Restricted Mean Time in Favor of Treatment

• RMT-IF is an estimand that measures the net average time a treatment arm spends in a better health state compared to a control over a prespecified time horizon.
• It generalizes the restricted mean survival time (RMST) difference by incorporating ordered, irreversible multistate outcomes and accounts for censoring using methods like IPCW and Aalen–Johansen.
• Advanced estimation frameworks, including stacked survival models, weighting methods, and doubly robust approaches, enhance accuracy in both randomized and observational settings.

The restricted mean time in favor of treatment (RMT-IF) is an interpretable and robust estimand for quantifying treatment effects in time-to-event and progressive multistate settings. It generalizes the restricted mean survival time (RMST) difference to ordered, irreversible event processes, and provides both a clinically meaningful and statistically valid alternative to conventional hazard ratio-based analysis. RMT-IF measures the net average time a subject in the treatment arm spends in a more favorable disease state, as compared to a subject in the comparator arm, over a prespecified time horizon.

1. Fundamental Definitions and Multistate Formulation

For two treatment arms indexed by $a \in \{0,1\}$, let $Y^{(a)}(t) \in \{0,1,\dots,K+1\}$ represent the (potential) state trajectory up to time $t \in [0, \tau]$, where states are totally ordered ($0$ is best, $K+1$ is absorbing worst, e.g., death). The treatment-specific transition (hitting) times are

$$T_k^{(a)} = \inf\{t:Y^{(a)}(t) \geq k\}, \quad k = 1, \dots, K+1$$

and right censoring is denoted $C^{(a)}$, yielding observed $X_k^{(a)} = T_k^{(a)} \wedge C^{(a)}$.

The RMT-IF estimand is defined by the net average time (over $t \in [0,\tau]$) that an individual on treatment is in a strictly better state than an individual on control, minus the converse: $$\mathrm{RMT\text{-}IF}(\tau) = \int_0^\tau \left\{ P(Y^{(1)}(t) < Y^{(0)}(t)) - P(Y^{(0)}(t) < Y^{(1)}(t)) \right\} dt$$ Alternatively, with $$\mathcal{W}\{Y^{(1)},Y^{(0)}\}(\tau) = \int_0^\tau \mathbf{1}\{Y^{(1)}(t) < Y^{(0)}(t)\} dt$$, this is expressed as

$$\mathrm{RMT\text{-}IF}(\tau) = \mathbb{E}[ \mathcal{W}\{Y^{(1)}, Y^{(0)}\}(\tau) ] - \mathbb{E}[ \mathcal{W}\{Y^{(0)}, Y^{(1)}\}(\tau) ]$$

In the classic two-state (alive/dead) case, RMT-IF reduces to the RMST difference: $$\mathrm{RMT\text{-}IF}(\tau) = \int_0^\tau [ S^{(1)}(t) - S^{(0)}(t) ] dt$$ where $S^{(a)}(t) = P(T^{(a)} > t)$ is the arm-specific survival function (Sun et al., 24 Jan 2026, Wey et al., 2014, Orsini et al., 7 Mar 2025).

2. Estimation Frameworks: Univariate and Multistate Settings

Estimation of RMT-IF depends on the structure of the outcome:

• Univariate (time-to-first-event) Setting: For each treatment arm, estimate $S^{(a)}(t)$ using either nonparametric or regression-based methods, then compute

$$\widehat{\mathrm{RMT\text{-}IF}}(\tau) = \int_0^\tau [ \widehat{S}^{(1)}(t) - \widehat{S}^{(0)}(t) ] dt$$

• Multistate Setting: Employ a pairwise comparison approach based on patient trajectories. A nonparametric U-statistic estimator at each $t$ is

$$\widehat\Delta(t) = \frac{1}{n_1 n_0} \sum_{i=1}^{n_1} \sum_{j=1}^{n_0} \mathbf{1}\{\widehat{Y}_i^{(1)}(t) < \widehat{Y}_j^{(0)}(t)\} - \mathbf{1}\{\widehat{Y}_j^{(0)}(t) < \widehat{Y}_i^{(1)}(t)\}$$

and integrate over $[0,\tau]$:

$$\widehat{\mathrm{RMT\text{-}IF}}(\tau) = \int_0^\tau \widehat\Delta(t) dt$$

Variance estimation can be analytic (via influence function) or bootstrap-based (Sun et al., 24 Jan 2026, Fang et al., 20 Jan 2026).

Estimation requires proper handling of censoring. Methods include Aalen–Johansen or inverse-probability-of-censoring weighting (IPCW). Doubly robust approaches are available to address covariate-dependent censoring and increase efficiency (Fang et al., 20 Jan 2026).

3. Covariate Adjustment and Model Averaging

In observational or multiregional contexts, covariate imbalance can bias RMT-IF estimation. Covariate adjustment is accomplished by modeling the covariate-conditional survival (or transition) functions, marginalizing over an appropriate distribution. Several strategies have been developed:

• Stacked Survival Models: Combine multiple survival models (AFT, Cox, random survival forests, etc.) for each arm via convex combination, with weights optimized to minimize cross-validated inverse-probability-weighted Brier-score loss:

$$\widehat{S}^{\mathrm{stack},(a)}(t|x) = \sum_{k=1}^m w_k \widehat{S}_k^{(a)}(t|x)\ \text{with}\ w_k \geq 0,\ \sum_k w_k = 1$$

Covariate-averaged RMT-IF is then estimated via Riemann sum discretization:

$$\widehat{\mu}(\tau,a) \simeq \frac{1}{n} \sum_{i=1}^n \sum_{j=1}^{N_\tau} \left\{ t_{(j)} - t_{(j-1)} \right\} \widehat S^{\mathrm{stack},(a)}(t_{(j-1)}|X_i)$$

and $$\widehat{\mathrm{RMT\text{-}IF}}(\tau) = \widehat{\mu}(\tau,1) - \widehat{\mu}(\tau,0)$$ (Wey et al., 2014).

• Weighting-based Methods in Multi-Regional Clinical Trials (MRCTs): Inverse probability of sampling weighting (IPSW) and calibration weighting (CW) are employed to reweight samples such that covariate distributions are aligned with a prespecified target. These weights are incorporated into RMST/RMT-IF estimators via weighted Kaplan–Meier, Hajek (IPCW), or doubly robust augmentations (Hua et al., 2024).
• Bayesian Pseudo-Observation GMM: Construct pseudo-observations for RMST via leave-one-out Kaplan–Meier integration; fit identity link regression with priors on coefficients. The treatment indicator coefficient directly estimates RMT-IF, and tail probabilities for clinically relevant thresholds can be obtained (Orsini et al., 7 Mar 2025).

4. Doubly Robust Estimation and Cluster-Randomized Designs

For ordered multistate outcomes subject to right censoring and complex randomization schemes, doubly robust influence function-based estimation enables valid inference under minimal assumptions:

• AIPW Estimator: For each transition $q$, combine inverse-probability (IPW) and augmentation terms involving outcome regression and censoring models:

$$\widehat S^{q,(a)}(t) = \frac{1}{n} \sum_{i=1}^n \left[ \frac{\mathbf{1}\{A_i=a\} \mathbf{1}\{U_i^q \geq t\} \delta_i^q }{\pi^{(a)} K_c^{(a)}(t|Z_i)} - \frac{\mathbf{1}\{A_i=a\}}{\pi^{(a)}} \widehat P\{T_i^{q,(a)} \geq t|Z_i\} + \cdots \right]$$

Marginal RMT-IF aggregates stagewise differences, which have explicit forms. Consistency is guaranteed if either outcome regression or censoring model is correct. Extensions exist for cluster-randomized trials (CRT), defining both individual-level and cluster-level RMT-IF, and handling informative cluster-size (Fang et al., 20 Jan 2026).

• Variance Estimation: Use leave-one-group-out (individual trials) or leave-one-cluster-out (CRT) jackknife for variance; pointwise and joint confidence intervals follow. The methodology is implemented in the R package DRsurvCRT (Fang et al., 20 Jan 2026).

5. Relationship to Alternative Estimands

RMT-IF is distinguished by several key features:

• In the two-state (time-to-death) setting, RMT-IF coincides with the difference in RMST, both representing mean survival benefit up to time $\tau$ (Wey et al., 2014, Orsini et al., 7 Mar 2025, Hua et al., 2024).
• In progressive multistate processes (e.g., CKD, sequential hospitalizations), RMT-IF captures net time spent in more favorable health states, incorporating all transitions and the ordering of event severity (Sun et al., 24 Jan 2026, Fang et al., 20 Jan 2026). This contrasts with:
  • RMST difference—does not account for intermediate events or recurrent states.
  • Area Under the Curve (AUC) of cumulative severity score—requires pre-specified weights for each state; RMT-IF is rank-based, avoids assignment of arbitrary severity weights, and is nonparametric.
• RMT-IF naturally decomposes contributions from each event level, providing detailed insight into disease trajectory modification beyond survival alone (Sun et al., 24 Jan 2026).

6. Simulation Results and Practical Findings

Key empirical findings across several studies:

Method	Relative Bias (%)	MSE Ratio (vs. Cox)	Coverage
Cox PH	+10	1.00	0.94
Splines PH	+3	0.86	0.95
Stacked Survival	+5	0.77	0.95

• Stacked estimators reduce MSE by ~20% compared to Cox models under non-proportional hazards or nonlinear covariate effects, while achieving correct coverage (Wey et al., 2014).
• Weighted estimators using calibration weights (CW) or doubly robust augmentation display unbiasedness and consistently lower variance in the presence of covariate imbalance in MRCT/MRCT-like settings (Hua et al., 2024).
• The doubly robust AIPW estimator exhibits negligible bias under single-model correctness, with substantial efficiency gains and robustness to covariate-dependent censoring in both independent and cluster-randomized studies (Fang et al., 20 Jan 2026).
• Application to CKD trials: over six years, RMT-IF showed a net gain of 0.116 years (42 days) in favor of dulaglutide, mainly attributable to extended survival, providing a clinically relevant and more interpretable result than standard composite endpoints (Sun et al., 24 Jan 2026).

7. Interpretation, Software, and Recommendations

RMT-IF estimands have direct clinical interpretation: a value of $\mu(\tau)>0$ indicates average additional time in superior health states for treatment over $[0,\tau]$, while $\mu(\tau)<0$ indicates the opposite. Selection of truncation time $\tau$ should match the study design or follow-up, with sensitivity analysis advised.

Computation requires efficient integration, careful handling of censoring, and may be sensitive to the choice of models if not using doubly robust approaches. Model selection via stacking and cross-validation, robust variance estimation via jackknife or bootstrap, and explicit normalization of covariates are necessary for accurate inference.

Implementation is available in several R packages, notably DRsurvCRT for doubly robust AIPW estimation in IRT and CRT designs (Fang et al., 20 Jan 2026), and standard survival analysis libraries for stacked models (Wey et al., 2014).

RMT-IF provides an interpretable, model-agnostic summary of treatment effect that is applicable across randomized and observational studies, univariate and multistate settings, and under minimal modeling assumptions. In multistate processes, it offers sensitivity to all clinically meaningful aspects of disease progression, overcoming limitations of hazard-based or first-event–based analyses for both clinical interpretation and regulatory assessment (Wey et al., 2014, Hua et al., 2024, Orsini et al., 7 Mar 2025, Sun et al., 24 Jan 2026, Fang et al., 20 Jan 2026).