CMH Test Basic Concepts

The Cochran-Mantel-Haenszel (CMH) test is used to test multiple 2 ⨯ 2 contingency tables across different values of a confounding variable. The test determines whether there is a significant difference between the odds ratios across the different values of the confounding variable.

We represent each of the k 2 ⨯ 2 contingency as shown in Figure 1.

Figure 1 – Contingency table

Odds Ratio

The odds ratio (see Effect Size for Chi-square Tests) for contingency table i is then

For example, if Factor B represents Has a Heart Attack (Yes in column 1 and No in column 2) and Factor A represents Statin Taken (Yes in row 1 and No/placebo in row 2), then the odds ratio measures whether taking statins has an impact on the rate of heart attacks.

An odds ratio greater than one indicates that the drug has a positive effect in reducing the rate of heart attacks, while a value less than one indicates that the drug has a negative effect by increasing heart attacks. Actually, the phrasing just used is misleading since it suggests a cause and effect. Instead, we are just measuring the association between the use of a statin and the rate of heart attacks. Finally, an odds ratio of one shows that there is no association between statins and the rate of heart attacks.

Hypothesis Testing

The CMH test is used to test the null hypothesis that

OR_i = 1 for all i = 1, 2, …, k

We now define the common odds ratio r  by

This is the odds ratio of factor A to factor B after removing the influence of the confounding factor (which has k values).

The null and alternative hypotheses for the CMH test can be viewed as H₀: r = 1 and H₁: r ≠ 1. Now let

The test statistic for the CMH test is

The test can be run with or without the Yates 0.5 continuity correction, although the correction gives slightly better results.

A significant result means that there is an association between the two factors represented in the contingency tables.

Confidence Interval

We can also use the following formulas to estimate a 1 – α confidence interval for r.

The confidence interval is then defined as follows:

where z_crit = NORM.S.INV(1–α/2).

Woolf’s Heterogeneity Test

There is also a Woolf’s Heterogeneity Test which tests the null hypothesis

OR₁ = OR₂ = …= OR_k

Here the test statistic is

whereReferences

PennState (2017) Cochran-Mantel-Haenszel test
https://online.stat.psu.edu/stat504/lesson/5/5.3/5.3.5

SAS Help Center (2019) Cochran-Mantel-Haenszel statistics
https://documentation.sas.com/doc/en/pgmsascdc/9.4_3.4/statug/statug_freq_details92.htm

Moral de la Rubia, J., Valle de la O, A. (2023) Everything you wanted to know but could never find from the Cochran-Mantel-Haenszel test
https://www.scirp.org/pdf/jdaip_2023082810224634.pdf