Optimal designs for contingent response models with application to toxicity and efficacy

Abstract

We generalize results in the literature to obtain a general family contingent response models. These models have ternary outcomes constructed from two Bernouli outcomes, where one outcome is only observed if the other outcome is positive. This family is represented in a canonical form which yields general results for its Fisher information. D and c optimal designs found numerically for a contingent response model with expected response probabilities taken from a bivariate extreme value distribution illustrate the model and motivate limiting results. Optimal designs for even modestly complex nonlinear response models cannot be expressed in closed form; and this includes the contingent response model. Limiting D optimal designs obtained in closed form can be used to approximate exact D and c optimal designs, as they are shown to be efficient over a wide range of parameter values, or they can be used to provide starting values in numerical searches for exact optimal designs. To provide a motivating context, we describe the two binary outcomes that compose the contingent responses as toxicity and no efficacy. Efficacy or lack thereof is assumed only to be observable in the absence of toxicity, resulting in the ternary response {toxicity, efficacy without toxicity, neither efficacy nor toxicity}. The rate of toxicity, and the rate of efficacy conditional on no toxicity, are assumed to increase with dose. The results provided in this paper are useful for the construction of efficient designs under a broad class of such models