A logistic model for trend in 2 x 2 x kappa tables with applications to meta-analyses

There recently has been an increased interest in examining the relationship between the baseline (control) risk of an adverse outcome and the magnitude of the treatment effect (Brand and Kragt, 1992, Statistics in Medicine 11, 2077-2082; Davey Smith, Song, and Sheldon, 1993, The British Medical Journal 306, 1367-1373; Senn, 1994, Statistics in Medicine 13, 293-294). To facilitate such an examination, we propose a logistic model in which the relationship between the treatment effect, as measured by the log odds ratio, and the baseline risk is specified parametrically. This procedure is founded on a product-binomial likelihood and generates maximum likelihood estimates of the baseline event rates and two parameters characterizing the trend in the treatment effect. We fit this model to data from a meta-analysis involving the treatment of women at risk of preterm labor and contrast our findings with those of an earlier analysis.