Deriving Approximations in a Random Effects Model for Multicenter Clinical Trials with Binary Response

The design and analysis of multicenter trials based on a random effects model is well developed for a continuous response, but is less well developed for a binary response. Here we describe a random effects model for a binary response for two treatments and show how maximum likelihood estimates for the unknown treatment difference can be derived using a novel approximation to the likelihood. The suggested approximation is easy to use and seems to be better suited to the problem than the Laplace approximation and the approximation based on adaptive Gaussian quadratures. We also derive an approximation for the Fisher information matrix of the treatment parameters. The results extend those previously reviewed by Agresti and Hartzel (2000 Agresti , A. , Hartzel , J. ( 2000 ). Strategies for comparing treatments on a binary response with multi-centre data . Statist. Med. 19 : 1115 – 1139 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).