is influenced by covariates that can be controlled. It is often used in observational studies where random assignment of subjects to different treatments cannot be controlled, but influencing covariates can.

In the CMH test, the data are arranged in a series of associated 2 × 2 contingency tables, the null hypothesis is that the observed response is independent of the treatment used in any 2 × 2 contingency table. The CMH test's use of associated 2 × 2 contingency tables increases the ability of the test to detect associations (the power of the test is increased).[citation needed]

## Notes

1. ^ SAS/STAT(R) 9.2 User's Guide Cochran-Mantel-Haenszel Statistics
2. ^ Wittes J, Wallenstein S (December 1993). "The Power of the Mantel-Haenszel Test". Biometrics 49 (4): 1077–87. doi:10.2307/2532249. PMID 8117902.
3. ^ 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.
4. ^ 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.
5. ^ 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 http://www.jstor.org/stable/2282717.