Extending Goodman-Kruskal's gamma to the comparison of ordered treatments in multicenter clinical trials

In a randomized, multicenter clinical trial setting, the treatments may consist of increasing doses of a drug and placebo and the response variable may be ordinal (e.g., physician's global evaluation of treatment effectiveness). Within each center (e.g., hospital), patients are randomly assigned to treatments (rows) such that the row totals are fixed and the rows form a product-multinomial sample of the ordinal response variable. Gamma, a measure of ordinal association in two-way contingency tables, and its asymptotic standard error can be estimated from the data in each center. We use these independent estimates of gamma for testing the hypothesis of homogeneity of gammas, controlling for center effect. If this hypothesis is not rejected, the within-center estimates of gamma can be combined to form a common gamma across the centers.