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Unified implementation and comparison of Bayesian shrinkage methods for treatment effect estimation in subgroups

Marcel Wolbers, Miriam Pedrera Gómez, Alex Ocampo, Isaac Gravestock

Abstract

Evaluating treatment effect heterogeneity across patient subgroups is a fundamental aspect of clinical trial analysis. Yet, these analyses have inherent limitations due to small sample sizes and the substantial number of subgroups investigated. Statisticians in regulatory agencies and pharmaceutical companies have begun considering shrinkage methods grounded in Bayesian statistical theory. These methods incorporate priors on treatment effect heterogeneity, which operationally shrink raw subgroup treatment effect estimates towards the overall treatment effect. Various shrinkage estimators and priors have been proposed, yet it remains unclear which methods perform best. This work provides a unified presentation, software implementation (in the R package bonsaiforest2), and simulation comparison of one-way and global shrinkage methods for continuous, binary, count, and time-to-event endpoints. One-way models fit a separate shrinkage model for each subgrouping variable, whereas global models fit a model including all subgroup indicators at once. Both can derive standardized subgroup-specific treatment effects. Across all simulation scenarios, shrinkage methods outperformed the standard subgroup estimator without shrinkage in terms of mean squared error. They were also more efficient in identifying a non-efficacious subgroup. Global shrinkage models tended to have smaller mean squared error and less dependence on hyperprior parameters than one-way models, but also exhibited slightly larger bias and worse frequentist coverage of associated credible intervals. For both models, hyperprior choices anchored in trial assumptions about the anticipated size of the overall treatment effect performed well. We conclude that some degree of shrinkage is preferable to none and advocate for the routine inclusion of shrunken estimates in clinical forest plots to facilitate more robust decision-making.

Unified implementation and comparison of Bayesian shrinkage methods for treatment effect estimation in subgroups

Abstract

Evaluating treatment effect heterogeneity across patient subgroups is a fundamental aspect of clinical trial analysis. Yet, these analyses have inherent limitations due to small sample sizes and the substantial number of subgroups investigated. Statisticians in regulatory agencies and pharmaceutical companies have begun considering shrinkage methods grounded in Bayesian statistical theory. These methods incorporate priors on treatment effect heterogeneity, which operationally shrink raw subgroup treatment effect estimates towards the overall treatment effect. Various shrinkage estimators and priors have been proposed, yet it remains unclear which methods perform best. This work provides a unified presentation, software implementation (in the R package bonsaiforest2), and simulation comparison of one-way and global shrinkage methods for continuous, binary, count, and time-to-event endpoints. One-way models fit a separate shrinkage model for each subgrouping variable, whereas global models fit a model including all subgroup indicators at once. Both can derive standardized subgroup-specific treatment effects. Across all simulation scenarios, shrinkage methods outperformed the standard subgroup estimator without shrinkage in terms of mean squared error. They were also more efficient in identifying a non-efficacious subgroup. Global shrinkage models tended to have smaller mean squared error and less dependence on hyperprior parameters than one-way models, but also exhibited slightly larger bias and worse frequentist coverage of associated credible intervals. For both models, hyperprior choices anchored in trial assumptions about the anticipated size of the overall treatment effect performed well. We conclude that some degree of shrinkage is preferable to none and advocate for the routine inclusion of shrunken estimates in clinical forest plots to facilitate more robust decision-making.
Paper Structure (16 sections, 6 equations, 5 figures, 8 tables)

This paper contains 16 sections, 6 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Outcome model
  4. Prior distributions
  5. Standardization
  6. Case study: The GALLIUM trial
  7. Simulation study
  8. Candidate estimators
  9. Evaluation criteria
  10. Simulation for a time-to-event endpoint
  11. Simulation for a continuous endpoint
  12. Discussion
  13. Appendix: Supplementary Tables and Figures
  14. Time-to-event endpoint simulation: Results for global models with a normal prior
  15. Continuous endpoint simulation: Results for global models with a normal prior
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Standard and shrinkage treatment effect estimates (average HR with 95% confidence or credible intervals) in pre-specified subgroups for the GALLIUM trial
  • Figure 2: Standard and shrinkage treatment effect estimates (average HR with 95% confidence or credible intervals) in the FLIPI low subgroup of the GALLIUM trial. The standard estimator without shrinkage is in the direction of harm; all shrinkage estimators are in the direction of benefit.
  • Figure 3: Time-to-event endpoint simulation: Standardized overall root mean squared error by simulation scenario.
  • Figure 4: Continuous endpoint simulation: Standardized overall root mean squared error by simulation scenario.
  • Figure S1: Continuous endpoint simulation: RMSE by simulation scenario for covariate-adjusted estimators.