What is the Cochran-Mantel-Haenszel test?

To perform the Cochran-Mantel-Haenszel (CMH) test, choose Stat > Tables > Cross Tabulation and Chi-Square and click Other Stats.

Use the CMH test to test the conditional association of two binary variables in the presence of a third categorical variable. For example, you are analyzing the results of an election in three different states between Candidates A and B. The first table shows the votes from all three states combined, tabulated by gender. Fisher's exact test reports a significant p-value of 0.008 for this table, indicating that gender and vote are dependent.

Gender Candidate A Candidate B
Female 942 737
Male 737 699
Combined voting results from three states

Fisher's exact test: P-value = 0.0076587

However, you want to know whether the state in which a voter resides acts as a lurking variable in this association. You separate the combined table, tallying votes by gender for each state in the latter three tables. The CMH test determines whether the apparent difference between the votes of men and women is actually due to an effect of gender, or whether it occurs from the lurking variable of a voter's state of residence. In this example, the test analyzes the following three tables.

Gender Candidate A Candidate B
Female 524 227
Male 240 102
Voters who reside in State X
Gender Candidate A Candidate B
Female 160 250
Male 243 355
Voters who reside in State Y
Gender Candidate A Candidate B
Female 258 260
Male 254 242
Voters who reside in State Z

The CMH test assesses the degree of association between vote and gender while controlling for the state of residence. It calculates a common-odds ratio across the tables and a p-value to assess its significance.

In the example, the CMH test produces a common-odds ratio of 0.95. This observed statistic states that, across all states, the odds that a woman votes for Candidate A is 0.95 times the odds that a man votes for Candidate A; in other words, the odds of voting for Candidate A are almost equal for men and women. The CMH test also calculates a p-value to assess the statistical significance of the common-odds ratio: the p-value of 0.55 is insignificant. Therefore, you conclude that although vote and gender seem to be associated in the combined table, controlling for state of residence reveals that vote and gender are independent, within each state. It is possible that the true difference in voting patterns exists between states, not between genders. Additional analysis should concentrate on the effect that a voter's state of residence has on his or her vote, because this CMH test determined that the effect of gender is not statistically significant.

The CMH test assumes that no 3-way interaction exists.