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Confidence intervals for two-stage adaptive designs with subpopulation selection

Enyu Li, Nigel Stallard, Ekkehard Glimm, Dominic Magirr, Peter K. Kimani

Abstract

We consider clinical trials in which an experimental treatment is compared with a control in pre-specified patient subpopulations. In such settings, adaptive enrichment designs allow the enrolled population to be modified at an interim analysis, with subpopulations selected according to preplanned rules. Since these interim decisions are data-dependent, valid statistical inference must account for them. We focus on constructing confidence intervals for the treatment effect in the selected population. Confidence interval methods that ignore the possibility of population modification may fail to achieve the desired coverage probability. We propose a new approach that constructs confidence intervals with exact nominal coverage conditional on the interim decision. Importantly, our method applies to a broad class of adaptive enrichment designs, rather than a single specific design. Our method involves deriving the distribution of the naive estimator of the treatment effect in the selected population conditional on the interim decision and inverting uniformly most accurate unbiased tests to obtain the confidence interval. We provide an efficient computational procedure and show through extensive simulations that the resulting confidence intervals satisfy the theoretical coverage guarantees.

Confidence intervals for two-stage adaptive designs with subpopulation selection

Abstract

We consider clinical trials in which an experimental treatment is compared with a control in pre-specified patient subpopulations. In such settings, adaptive enrichment designs allow the enrolled population to be modified at an interim analysis, with subpopulations selected according to preplanned rules. Since these interim decisions are data-dependent, valid statistical inference must account for them. We focus on constructing confidence intervals for the treatment effect in the selected population. Confidence interval methods that ignore the possibility of population modification may fail to achieve the desired coverage probability. We propose a new approach that constructs confidence intervals with exact nominal coverage conditional on the interim decision. Importantly, our method applies to a broad class of adaptive enrichment designs, rather than a single specific design. Our method involves deriving the distribution of the naive estimator of the treatment effect in the selected population conditional on the interim decision and inverting uniformly most accurate unbiased tests to obtain the confidence interval. We provide an efficient computational procedure and show through extensive simulations that the resulting confidence intervals satisfy the theoretical coverage guarantees.
Paper Structure (20 sections, 5 theorems, 48 equations, 3 figures, 3 tables)

This paper contains 20 sections, 5 theorems, 48 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Definition and Notation
  4. Data description, notation, and assumptions
  5. Definition of test statistics and naive confidence interval
  6. Decision rule
  7. Example designs with decision rule in class $\mathcal{D}$
  8. The interim decision rule proposed by rosenblum2013confidence
  9. An interim decision rule with stopping for futility
  10. Co-primary analysis
  11. Construction of Conditional Uniformly Most Accurate Unbiased Confidence Intervals
  12. General method and theorem
  13. Numerical method for calculating the C-UMAU confidence interval
  14. Application on two-stage enrichment designs with interim decision rules in $\mathcal{D}$
  15. A computationally simpler alternative based on inversion of two one-sided tests
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose $\hat{\Delta}^{(1)} \sim \mathcal{N}(\Delta, \sigma_{(1)}^2)$ and $\hat{\Delta}^{(2)} \sim \mathcal{N}(\Delta, \sigma_{(2)}^2)$, where $\sigma_{(1)}^2$ and $\sigma_{(2)}^2$ are known. Let $\tau_{(1)} = 1/{\sigma_{(1)}^2},\ \tau_{(2)} = 1/{\sigma_{(2)}^2},\ \hat{\Delta}=\tau_{(1)}/(\tau_{(1)}

Figures (3)

  • Figure 1: Schematic diagram of the interim decision rule $D_1$ (proposed by rosenblum2013confidence)
  • Figure 2: Schematic diagram of the interim decision rule $D_2$
  • Figure 3: Schematic diagram of the interim decision rule in kimani2015estimation

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 1
  • Definition 1
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • ...and 1 more