# Generalised Cochran-Mantel-Haenszel tests

Three generalised tests for association between row and column classes are offered for stratified r by c tables produced in the crosstabs function when you specify a third (stratum, controlling for) classifier (Agresti, 2002; Landis et al., 1978, 1979).

The first test (ordinal association) assumes that there is meaningful order to both the columns and rows of each r by c table.

The second test (ordinal columns vs. nominal rows) assumes that there is meaningful order in the columns of each r by c table.

The third test (nominal association) does not assume any order in rows or columns; it provides a general test of association between the row and column classifiers.

The reliability of the tests increases with sample size, but unlike the Pearson chi-square statistic for single r by c tables, small counts in a few cells are unlikely to invalidate the tests.

You could control for more than one factor by making a stratum variable consisting of several factors (e.g. UK male, US male, UK female, US female to control for gender and country of residence).

Note that there are other approaches to these analyses, namely ordinal and nominal logistic regression. You should consult with a statistician before using these methods in important studies.

Data entry

Note the potentially confusing terminology over row and column scores: The 'row scores' are the scores associated with the column classification; these are applied to the entries (by column) in each row. The 'column scores' are the scores associated with the row classification; these are applied to the entries (by row) in each column.

Example

From Agresti (2002).

The data can be found in the Tables worksheet of the Test workbook. Use the menu item Analysis_Crosstabs to generate a cross tabulation of the job satisfaction, income and gender variables. Use the row (income) scores as 3, 10, 20 and 35. Use the column (job satisfaction) scores as 1, 3, 4 and 5.

For this example:

Generalised Cochran-Mantel-Haenszel tests

Row variable (first classifier): Income

Column variable (second classifier): Job Satisfaction

Stratum variable (third classifier, controlling for): Gender

Income scores: 3, 10, 20, 35

Job Satisfaction scores: 1, 3, 4, 5

 Alternative hypothesis Statistic DF Probability Ordinal association 6.156301 1 P = 0.0131 Nominal rows vs. ordinal columns association 9.034222 3 P = 0.0288 Nominal association 10.200089 9 P = 0.3345

Sample size = 104

From the results above you can see that the strongest effect detected is ordinal association (i.e. association between greater job satisfaction with greater income), after controlling for gender.