Addressing truncation by death: direct regression of survival-incorporated outcome sub-distributions

Background. Nonfatal outcomes measured at a landmark time are defined only among survivors, and survivor-restricted comparisons may be affected by selection bias. We propose a framework for directly regressing the joint probability of surviving to the landmark and attaining a given outcome level on an exposure or treatment indicator and baseline covariates. Methods. When deaths are retained in the denominator, the empirical distribution function of the outcome levels off at the proportion of survivors rather than at one: it is the empirical distribution function of an improper random variable and estimates its sub-distribution, the joint probability of surviving to the landmark and reaching a given outcome level. Under right censoring on the survival or the outcome axis, the same function is estimated by an inverse-probability-of-censoring weighted empirical distribution function. We model the sub-distribution directly with a log-link regression over a grid of outcome thresholds; anchored at the structural bound of the outcome scale, the model yields a marginal survival component and a threshold-varying survivor-conditional component. The model was estimated by three direct estimators built on the weighted estimator, binomial regression with inverse probability of censoring weights applied to the outcome or to the estimating equations, and jackknife pseudo-observations, which coincide without censoring. All three were compared on the same simulated datasets in a Monte Carlo study with four effect scenarios and four censoring settings, and the framework was illustrated in the ENIGMA-shock registry, relating cardiogenic shock severity (SCAI stage) to the guideline-directed medical therapy score and to left ventricular ejection fraction at discharge. Results. Over the adequately supported region of the outcome scale, all three estimators showed negligible bias and confidence-interval coverage close to nominal for the risk ratios, including with censoring on both the survival and the outcome axes; in the independent censoring settings considered, pseudo-observations were at least as precise as the weighted estimators. Estimates were unstable in the sparsely supported tails. In ENIGMA-shock, the joint association with ejection fraction was predominantly survival-driven, whereas for the therapy score a survivor-conditional component added to the survival component. Conclusions. Retaining deaths turns the distribution of a survivor-defined outcome into a sub-distribution, which can be estimated nonparametrically and regressed on covariates with established direct methods for cumulative incidence functions. The resulting regression is defined in the observable population and separates survival from survivor-conditional outcome patterns; its use requires adequate support across the threshold range and a correctly specified censoring mechanism.