A Bayesian Framework for Causal Analysis of Recurrent Events with Timing Misalignment

TL;DR

The approach addresses misalignment by casting it as a time-varying treatment problem: some patients are on treatment at eligibility while others are off treatment but may switch to treatment at a specified time-if they survive long enough.

Abstract

Observational studies of recurrent event rates are common in biomedical statistics. Broadly, the goal is to estimate differences in event rates under two treatments within a defined target population over a specified followup window. Estimation with observational data is challenging because, while membership in the target population is defined in terms of eligibility criteria, treatment is rarely observed exactly at the time of eligibility. Ad-hoc solutions to this timing misalignment can induce bias by incorrectly attributing prior event counts and person-time to treatment. Even if eligibility and treatment are aligned, a terminal event process (e.g. death) often stops the recurrent event process of interest. In practice, both processes can be censored so that events are not observed over the entire followup window. Our approach addresses misalignment by casting it as a time-varying treatment problem: some patients are on treatment at eligibility while others are off treatment but may switch to treatment at a specified time - if they survive long enough. We define and identify an average causal effect estimand under right-censoring. Estimation is done using a g-computation procedure with a joint semiparametric Bayesian model for the death and recurrent event processes. We apply the method to contrast hospitalization rates among patients with different opioid treatments using Medicare insurance claims data.

Figures (3)

• Figure 1: Example patient trajectories. Triangle depicts time of treatment switching $W$, a pipe $|$ indicates time of $k^{th}$ occurrence of recurrent event at time $V_k$, an $\times$ indicates death at time $U$, and $\circ$ indicates censoring event at time $C$. Subjects 1 and 2 initiate treatment within followup, Subject 3 initiates at the beginning of followup, and Subjects 4 and 5 never initiate. Subject 3 reaches the end of the followup window, leading to complete information on survival and event count. Similarly, information is complete for subjects who die within followup since events cannot occur after death. Survival and recurrent event information is missing for Subjects 2 and 5 who are censored before $\tau$. The variables $\tilde{C} = \min(C, \tau)$, $\tilde{U} = \min (U, \tilde{C})$, and $\tilde{W} = \min (W, \tilde{U})$ are transformations of the variables in this figure representing a subject's censoring time, end of followup (due to first of either death or censoring), and first of either treatment switching time or end of follow-up, respectively. The dashed lines show the partition of the followup $(0,\tau]$ into $k=1,2,\dots, K=10$ intervals given by $I_k = (\tau_{k-1},\tau_k]$ as described in Section \ref{['sc:obs_data']}. Subject 1, for example, was not alive in interval 9 and 10 and thus has survival trajectory $\bar{T}_{10} = (0,0,0,0,0,0,0,0,1,1)$. Since they were uncensored, $\bar{C}_{10}$ is the length-10 zero vector. Since they switched to treatment in interval 4, $\bar{A}_{10} = (0,0,0,1,1,1,1,1,1,1)$. Finally, their recurrent event trajectory is given by $\bar{Y}_{10} = (0,1,1,1,1,1,2,0,0,0)$.
• Figure 2: Posterior $gAR1$ smoothing of baseline hazard of logistic regression (left) and baseline mean of Poisson regression (right) demonstrated in a simulated example with at-risk counts on the x-axis label. Blue points and shading depict posterior mean and 95% credible interval estimates, respectively. At earlier time points with many subjects at risk empirical and Bayesian estimates agree. By interval 35, there are fewer than 10 subjects at risk, leading to erratic empirical hazard and mean event count estimates. For exmaple, in many intervals the empirical hazard is exactly zero. This is not due to the true hazard actually being zero in this interval, but because there are too few at-risk subjects for reliable estimation. In contrast, the Bayesian estimates remain stable in these intervals due to prior shrinkage.
• Figure 3: Left: Checking positivity. Each boxplot shows the distribution of estimated probability of switching to opioid at week $s$. Distributions centered above zero at a given $s$ indicates positivity holds for that $s$. For example, very few patients switch to opioids after week 34 and boxplots tend to be right at the bound. So, estimation of $\Psi(s,1)$ is sensitive to extrapolation error for $s\geq 34$. Right: Posterior mean and 95% credible interval estimates of the causal incidence rate difference $\Psi(s, 1)$ - the difference in event rates had everyone switched to opioids $s$ weeks after diagnosis versus immediately at diagnosis. The units are hospitalizations per year. Delaying opioids tends to reduce hospitalization rates. For example, starting opioids 26 weeks after chronic pain diagnosis leads to about .05 fewer hospitalizations per year.