HETEROGENEITY IN MANTEL-HAENSZEL-TYPE MODELS

The Mantel-Haenszel problem involves inferences about a common odds ra tio in a set of 2 x 2 tables. Although it is a fairly standard practic e to test whether the odds ratios are indeed constant, there is remark ably little methodology available for proceeding when there is evidenc e of some heterogeneity. Our interest is in models where the log odds ratios theta(k), for tables k = 1, 2,..., K, are thought of as a sampl e from a population with mean theta and standard deviation sigma, and inferences are desired regarding the parameters (theta,sigma). By Mant el-Haeaenszetyty models we mean to include the generalization involvin g pairs of Poisson observations rather than 2 x 2 tables, where the th eta(k) are logarithms of ratios of the Poisson means within pairs. Dir ect computation of the likelihood function for (theta, sigma) in these settings involves numerical integration, and the main point here is a simple approximation to this likelihood. The approximation is based o n Laplace's method, and is very accurate for practical applications, I nference regarding the parameter theta of primary interest may be made from the profile likelihood function. An alternative approach to like lihood methods, based on approximations to the marginal means and vari ances, is also considered. The methods explored here can be readily ge neralized to settings where the parameters theta(k) depend on covariab les as well.