Bayesian analysis of heterogeneous treatment effects for patient-centered outcomes research

Abstract

Evaluation of heterogeneity of treatment effect (HTE) is an essential aspect of personalized medicine and patient-centered outcomes research. Our goal in this article is to promote the use of Bayesian methods for subgroup analysis and to lower the barriers to their implementation by describing the ways in which the companion software beanz can facilitate these types of analyses. To advance this goal, we describe several key Bayesian models for investigating HTE and outline the ways in which they are well-suited to address many of the commonly cited challenges in the study of HTE. Topics highlighted include shrinkage estimation, model choice, sensitivity analysis, and posterior predictive checking. A case study is presented in which we demonstrate the use of the methods discussed.

Keywords: Bayesian subgroup analysis; Heterogeneity of treatment effect; Hierarchical modeling; Personalized medicine; Precision medicine; Treatment–covariate interaction.

Fig. 1

Basic shrinkage model. SOLVD data. Posterior means and frequentist estimates for each of the 12 subgroups defined by the variables gender, age, and ejection fraction. Frequentist estimates ${\widehat{\mathit{\theta }}}_g$ and associated $95\%$ confidence intervals are in black while Bayes estimates and associated $95\%$ credible intervals are in red. The solid vertical line represents the estimated overall treatment effect from the basic shrinkage model, namely, the posterior mean of $\mathit{\tau }$ (Color figure online)

Fig. 2

Half-normal densities plotted for several values of the scale parameter: ${\mathit{\sigma }}_{\mathit{\omega }}=1/2$, ${\mathit{\sigma }}_{\mathit{\omega }}=1$, ${\mathit{\sigma }}_{\mathit{\omega }}=2$, and ${\mathit{\sigma }}_{\mathit{\omega }}=5$

Fig. 3

Basic shrinkage model—sensitivity to choice of prior. SOLVD data. Posterior means and associated credible intervals for the following choices of the prior for $\mathit{\omega }$: $\mathit{\omega }\sim \text{Half-Normal}(0.1)$, $\mathit{\omega }\sim \text{Half-Normal}(1)$, $\mathit{\omega }\sim \text{Half-Normal}(100)$, and $\mathit{\omega }\sim \text{Jeffreys}$. The approximate Jeffreys prior for ${\mathit{\omega }}^2$ employed here is $p({\mathit{\omega }}^2)\propto {\mathit{\omega }}^{-2}$ for ${\mathit{\omega }}^2\ge 0.005$ and $p({\mathit{\omega }}^2)=200$ otherwise

Fig. 4

Extended Dixon–Simon model. SOLVD data. Posterior means and credible intervals for each of the 12 subgroups defined by the variables: gender, age, and baseline ejection fraction. Point estimates and uncertainty intervals from the basic shrinkage model and from the fully stratified frequentist analysis are also shown

Fig. 5

Posterior predictive checks. Samples from the posterior predictive distribution using both the basic shrinkage model and the extended Dixon–Simon model on the SOLVD data. Samples from the posterior predictive distribution of various test statistics T(y) are shown: median, standard deviation, minimum, and maximum. For each panel, the solid vertical line represents the observed value of the statistic T(y)