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Toxicity Monitoring Rule for a Two-Cohort Phase II Clinical Trial with Bivariate Beta Prior

Yu Wang, Aniko Szabo

Abstract

Toxicity monitoring is essential in Phase II clinical trials to ensure participant safety. While monitoring rules are well-established for single-arm trials, two-cohort trials present unique challenges because toxicities are expected to be similar between cohorts but may still differ. Current approaches either monitor the two cohorts independently, which ignores their similarity, or pool them together as a single arm, which neglects heterogeneity between cohorts. We propose a Bayesian method based on a bivariate beta prior that provides a compromise between these two approaches. The marginal posterior distribution is derived as a mixture of beta distributions, enabling exact calculations of the proposed method's operating characteristics. Examples demonstrate that joint monitoring offers a balanced approach between the independent and pooled methods. Keywords: Toxicity; Two-cohort; Phase II clinical trial; Monitoring rules; Bivariate Beta; Exact Operating characteristics

Toxicity Monitoring Rule for a Two-Cohort Phase II Clinical Trial with Bivariate Beta Prior

Abstract

Toxicity monitoring is essential in Phase II clinical trials to ensure participant safety. While monitoring rules are well-established for single-arm trials, two-cohort trials present unique challenges because toxicities are expected to be similar between cohorts but may still differ. Current approaches either monitor the two cohorts independently, which ignores their similarity, or pool them together as a single arm, which neglects heterogeneity between cohorts. We propose a Bayesian method based on a bivariate beta prior that provides a compromise between these two approaches. The marginal posterior distribution is derived as a mixture of beta distributions, enabling exact calculations of the proposed method's operating characteristics. Examples demonstrate that joint monitoring offers a balanced approach between the independent and pooled methods. Keywords: Toxicity; Two-cohort; Phase II clinical trial; Monitoring rules; Bivariate Beta; Exact Operating characteristics

Paper Structure

This paper contains 20 sections, 2 theorems, 22 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Bayesian Monitoring for One Arm
  4. Bivariate Beta Distribution
  5. Bivariate Beta-Binomial Model
  6. Model Components
  7. Marginal Posterior
  8. Two-Cohort Toxicity Monitoring
  9. Hypotheses
  10. Prior Distribution Parameters
  11. Calculating the Exceedance Probability
  12. Posteriors for Independent and Pooled Monitoring
  13. Stopping Tables
  14. Frequentist Operating Characteristics of Stopping Rules
  15. Comparison Among Three Monitoring Rules
  16. ...and 5 more sections

Key Result

Theorem 1

The posterior distribution is a mixture of bivariate beta distributions: where $\bm z(x_1,x_2,y_1,y_2) = (x_1+y_1, k_1-x_1+y_2, x_2+k_2-y_1, n_1-k_1-x_2+n_2-k_2-y_2)$, $w(x_1, x_2, y_1, y_2) = c_0 \bm B (\alpha+\bm z) {k_1 \choose x_1} {{n_1-k_1} \choose x_2} {k_2 \choose y_1} {{n_2-k_2} \choose y_2}$, and $c_0$ is a scaling constant.

Figures (5)

  • Figure 1: Type I error in cohort 1 ($\rho=0.5$, $N=20$, $\tau=0.98$)
  • Figure 2: Probability of early stopping in cohort 1 ($\rho=0.5$, $N=20$, $\tau=0.98$, $ESS=3$)
  • Figure 3: Expected number of patients enrolled at stopping ($\rho=0.5$, $N=20$, $\tau=0.98$, $ESS=3$)
  • Figure 4: Expected number of events at stopping ($\rho=0.5$, $N=20$, $\tau=0.98$, $ESS=3$)
  • Figure 5: Expected number of events at early stopping in cohort 1 ($\rho=0.5$, $N=20$, $\tau=0.98$, $ESS=3$)

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2