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On the interplay between prior weight and variance of the robustification component in Robust Mixture Prior Bayesian Dynamic Borrowing approach

Marco Ratta, Gaelle Saint-Hilary, Mauro Gasparini, Pavel Mozgunov

Abstract

Robust Mixture Prior (RMP) is a popular Bayesian dynamic borrowing method, which combines an informative historical distribution with a less informative component (referred as robustification component) in a mixture prior to enhance the efficiency of hybrid-control randomized trials. Current practice typically focuses solely on the selection of the prior weight that governs the relative influence of these two components, often fixing the variance of the robustification component to that of a single observation. In this study we demonstrate that the performance of RMPs critically depends on the joint selection of both weight and variance of the robustification component. In particular, we show that a wide range of weight-variance pairs can yield practically identical posterior inferences (in particular regions of the parameter space) and that large variance robust components may be employed without incurring in the so called Lindley's paradox. We further show that the use of large variance robustification components leads to improved asymptotic Type I error control and enhanced robustness of the RMP to the specification of the location parameter of the robustification component. Finally, we leverage these theoretical results to propose a novel and practical hyper-parameter elicitation routine.

On the interplay between prior weight and variance of the robustification component in Robust Mixture Prior Bayesian Dynamic Borrowing approach

Abstract

Robust Mixture Prior (RMP) is a popular Bayesian dynamic borrowing method, which combines an informative historical distribution with a less informative component (referred as robustification component) in a mixture prior to enhance the efficiency of hybrid-control randomized trials. Current practice typically focuses solely on the selection of the prior weight that governs the relative influence of these two components, often fixing the variance of the robustification component to that of a single observation. In this study we demonstrate that the performance of RMPs critically depends on the joint selection of both weight and variance of the robustification component. In particular, we show that a wide range of weight-variance pairs can yield practically identical posterior inferences (in particular regions of the parameter space) and that large variance robust components may be employed without incurring in the so called Lindley's paradox. We further show that the use of large variance robustification components leads to improved asymptotic Type I error control and enhanced robustness of the RMP to the specification of the location parameter of the robustification component. Finally, we leverage these theoretical results to propose a novel and practical hyper-parameter elicitation routine.

Paper Structure

This paper contains 30 sections, 4 theorems, 47 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Setting
  4. Bayesian Design of a Randomized Controlled Trial (RCT)
  5. Frequentist and Bayesian Operating Characteristics
  6. Posterior Estimation Metrics
  7. Robust Mixture Prior (RMP)
  8. Normal Robust Mixture Prior
  9. Motivation for the Work
  10. Background
  11. Illustration in a Hypothetical Trial
  12. Analysis
  13. Research Questions
  14. Analytical results
  15. Asymptotic inflation of type I error rate
  16. ...and 15 more sections

Key Result

Theorem 1

Consider a RCT where mean control and treatment responses are normal $X_c \sim \mathcal{N} \left(\theta_c, \sigma^2_c \right)$, $X_t \sim \mathcal{N} \left(\theta_t, \sigma^2_t \right)$, and assume $\sigma^2_t = K\sigma^2_c$ (where $K^{-1}$ is the randomization ratio, assumed > 1). Assume a RMP $\pi

Figures (11)

  • Figure 1: type I error rate $\alpha(D)$ under different RMP parameterizations. Red curves: improper priors ($\sigma^2_{\text{rob}} = 10^{100}$). Black curves: unit-information priors ($\sigma^2_{\text{rob}} = 1$). Line styles denote values of $\mu_{\text{rob}}$. Panel (\ref{['w=0.5.fig']}): $\omega=0.5$; Panel (\ref{['w=0.9.fig']}): $\omega=0.9$.
  • Figure 2: Posterior weight $\Tilde{\omega}$ as a function of effecive sample size of the robust component $n_0$, prior weight $\omega$ and observed control response $x_c$. The red curve in the $(n_0, \omega)$ represents all RMPs with $\beta^{*}=5.83$.
  • Figure 3: Panel (a): type I error rate. Panel (b): power under $\delta^{*}=0.31$. Colors represent different couples of $(\omega, n_0)$, corresponding to $\beta=5.83$.
  • Figure 4: Panel (a): bias; Panel (b): variance; Panel (c): mean squared error, all computed using the posterior mean of the treatment effect parameter $\delta$. Colors denote different pairs of $(\omega, n_0)$, each corresponding to $\beta^{*} = 5.83$.
  • Figure 5: For each panel representing a different couples of $(\omega, n_0)$, type I error rate as a function of the prior-data conflict $D$ is displayed for five different values of the location of the robustification component $\mu_{\text{rob}}$.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • proof