Principal stratification with recurrent events truncated by a terminal event: A nested Bayesian nonparametric approach

Abstract

Recurrent events often serve as key endpoints in clinical studies but may be prematurely truncated by terminal events such as death, creating selection bias and complicating causal inference. To address this challenge, we develop a Bayesian nonparametric framework to address potential selection bias due to truncation by death within the continuous-time principal stratification framework. We introduce causal estimands for recurrent events in the presence of a terminal event and derive a partial identification result for the estimand under a dual-frailty framework, enabling transparent sensitivity analysis for non-identifiable parameters. We then propose a flexible Bayesian nonparametric prior, the enriched dependent Dirichlet process, specifically designed for joint modeling of recurrent and terminal events, addressing a limitation where standard Dirichlet process priors create random partitions dominated by recurrent events, yielding poor predictive performance for terminal events. Simulations are carried out to show that our method has superior performance compared to existing methods. We apply the proposed new Bayesian nonparametric methods to infer the causal effect of a structured exercise program on rehospitalizations, which are subject to truncation by death.

Figures (5)

• Figure 1: Posterior mean and 95% posterior interval for $\mathrm{SANR}(t;r)$ with $r=t$, $\rho=0.5$.
• Figure 2: Posterior mean and 95% posterior interval for $\mathrm{SANR}(t;r)$ as a function of $t$ within fixed always-survivor stratum $\mathcal{AS}(r)$ with $r\in\{720,1080,1440\}$, $\rho=0.5$.
• Figure 3: Estimated always-survivor (AS) rate $p(r<D_i^0,r<D_i^1)$ over time (days) in HF-ACTION. Points connected by the solid line indicate the posterior mean AS rate at each time index; the shaded band denotes the corresponding $95\%$ posterior interval.
• Figure 4: Posterior mean and $95\%$ posterior interval for $\mathrm{SANR}(t;r)$ in HF-ACTION under five sensitivity parameters $\rho$. In each row, the left panel shows the diagonal slice $r=t$, and the right panel shows fixed always-survivor stratum $r\in\{720,1080,1440\}$.
• Figure 5: Contour plots of posterior mean (left), $95\%$ posterior lower bound (middle), and upper bound (right) for SANR. The posterior values are computed for different cut-off values of $t$ and $r$ from $360$ days to $1440$ days with an increment of $90$ such that $t \leq r$, and interpolated between the grid values with a linear spline. The regions with the estimate greater than $1$ are represented using warm colors (yellow to red), while regions with the estimate less than $1$ are indicated using cool colors (blue).

Theorems & Definitions (5)

• Theorem 1
• Remark 1: Frailty vs. copula for cross-world dependence in the recurrent-event setting
• Proposition 1
• proof
• proof