Behaviour of the likelihood ratio test statistic under a Bahadur model for exchangeable binary data

Bahadur proposed a now well-known although not very frequently used model for correlated binary data. In its general form, it combines marginal logits with pairwise and higher order correlations, to describe the joint response distribution. In many applications, the third and higher order correlations are set equal to zero. We show that this simplification leads to a considerably inflated likelihood ratio test statistic. This phenomenon is intimately related to the constraints on the parameter space. Clustered data from developmental toxicity studies further illustrate the implications of this issue.