Optimal design to discriminate between rival copula models for a bivariate binary response

We consider a bivariate logistic model for a binary response, and we assume that two rival dependence structures are possible. Copula functions are very useful tools to model different kinds of dependence with arbitrary marginal distributions. We consider Clayton and Gumbel copulae as competing association models. The focus is on applications in testing a new drug looking at both efficacy and toxicity outcomes. In this context, one of the main goals is to find the dose which maximizes the probability of efficacy without toxicity, herein called P-optimal dose. If the P-optimal dose changes under the two rival copulae, then it is relevant to identify the proper association model. To this aim, we propose a criterion (called PKL) which enables us to find the optimal doses to discriminate between the rival copulae, subject to a constraint that protects patients against dangerous doses. Furthermore, by applying the likelihood ratio test for non-nested models, via a simulation study we confirm that the PKL-optimal design is really able to discriminate between the rival copulae.