Bayesian hierarchical EMAX model for dose-response in early phase efficacy clinical trials

Abstract

A primary goal of a phase II dose-ranging trial is to identify a correct dose before moving forward to a phase III confirmatory trial. A correct dose is one that is actually better than control. A popular model in phase II is an independent model that puts no structure on the dose-response relationship. Unfortunately, the independent model does not efficiently use information from related doses. One very successful alternate model improves power using a pre-specified dose-response structure. Past research indicates that EMAX models are broadly successful and therefore attractive for designing dose-response trials. However, there may be instances of slight risk of nonmonotone trends that need to be addressed when planning a clinical trial design. We propose to add hierarchical parameters to the EMAX model. The added layer allows information about the treatment effect in one dose to be "borrowed" when estimating the treatment effect in another dose. This is referred to as the hierarchical EMAX model. Our paper compares three different models (independent, EMAX, and hierarchical EMAX) and two different design strategies. The first design considered is Bayesian with a fixed trial design, and it has a fixed schedule for randomization. The second design is Bayesian but adaptive, and it uses response adaptive randomization. In this article, a randomized trial of patients with severe traumatic brain injury is provided as a motivating example.

Keywords: EMAX; dosing design, Bayesian models; hierarchical models; logistic.

© 2019 John Wiley & Sons, Ltd.

Figures

Figure 1.

Illustrative data for the exploration of posterior distributions for assumed responses.

Figure 2.

Results for fitting models in the large effect example. The ‘▭’ in the first three frames represent the observed rate and the shaded regions are the 2.5%-tile and 97.5%-tile from models (e.g. 95% intervals) for Pd for all models. The last frame shows the 50%-tile (point estimate) and 2.5%-tile and 97.5%-tile for ψd in the hierarchical EMAX model.

Figure 3.

Results for fitting models in the NBH only effect example. The ‘▭’ in the first three frames represent the observed rate and the shaded regions are the 2.5%-tile and 97.5%-tile from models (e.g. 95% intervals) for Pd for all models. The last frame shows the 50%-tile (point estimate) and 2.5%-tile and 97.5%-tile for ψd in the hierarchical EMAX model.

Figure 4.

Results for fitting models in the over dose effect example. The ‘▭’ in the first three frames represent the observed rate and the shaded regions are the 2.5%-tile and 97.5%-tile from models (e.g. 95% intervals) for Pd for all models. The last frame shows the 50%-tile (point estimate) and 2.5%-tile and 97.5%-tile for ψd in the hierarchical EMAX model.

Figure 5.

Large monotone effect.

Figure 6.

NBH only.

Figure 7.

Over dose.

Figure 8.

Presented is the probability of identifying correct dose minus the probability of identifying incorrect dose as a function of possibility of a non-monotone scenario. The possibility of non-monotone pattern produces a combination of the effects Large, NBH Only, and Over Dose. Let π be the probability of a non-monotone pattern (this probability is split between the two non-monotone patterns NBH Only and Over Dose), then the difference in probability correct (Pc) and probability of incorrect (PI), where I=Pc-PI, for each model is calculated as a function of the probability of the effects, therefore this operating characteristic becomes πILarge + (π/2)INBH + (π/2)Iover Dose. Notice that no model is best across all possibilities of non-monotone patterns however hierarchical EMAX model works very well across a broad range.