Pseudo-RMST targets

From censored observations to an estimand

For observation i, let X_i denote baseline covariates, T_i the event time and C_i the censoring time. Only the observed follow-up and event indicator are available:

$$O_i=(X_i,Y_i,{\Delta }_i),Y_i=\min (T_i,C_i),{\Delta }_i=𝕀(T_i\le C_i).$$

At restriction horizon τ, the prediction target is covariate-conditional RMST:

$${\mu }_{\tau }\left(x\right)=𝔼\left\{\min \right(T,\tau \left)\right|X=x\}={\int }_0^{\tau }S(t|X=x)dt.$$

This is a mean restricted event-free time for observations with baseline profile x. It is not an individually observed survival time. Under squared-error loss with complete event times, this conditional mean is the optimal horizon-specific regression function.

Kaplan–Meier RMST functional

Within one outer training fold 𝒟, let d_𝒟(u) and r_𝒟(u) be the numbers of events and observations at risk at event time u. The marginal Kaplan–Meier estimator is

$${\hat{S}}_{𝒟}\left(t\right)=\prod _{u\le t}\{1-\frac{d_{𝒟}\left(u\right)}{r_{𝒟}\left(u\right)}\},$$

and its restricted integral is

$${\hat{\theta }}_{\tau }\left(𝒟\right)={\int }_0^{\tau }{\hat{S}}_{𝒟}\left(t\right)dt.$$

Delete-one pseudo-observation

Let 𝒟_−i be the same training fold with observation i removed. Its jackknife pseudo-RMST target is

$$Z_i(\tau ;𝒟)=n_{𝒟}{\hat{\theta }}_{\tau }\left(𝒟\right)-(n_{𝒟}-1){\hat{\theta }}_{\tau }\left({𝒟}_{-i}\right).$$

The jackknife maps a censored fold-level functional to one quantitative label per training observation. Under the marginal independent-censoring and regularity conditions described below, its conditional expectation motivates regression toward conditional RMST asymptotically rather than as a finite-sample identity.

Exact implementation

The package implements the same delete-one estimator exactly. It computes the training event-time grid once, then evaluates leave-one-out curves in bounded observation batches. This changes computational organization, not the target: it is not an influence-function, subsampled, grouped or cross-fitted approximation. The accelerated implementation is tested against the direct leave-one-out reference.

from survfm import jackknife_pseudo_rmst_fast_exact

targets = jackknife_pseudo_rmst_fast_exact(
    time_train,
    event_train,
    tau=365.0,
    clip=True,
    batch_size=256,
)

Target and prediction clipping

Finite-sample pseudo-observations may fall outside the natural RMST range. The manuscript workflow bounds targets with the projection

$${\tilde{Z}}_i\left(\tau \right)={\Pi }_{[0,\tau ]}\left\{Z_i\right(\tau \left)\right\},{\Pi }_{[0,\tau ]}\left(u\right)=\min \{\tau ,\max (0,u\left)\right\}.$$

The same projection is applied separately to final predictions. These are finite-sample stabilizations, not theoretical properties of classical pseudo-observations.

Assumptions and known boundary

The implemented target uses an unstratified marginal Kaplan–Meier estimator. Its standard regression justification assumes censoring is independent of both event time and the covariates entering the regression, together with positivity and regularity conditions. Conditional independence of event and censoring given covariates does not by itself justify this marginal construction when censoring varies with prognostic covariates. Conditional, stratified or weighted targets are future method development rather than hidden behavior in this API.