Heterogeneity in Mantel-Haenszel-Type Models
研究了当多个2×2表格或配对泊松观测的比值比存在异质性时，如何对总体均值和标准差进行推断，提出了一种基于拉普拉斯近似的似然函数计算方法。
The Mantel-Haenszel problem involves inferences about a common odds ratio in a set of 2 x 2 tables. Although it is a fairly standard practice to test whether the odds ratios are indeed constant, there is remarkably little methodology available for proceeding when there is evidence of some heterogeneity. Our interest is in models where the log odds ratios θk, for tables k = 1, 2, …, K, are thought of as a sample from a population with mean θ and standard deviation σ, and inferences are desired regarding the parameters (θ, σ). By Mantel-Haenszel-type models we mean to include the generalization involving pairs of Poisson observations rather than 2 ×2 tables, where the θk are logarithms of ratios of the Poisson means within pairs. Direct computation of the likelihood function for (θ, σ) in these settings involves numerical integration, and the main point here is a simple approximation to this likelihood. The approximation is based on Laplace's method, and is very accurate for practical applications. Inference regarding the parameter θ of primary interest may be made from the profile likelihood function. An alternative approach to likelihood methods, based on approximations to the marginal means and variances, is also considered. The methods explored here can be readily generalized to settings where the parameters θk depend on covariables as well.