---
title: Recurrent Misalignment from FOEWPT
url: https://www.emergentmind.com/topics/recurrent-misalignment-from-foewpt
type: topic
---

# Recurrent Misalignment from FOEWPT

Recurrent misalignment arising from First-Observed Eligibility with Waiting-Period Treatment (FOEWPT) refers to the bias generated when treatment initiation in recurrent-event studies is not synchronized with study eligibility, leading to incorrect attribution of risk intervals and event counts. This misalignment creates fundamental challenges for causal inference with observational recurrent event data, as the gap between eligibility and actual treatment initiation can produce substantial immortal-risk bias if not properly addressed. The formal treatment and resolution of FOEWPT misalignment, particularly in right-censored settings with terminal events, are provided in a Bayesian framework for causal analysis of recurrent events [2304.03247].

## 1. Formal Problem Definition and Manifestation of FOEWPT Misalignment

In observational studies of recurrent events, each subject $i=1,\ldots,n$ is observed over $K$ discrete intervals $[0,\tau)=\bigcup_{k=1}^K I_k$, with baseline covariates $L_i\in\mathcal L$, a random treatment switching time $W_i\in\{0,1,...,\tau\}$, a death time $T_i$, a censoring time $C_i$, and a recurrent event counting process $Y_i(t)$ up to $\tilde T_i = \min(T_i,C_i,\tau)$.

A key indicator for FOEWPT misalignment is that the time-varying treatment indicator $A_{k,i}=\mathbf{1}\{W_i \leq \tau_k\}$ remains $0$ before actual switch and $1$ thereafter. If analysis naively compares ever-treated versus never-treated, it misattributes both events and exposure time from the eligibility-to-switch window to the treatment, inducing substantial immortal-risk bias. The primary manifestation is that time and events accrued during guaranteed survival (before treatment switch) are incorrectly assigned to the treatment group.

## 2. Causal Estimand and Nonparametric Identification

The primary causal target is the contrast in recurrent-event incidence rates under different treatment switching regimes. Specifically, under a hypothetical intervention $z(s)=(\bar C_K=0,\bar A_K=a(s))$ where all subjects are uncensored and switch treatments exactly at interval $s$, define:

\[
R^{z(s)} = \frac{\sum_{k=1}^K Y_k^{z(s)}}{K - \sum_{k=1}^K T_k^{z(s)}}
\]
  
where $Y_k^{z(s)}$ and $T_k^{z(s)}$ are the potential recurrent event count and death indicator at interval $k$ under $z(s)$. The estimand is the difference in mean rates under two regimes:

\[
\Psi(s,s') = E\{R^{z(s)}\} - E\{R^{z(s')}\}
\]

Special cases include $\Psi(1,K+1)$ (immediate switch vs. never switch). Nonparametric identification is established under right-censoring, sequential ignorability, positivity, and SUTVA. The distribution of potential outcomes is expressed by a discrete-time $g$-formula:

\[
P^*\bigl(\bar Y_K^{z(s)}=y,\;\bar T_K^{z(s)}=t\bigr) = \sum_{l\in\mathcal L} f_L(l)\; \prod_{k=1}^K \left\{f(y_k|a_k^s,\bar y_{k-1},l)\,[1-\lambda_k(a_k^s,\bar y_{k-1},l)]\right\}
\]

where $\lambda_k$ is the discrete-time death hazard and $f(y_k|\cdot)$ is the count distribution, conditional on survival.

## 3. Semiparametric Bayesian Model and Prior Specification

The joint model factorizes the likelihood into components for the terminating event and the recurrent process. For each interval $k$:
\[
\begin{aligned}
\lambda_k(a_k, \bar y_{k-1}, l;\beta) &= \expit(\beta_{0k} + \beta_L(l, y_{k-1}) + \beta_A(l, y_{k-1}) a_k),\\
\mu_k(a_k, \bar y_{k-1}, l;\theta)     &= \exp(\theta_{0k} + \theta_L(l, y_{k-1}) + \theta_A(l, y_{k-1}) a_k).
\end{aligned}
\]

Time-varying intercepts receive AR(1)-type shrinkage Gaussian priors to promote regularization in late intervals:

\[
\beta_{0,1}\sim N(\beta_0,\sigma^2_\beta), \quad \beta_{0,k} = \rho_\beta \beta_{0,k-1} + (1-\rho_\beta)\beta_0 + \sigma_\beta \varepsilon_k,
\]
with $\varepsilon_k \sim N(0,1)$ and weakly informative hyperpriors on $\beta_0$, $\rho_\beta$, and $\sigma_\beta$. The confounder distribution $f_L$ is estimated nonparametrically via the Bayesian bootstrap, i.e., Dirichlet-weighted empirical distribution over $L_i$.

## 4. Identification Assumptions

Three assumptions support identification via the $g$-formula:

1. **SUTVA**: Potential outcomes are well-defined and there is no interference among subjects.
2. **Sequential Ignorability**: At every interval $k$,
   \[
   (\underline Y_k^{z(s)}, \underline T_k^{z(s)}) \perp (A_k, C_k)\ |\ \bar A_{k-1}, \bar Y_{k-1}, L, T_{k-1}=C_{k-1}=0,
   \]
   requiring adjustment for past recurrent events to account for time-varying confounding.
3. **Positivity**: For all histories $(l,\bar y_{s-1})$,
   \[
   P(A_s=1|\bar A_{s-1}=0, \bar Y_{s-1}, L=l)>0,\qquad P(C_k=0|\cdots)>0,\ \forall k.
   \]
   Notably, $P(A_1=1|l)\approx 0$ violations correspond to the exclusions induced by naive ever-treated comparators.

These assumptions enable substituting the distribution of counterfactuals with the $g$-formula based on observed data.

## 5. G-Computation Algorithm for Immortal-Risk Bias Correction

Correction for FOEWPT-induced immortal-risk bias leverages a $g$-computation procedure using posterior samples from MCMC on the joint Bayesian model. For each posterior draw $\omega^{(m)}$ and target switch time $s$:

1. For each $L_i$ and for $b=1,\ldots,B$ Monte Carlo replicates, simulate full event and death paths under intervention $A_k^s=\mathbf1\{k\geq s\}$:
   - Draw $Y_{k,i}$ from the fitted recurrent event pmf,
   - Sample death by $T_{k,i} = \mathbf1\{U \leq \lambda_k(A_k^s, \bar Y_{k-1,i}, L_i; \beta^{(m)})\},\ U\sim U(0,1)$.
2. Compute each subject's incidence rate $R_i^{(b,m)}$ as above.
3. Average over simulation replicates to obtain $E\{R^{z(s)}|L_i\}$, and then over bootstrap weights $\pi_i^{(m)}$.
4. Form $\Psi^{(m)}(s,s') = \widehat E^{(m)}[R^{z(s)}] - \widehat E^{(m)}[R^{z(s')}]$.
5. Aggregate over $m=1,...,M$ to obtain posterior summaries for $\Psi(s,s')$.

## 6. Empirical Illustration: Simulation and Comparative Performance

A large-scale simulation ($n=500$, $K=12$) contrasted Bayesian gAR1 models, frequentist discrete-time GLMs, and naive approaches ("grace-period", and "ever vs. never") in estimating $\Psi(6,13)$ (switch at week 6 vs. never). The naive "ever-never" stratification exhibited over 1000% bias (immortal-risk bias) even under low censoring. The Bayesian gAR1 achieved mapping bias ≈3% (vs 1% for the frequentist GLM), but exhibited substantially reduced mean square error due to shrinkage regularization. Under heavy censoring (positivity-challenged regime), Bayesian credible intervals were near nominal (93%) and notably narrower than frequentist bootstrap intervals, while naive methods failed altogether. Detailed statistics for bias, variance, mean squared error, and coverage rates are provided in Table 2 of the source [2304.03247].

## 7. Practical Implications and Scope of the Bayesian g-Computation Framework

The Bayesian gAR1 g-computation resolves FOEWPT timing misalignment, providing robust correction for immortal-risk bias in recurrent-event settings with right-censoring and terminal events. This approach enables valid and stable inference even in realistic data regimes with extreme sparsity or informative censoring, outperforming naive and conventional approaches under classical failures of positivity. A plausible implication is that the outlined framework generalizes to address misalignment problems in a variety of recurrent-event settings where treatment initiation is asynchronous with eligibility, and where rich time-varying confounding and censoring mechanisms are present.

Source: https://www.emergentmind.com/topics/recurrent-misalignment-from-foewpt