# A Bayesian Framework for Causal Analysis of Recurrent Events in Presence of Immortal Risk

**Authors:** Arman Oganisian, Anthony Girard, Jon A. Steingrimsson, Patience Moyo  
**arXiv:** [2304.03247](https://arxiv.org/abs/2304.03247)  
**Date:** 2023-04-06  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2304.03247v1/

## Key points

- Observational studies of recurrent event rates are common in biomedical statistics
- Goal is to estimate differences in event rates under two treatments within a defined target population over a specified follow-up window
- Proposed all-in-one Bayesian approach simultaneously handles timing misalignment and terminal events while modeling counts
- Defines a monotone time-varying treatment indicator that emulates a trial
- A g-formula nonparametrically identifies the causal target estimand of interest as the difference in average potential incidence rates had subjects switched to treatment at time s versus s0
- Sequence of two semiparametric Bayesian models is used to address the modeling challenge: a discrete-time survival model and a model for the count of events at a time interval given survival up to that interval
- Tailored priors are developed to regularize the models when the at-risk count is low
- Computing the estimand across various switching times allows us to assess the impact of treatment's timing
- Method is applied to contrast hospitalization rates under different opioid treatment strategies among patients with chronic back pain using Medicare claims data
- Approach targets a marginal estimand with clean interpretation and considers both timing misalignment and terminal events simultaneously while accounting for censoring
- Contributes to literature on Bayesian causal inference by developing a framework for causal analysis of recurrent events in presence of immortal risk bias
- Proposed approach can be used to adjust for confounding, avoid biases due to opioid initiation timing, and jointly model recurrent and terminal events in presence of censoring

## Summary

Observational studies of recurrent event rates are common in biomedical statistics, where the goal is to estimate differences in event rates under two treatments within a defined target population over a specified follow-up window. To address challenges such as timing misalignment and terminal events, this paper proposes an all-in-one Bayesian approach that simultaneously handles these issues while modeling counts. The approach defines a monotone time-varying treatment indicator that emulates a trial in which patients are enrolled at eligibility and randomized to switch to treatment at some specified time if they survive long enough. Under assumptions on the joint treatment-censoring mechanism, a g-formula nonparametrically identifies the causal target estimand of interest as the difference in average potential incidence rates had subjects switched to treatment at time s versus s0. A sequence of two semiparametric Bayesian models is used to address the modeling challenge: a discrete-time survival model and a model for the count of events at a time interval given survival up to that interval. Tailored priors are developed to regularize the models when the at-risk count is low. Computing the estimand across various switching times allows us to assess the impact of treatment's timing. The method is applied to contrast hospitalization rates under different opioid treatment strategies among patients with chronic back pain using Medicare claims data. Compared with other proposals in literature, this approach targets a marginal estimand with clean interpretation and considers both timing misalignment and terminal events simultaneously while accounting for censoring. This paper contributes to the literature on Bayesian causal inference by developing a framework for causal analysis of recurrent events in presence of immortal risk bias. The proposed approach can be used to adjust for confounding, avoid biases due to opioid initiation timing, and jointly model recurrent and terminal events in presence of censoring. The research is supported by a National Institute on Drug Abuse grant R03DA051778.