Treatment effect heterogeneity for univariate subgroups in clinical trials: Shrinkage, standardization, or else

Abstract

Treatment effect heterogeneity is a well-recognized phenomenon in randomized controlled clinical trials. In this paper, we discuss subgroup analyses with prespecified subgroups of clinical or biological importance. We explore various alternatives to the naive (the traditional univariate) subgroup analyses to address the issues of multiplicity and confounding. Specifically, we consider a model-based Bayesian shrinkage (Bayes-DS) and a nonparametric, empirical Bayes shrinkage approach (Emp-Bayes) to temper the optimism of traditional univariate subgroup analyses; a standardization approach (standardization) that accounts for correlation between baseline covariates; and a model-based maximum likelihood estimation (MLE) approach. The Bayes-DS and Emp-Bayes methods model the variation in subgroup-specific treatment effect rather than testing the null hypothesis of no difference between subgroups. The standardization approach addresses the issue of confounding in subgroup analyses. The MLE approach is considered only for comparison in simulation studies as the "truth" since the data were generated from the same model. Using the characteristics of a hypothetical large outcome trial, we perform simulation studies and articulate the utilities and potential limitations of these estimators. Simulation results indicate that Bayes-DS and Emp-Bayes can protect against optimism present in the naïve approach. Due to its simplicity, the naïve approach should be the reference for reporting univariate subgroup-specific treatment effect estimates from exploratory subgroup analyses. Standardization, although it tends to have a larger variance, is suggested when it is important to address the confounding of univariate subgroup effects due to correlation between baseline covariates. The Bayes-DS approach is available as an R package (DSBayes).

Keywords: Bayesian shrinkage estimate; Confounding; Empirical Bayes; Marginal subgroup analysis; Maximum likelihood estimate; Naïve estimate; Standardization; Subgroup analysis.

Figure 1

Univariate subgroup-specific treatment effect estimates for X₁ and X₂ across scenarios by five methods when treatment-by-X₁ interaction exists; N₁ = 15,750.

Figure 2

Univariate subgroup-specific treatment effect estimates for X₃ across scenarios by five methods; X₃ is weakly correlated with X₁ and X₂, and not interacts with treatment; N₁ = 15,750.

Figure 3

Univariate subgroup-specific treatment effect estimates for a hypothetical example with X₁ a potential predictor that is strongly correlated with X₂ and interacts with treatment, whereas X₃ is at most weakly correlated with X₁ (corr = 0.03) and X₂ (corr = 0.09).