Understanding and testing for heterogeneity across 2 x 2 tables: application to meta-analysis

A question of interest in a meta-analysis is whether the studies considered are homogeneous. This paper addresses two forms of heterogeneity. Additive heterogeneity occurs when a particular treatment performs differently in different studies. Interactive heterogeneity occurs when the relative performance of two treatments varies with the study. This paper discusses the use and calculation of Bayes factors to test for both forms of heterogeneity. I consider three hierarchical models. One exhibits both forms of heterogeneity, a second exhibits only additive heterogeneity, and the third model has neither form of heterogeneity. I use a method based on bridge sampling for calculating the Bayes factors between the models. I consider two examples: in one both forms of heterogeneity seem to be present and in the other there seems to be additive heterogeneity but not interactive heterogeneity.