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Generalized Cochran-Mantel-Haenszel tests

Generalized Cochran-Mantel-Haenszel test:

The Generalized Cochran-Mantel-Haenszel test is a family of tests aimed at detecting of association between two categorical variables observed in K strata.

The initial data are represented as a series of K RxC contingency table s, where K is the number of strata and at least one of the variables (“group”, “response”) takes on more than 2 values. Typically, in each table the rows correspond to the “Treatment group” values (e.g “Placebo”, “Low dose”, “High dose) and the columns to the “Response” values (e.g “Worsening”, “No change” “Improvement”).

The stratification of the subjects into K groups (according to the values of controlled variables – e.g. “Age group”) increases the power of the test to detect association. This increase in power comes from comparing like subjects to like subjects.

The alternative hypotheses vary across the generalized CMH tests, depending on the type of the scale of the two “treatment” and “response” variables.

The general feature of the test statistics used in the generalized CMH tests is that they all are quadratic forms of the differences

nijk mijk;   i=1,…,(R1);   j=1,…,(C1);   k=1,…,K;
nijk mijk;   i=1,…,(R1);   j=1,…,(C1);   k=1,…,K;nijk mijk;   i=1,…,(R1);   j=1,…,(C1);   k=1,…,K;
nijk mijk;   i=1,…,(R1);   j=1,…,(C1);   k=1,…,K;
nijk mijk;   i=1,…,(R1);   j=1,…,(C1);   k=1,…,K;

where nijk and mijk is the observed and expected counts in the (ij)th cell of the (k)th table. For details on the major three generalized CMH statistics, see the General association statistic, the Mean score statistic, and the Correlation statistic.

See also the Cochran-Mantel-Haenszel test – the prototype of the generalized CMH tests.

Note: The CMH tests and corresponding statistics should not be confused with Cochran Q statistic, Mantel-Cox test.

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