Doubly robust estimators of the restricted mean time in favor estimands in individual- and cluster-randomized trials

TL;DR

This work develops covariate-adjusted, doubly robust estimators for the restricted mean time in favor (RMT-IF) in progressive multi-state survival trials, addressing covariate-dependent censoring and extending to cluster-randomized designs. The estimators combine stage-specific outcome regression with arm- and cluster-specific censoring models within an augmented inverse-probability weighting framework, ensuring consistency if either nuisance model is correct. Variance is inferred via group or cluster jackknife methods, enabling valid pointwise confidence intervals in both individually randomized trials and CRTs. Through extensive simulations and two real trials (SPRINT and STRIDE), the authors demonstrate robustness to model misspecification, efficiency gains from covariate adjustment, and important distinctions between cluster- and individual-level estimands in CRTs with multistate outcomes.

Abstract

Progressive multi-state survival outcomes are common in trials with recurrent or sequential events and require treatment effect estimands that remain interpretable without proportional intensity or Markov assumptions. The restricted mean time in favor of treatment (RMT-IF) extends the restricted mean survival time to ordered multi-state processes and provides such an interpretable estimand. However, existing RMT-IF methods are nonparametric, assume covariate-independent censoring for independent observations, and do not accommodate cluster-randomized trials (CRTs), limiting both efficiency and applicability. We develop a class of doubly robust estimators for RMT-IF under right censoring using an augmented inverse-probability weighting framework that combines stage-specific outcome regression with arm-specific censoring models, yielding consistency when either nuisance model is correctly specified. We further extend the framework to CRTs by formalizing both cluster-level and individual-level average RMT-IF estimands to address informative cluster size and by constructing corresponding doubly robust estimators that account for within-cluster correlation. For inference, we employ model-agnostic jackknife variance estimators in both individually randomized and cluster-randomized settings. Extensive simulation studies demonstrate finite-sample performance, and the methods are illustrated using two randomized trial examples.

Figures (4)

• Figure 1: An example diagram for a 3 transition-state process; state 0 can be considered as an initial state of being alive and event free.
• Figure 2: Estimated RMT-IF, $\xi^{(a)}(t)$ (left panels), and the causal effect, $\Delta^{\text{rmt-if}}(t)$ (right panels), from the SPRINT data. For each method, the left panel displays the bouquet plot of the arm-specific RMT-IF, and the right panel shows the corresponding estimated causal effect. Panel (a) reports results from our proposed doubly robust method. Panel (b) reports the nonparametric estimator using the package from Mao mao2023restricted.
• Figure 3: Estimated individual-level RMT-IF, $\xi_I^{(a)}(t)$ (left panels), and the causal effect, $\Delta_I^{\text{rmt-if}}(t)$ (right panels), from the STRIDE data. For each method, the left panel displays the bouquet plot of the arm-specific individual-level RMT-IF, and the right panel shows the corresponding estimated causal effect. Panel (a) reports results from the proposed doubly robust method using stage-specific frailty Cox models for $P\{T_{ij}^{q,(a)} > t \mid \bm Z_i\}$ and for the censoring distribution $K_c^{(a)}(t \mid \bm V_{ij})$. Panel (b) reports results from the proposed doubly robust method using stage-specific marginal Cox models for $P\{T_{ij}^{q,(a)} > t \mid \bm V_i\}$ and for the censoring distribution. Panel (c) reports the nonparametric estimator of Mao mao2023restricted using Kaplan--Meier estimators for $P\{T_{ij}^{q,(a)} > t \mid \bm V_i\}$ and $K_c^{(a)}(t \mid \bm V_i)$ with equal weight for each individual.
• Figure 4: Estimated cluster-level RMT-IF, $\xi_C^{(a)}(t)$ (left panels), and the causal effect, $\Delta_C^{\text{rmt-if}}(t)$ (right panels), from the SPRIDE data. For each method, the left panel displays the bouquet plot of the arm-specific cluster-level RMT-IF, and the right panel shows the corresponding estimated causal effect. Panel (a) reports results from the proposed doubly robust method using stage-specific frailty Cox models for $P\{T_i^{q,(a)} > t \mid \bm V_i\}$ and for the censoring distribution $K_c^{(a)}(t \mid \bm V_i)$. Panel (b) reports results from the proposed doubly robust method using stage-specific marginal Cox models for $P\{T_i^{q,(a)} > t \mid \bm V_i\}$ and for the censoring distribution. Panel (c) reports the nonparametric estimator of Mao mao2023restricted using Kaplan--Meier estimators for $P\{T_i^{q,(a)} > t \mid \bm V_i\}$ and $K_c^{(a)}(t \mid \bm V_i)$ with weights $1/N_i$ applied at the cluster level.

Theorems & Definitions (10)

• Proposition 2.3: Mao, 2023
• Proposition 2.5
• Remark
• Remark
• Proposition 3.1
• Proposition 3.3
• Remark
• Remark
• proof
• proof