Robust estimation of occupation probabilities for coarsened multistate processes

Technical Summary: Robust Estimation of Occupation Probabilities for Coarsened Multistate Processes

Problem Statement
Multistate models provide a general framework for analyzing phenomena where entities transition between well-defined states over time, such as disease progression, employment status, or credit ratings. A primary parameter of interest in these models is the occupation probability—the probability that a process occupies a specific state at a given time $t_0$. In practice, these processes are often subject to two levels of coarsening: right-censoring (e.g., drop-out) and baseline exposure (treatment assignment).

Existing literature on multistate models often relies on restrictive assumptions, such as the Markov property for state transitions or independence of censoring from the process history given baseline covariates. This paper addresses the challenge of estimating occupation probabilities for counterfactual multistate processes under weaker assumptions. Specifically, the authors aim to handle non-Markovian state transitions and allow right-censoring mechanisms to depend on both the natural history of the process and time-dependent covariates, provided the Coarsening at Random (CAR) assumption holds.

Methodology
The authors derive Augmented Inverse Probability Weighted (AIPW) estimators for occupation probabilities using techniques from causal inference and missing data literature.

1. Data Structure and Identifiability:

   • The full data process is defined as a cadlag multistate process $Z_a$ under a hypothetical intervention $a \in \{0, 1\}$, with associated time-dependent covariates $L_a$ and baseline covariates $W$.
   • The observed data is coarsened by a latent right-censoring time $C_a$ and the observed treatment $A$.
   • Identifiability Criteria: The authors assume:
     • Consistency: Observed data is a consistent coarsening of the full counterfactual data.
     • Positivity: The probability of treatment assignment and remaining uncensored is strictly positive.
     • Exchangeability (CAR): The coarsening mechanism (treatment and censoring) is independent of the full counterfactual data given baseline covariates and the history up to the current time. Crucially, this allows censoring to depend on the process history, unlike standard independent censoring assumptions.

2. Efficient Influence Function (EIF):

   • The target parameter is identified via an Inverse Probability Weighted (IPW) estimand.
   • The authors derive the efficient influence function (EIF) for the observed data estimand. This involves projecting the score of the full-data mapping onto the tangent space of the observed data model and subtracting the orthogonal projections onto the nuisance tangent spaces associated with the treatment propensity score ($\pi$) and the censoring mechanism ($G$).
   • The EIF depends on nuisance parameters: the propensity score $\pi$, the censoring survival function $G$, and two conditional expectation functions, $Q$ and $\eta$, which represent expected outcomes weighted by the inverse of the censoring probability.

3. Proposed Estimators:
   Based on the EIF, five estimators are proposed:

   • IPW Estimator ($\hat{\psi}^0_n$): A simple inverse probability weighted estimator.
   • Augmented IPW Estimators ($\hat{\psi}^\nu_n$ and $\hat{\psi}^\mu_n$): These incorporate correction terms (augmentations) derived from the EIF projections. $\hat{\psi}^\mu_n$ is a one-step estimator that includes both the treatment and censoring augmentation terms.
   • Modified AIPW Estimators ($\hat{\psi}^{\nu, mod}_n$ and $\hat{\psi}^{\mu, mod}_n$): These modify the standard AIPW estimators by estimating regression coefficients (via OLS) for the augmentation terms. This is intended to force orthogonality when nuisance parameter estimators are biased, potentially improving efficiency.

4. Nuisance Parameter Estimation:

   • $\pi$ and $G$ are estimated using working models (logistic regression for propensity scores and Cox models for censoring hazards).
   • $Q$ and $\eta$ are estimated via regression of weighted observations (pseudo-outcomes) against summaries of the conditioning set.

Key Results
The performance of the five estimators was evaluated through a simulation study involving a three-state illness-death model with recurrence, where transition rates depended on baseline covariates and censoring depended on both baseline and time-dependent history.

• Correct Specification: When all nuisance parameters were consistently estimated, all estimators were unbiased. However, the one-step estimator ($\hat{\psi}^\mu_n$) demonstrated the lowest Root Mean Square Error (RMSE), confirming its efficiency advantage over the simple IPW and partially augmented estimators.
• Robustness to Misspecification:
  • Biased Propensity Score ($\pi$) or Censoring ($G$): The simple IPW estimator exhibited significant bias. The fully augmented estimators ($\hat{\psi}^\mu_n$) remained robust (unbiased) even when $\pi$ or $G$ were misspecified, provided the other components were correctly specified.
  • Biased Regression Terms ($Q$ or $\eta$): When the regression-based nuisance parameters were misspecified, the fully augmented estimators remained unbiased but lost some efficiency. The modified estimators ($\hat{\psi}^{\mu, mod}_n$) showed slight improvements in RMSE in specific scenarios, suggesting they can partially mitigate efficiency losses due to bias in the augmentation terms.
• Convergence: The simulation results confirmed the theoretical hierarchy of efficiency: $\hat{\psi}^\mu_n$ (one-step) > $\hat{\psi}^\nu_n$ (partially augmented) > $\hat{\psi}^0_n$ (IPW).

Significance and Claims
The paper claims that the proposed estimators offer a robust and efficient approach to estimating occupation probabilities in multistate models under weaker assumptions than previously common in the literature.

• Generality: Unlike previous methods tailored to specific models (e.g., illness-death without recurrence or competing risks under Markov assumptions), these estimators apply to multistate models with an arbitrary number of states and allow for non-Markovian transitions and time-dependent confounding.
• Relaxation of Censoring Assumptions: The framework explicitly allows right-censoring to depend on the internal history of the multistate process, a feature often required for realistic modeling (e.g., in illness-death models with recurrence) but difficult to handle under standard independence assumptions.
• Double Robustness: The one-step estimator exhibits double robustness properties; it remains consistent if either the treatment/censoring models ($\pi, G$) or the outcome regression models ($Q, \eta$) are correctly specified.
• Practical Utility: The authors note that while the one-step estimator requires sufficient sample size to estimate the complex nuisance parameters ($\eta$) at every censoring time, it remains stable for datasets of moderate size (e.g., 2,500–9,000 individuals) and offers superior performance over IPW in the presence of model misspecification.

The authors conclude that while the Coarsening at Random (CAR) assumption is a strong condition (implying cross-world assumptions), it is a necessary tool for enabling likelihood-based inference and constructing semi-parametric models in the presence of complex missing data structures. The proposed methods provide a practical implementation of these theoretical advances for real-world multistate data analysis.