Efficacy and toxicity monitoring via Bayesian predictive probabilities in phase II clinical trials

Bayesian monitoring strategies based on predictive probabilities are widely used in phase II clinical trials that involve a single efficacy binary variable. The essential idea is to control the predictive probability that the trial will show a conclusive result at the scheduled end of the study, given the information at the interim stage and the prior beliefs. In this paper, we present an extension of this approach to incorporate toxicity considerations in single-arm phase II trials. We consider two binary endpoints representing response and toxicity of the experimental treatment and define the result as successful at the conclusion of the study if the posterior probability of an high efficacy and that of a small toxicity are both sufficiently large. At any interim look, the Multinomial-Dirichlet distribution provides the predictive probability of each possible combination of future efficacy and toxicity outcomes. It is exploited to obtain the predictive probability that the trial will yield a positive outcome, if it continues to the planned end. Different possible interim situations are considered to investigate the behaviour of the proposed predictive rules and the differences with the monitoring strategies based on posterior probabilities are highlighted. Simulation studies are also performed to evaluate the frequentist operating characteristics of the proposed design and to calibrate the design parameters.

     

Optimal predictive sample size for case–control studies

Summary.  The identification of factors that increase the chances of a certain disease is one of the classical and central issues in epidemiology. In this context, a typical measure of the association between a disease and risk factor is the odds ratio. We deal with design problems that arise for Bayesian inference on the odds ratio in the analysis of case–control studies. We consider sample size determination and allocation criteria for both interval estimation and hypothesis testing. These criteria are then employed to determine the sample size and proportions of units to be assigned to cases and controls for planning a study on the association between the incidence of a non-Hodgkin's lymphoma and exposition to pesticides by eliciting prior information from a previous study.     

Likelihood and bayesian approaches to inference for the stationary point of a quadratic response surface

In response surface analysis, a second order polynomial model is often used for inference on the stationary point of the response function. The standard confidence regions for the stationary point are due to Box & Hunter (1954). The authors propose an alternative parametrization, in which the stationary point is the parameter of interest; likelihood techniques and Bayesian analysis are then easier to perform. The authors also suggest an approximate method to get highest posterior density regions for the maximum point (not simply for the stationary point). Furthermore, they study the coverage probabilities of these Bayesian regions through simulations.     

Accounting for uncertainty in the historical response rate of the standard treatment in single‐arm two‐stage designs based on Bayesian power functions

In phase II single‐arm studies, the response rate of the experimental treatment is typically compared with a fixed target value that should ideally represent the true response rate for the standard of care therapy. Generally, this target value is estimated through previous data, but the inherent variability in the historical response rate is not taken into account. In this paper, we present a Bayesian procedure to construct single‐arm two‐stage designs that allows to incorporate uncertainty in the response rate of the standard treatment. In both stages, the sample size determination criterion is based on the concepts of conditional and predictive Bayesian power functions. Different kinds of prior distributions, which play different roles in the designs, are introduced, and some guidelines for their elicitation are described. Finally, some numerical results about the performance of the designs are provided and a real data example is illustrated. Copyright © 2016 John Wiley & Sons, Ltd.     

Confidence regions for the stationary point of a quadratic response surface based on the asymptotic distribution of its MLE

Response surface methodology aims at finding the combination of factor levels which optimizes a response variable. A second order polynomial model is typically employed to make inference on the stationary point of the true response function. A suitable reparametrization of the polynomial model, where the coordinates of the stationary point appear as the parameter of interest, is used to derive unconstrained confidence regions for the stationary point. These regions are based on the asymptotic normal approximation to the sampling distribution of the maximum likelihood estimator of the stationary point. A simulation study is performed to evaluate the coverage probabilities of the proposed confidence regions. Some comparisons with the standard confidence regions due to Box and Hunter are also showed.     

A reference prior for the analysis of a response surface

Standard response surface methodology employs a second order polynomial model to locate the stationary point ξ$\mathit{\xi }$ of the true response function. To make Bayesian analysis more direct and simpler, we refer to an alternative and equivalent parametrization, which contains ξ$\mathit{\xi }$ as parameter of interest. The marginal reference prior of ξ$\mathit{\xi }$ is derived in its general form and particular cases are also given in detail, showing the Bayesian role of rotatability.