Bayesian treatment comparison using parametric mixture priors based on histograms elicited from expert physicians

Abstract

Introduction A pervasive problem when comparing treatments based on randomized clinical trials in children, rare diseases, or important disease subgroups, is that the sample size often is too small to obtain a confirmatory conclusion using conventional statistical methods. One reasonable course of action is seeking expert opinion as an additional source of information. In this work, a Bayesian methodology is proposed for constructing a parametric prior on two treatment effect parameters, based on information elicited from a group of expert physicians. The methodology may be applied to settings with binary, real-valued, or to event time or other non-negative-valued outcomes, by formulating appropriate parametric distributions. The motivating application is a 70-patient randomized trial to compare two treatments for idiopathic nephrotic syndrome (INS) in children. Methods To obtain prior expert opinion before the trial is begun, a group of physicians experienced in treating the disease are asked to provide their opinions about the treatment effect parameter for each arm, denoted by θ1 and θ2. This is done by applying the so-called “bins-and-chips” graphical method of Johnson et al.. For each physician, a marginal prior for each treatment parameter, characterized by location (μ) and precision (γ) hyper-parameters is fit to each elicited histogram. E.g., for a binary outcome such as response, as in the INS trial, a beta distribution may be used. To induce prior within-physician correlation, a bivariate expert-specific prior for (θ1, θ2) is constructed by averaging the product of each expert's two marginal parametric priors over a distribution for two correlated latent expert effects, one for μ and the other for γ. We consider two ways to formulate this latent effect distribution, either assuming homogeneity across experts (Method 1) or including expert covariates (Method 2). An overall joint prior for (θ1, θ2) is constructed as a mixture of the individual physicians’ priors, weighted by a covariate-based index of experience, or by agreement between the elicited histogram means and corresponding parameter estimates obtained from the trial data, or equally. A framework is provided for performing a sensitivity analysis of posterior inferences to prior bias and precision. Results A simulation study evaluating several versions of the methodology is presented for the binary outcome case. Methods 1 and 2, and the three weighting schemes, were evaluated under four scenarios. In each scenario, 500 replications of a 70-patient randomized trial, were performed. Posterior probabilities π1,2(e) = Pr(θ1 − e Conclusion The methodology provides a practical way to incorporate expert opinion to improve inferential reliability in settings with small to moderate sample sizes. The prior-to-posterior sensitivity analysis allows non-statisticians to draw their own conclusions in an informed way.