A joint QoL-Survival framework with debiased estimation under truncation by death

Evaluating quality-of-life (QoL) outcomes in populations with high mortality risk is complicated by truncation by death, since QoL is undefined for individuals who do not survive to the planned measurement time. We propose a framework that jointly models the distribution of QoL and survival without extrapolating QoL beyond death. Inspired by multistate formulations, we extend the joint characterization of binary health states and mortality to continuous QoL outcomes. Because treatment effects cannot be meaningfully summarized in a single one-dimensional estimand without strong assumptions, our approach simultaneously considers both survival and the joint distribution of QoL and survival with the latter conveniently displayed in a simplex. We develop assumption-lean, semiparametric estimators based on efficient influence functions, yielding flexible, root-n consistent estimators that accommodate machine-learning methods while making transparent the conditions these must satisfy. The proposed method is illustrated through simulation studies and two real-data applications.

💡 Research Summary

This paper tackles the pervasive problem of “truncation by death” that arises when quality‑of‑life (QoL) or other functional outcomes are only defined for subjects who survive to a pre‑specified landmark time. Traditional approaches either (i) extrapolate QoL beyond death, (ii) treat the observed data as ordinary missingness, (iii) focus on survivor‑only subpopulations (e.g., survivor average causal effects), or (iv) model QoL conditional on being alive without a causal interpretation. Each of these strategies relies on strong, often untestable assumptions and can lead to misleading clinical conclusions, especially when the treatment influences mortality.

The authors propose a joint modeling framework that simultaneously characterizes the distribution of survival time (T) and a continuous QoL marker (Y) at a landmark time (t). Inspired by the illness‑death multistate model, they define three mutually exclusive states at time (t): (1) alive with QoL ≤ y, (2) alive with QoL > y, and (3) dead. For each treatment arm (a\in{0,1}) the state occupation probabilities are denoted (Q_{a0}, Q_{a1}, Q_{aD}) and satisfy (Q_{a0}+Q_{a1}+Q_{aD}=1). Plotting ((Q_{a0},Q_{a1},Q_{aD})) on a simplex provides a visual summary of both survival and QoL effects, making it clear that a treatment can improve survival while having no impact—or even a detrimental impact—on QoL among survivors.

Formally, the key estimand is the cumulative joint distribution \