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Longitudinal Outcomes Truncated by Death: Causal Estimands and Bayesian Estimators

This paper proposes a framework to clarify causal estimands for longitudinal outcomes truncated by death, develops corresponding Bayesian estimators, and demonstrates through simulations and an ALS trial that the stratified average causal effect combined with restricted mean survival time offers a more comprehensive characterization of treatment effects by addressing the inherent challenges of ordering and distance in such outcomes.

Original authors: Juliette Ortholand, Young Lee, Marie-Abele C Bind

Published 2026-04-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Juliette Ortholand, Young Lee, Marie-Abele C Bind

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: When the Game Ends Early

Imagine you are watching a long race to see which team runs the fastest. You want to measure their speed every hour. But here's the catch: some runners get injured and have to leave the track before the race is over.

In medical trials for serious diseases (like ALS, or Lou Gehrig's disease), patients often pass away before the study ends. This creates a huge headache for scientists. If a patient dies, their "score" for how well they can move or think stops making sense. You can't compare the speed of a runner who finished the race to a runner who left early because they are no longer running.

The paper asks: How do we fairly compare two treatments when some people die before the study finishes?

The Four Ways to Look at the Data

The authors explain that there are four main ways scientists try to solve this puzzle, each with its own pros and cons. They use a "Science Table" to visualize these different approaches.

1. The "Naive" Mistake (Ignoring the Dead)

The Analogy: Imagine you only look at the runners who finished the race and say, "Look, Team A's finishers are faster!"
The Problem: This is unfair. Maybe Team A had more people drop out because they were slower to begin with, or maybe Team B had more people drop out because they got hurt. By ignoring the people who left, you are comparing two different groups of people, not the two treatments. The paper says this method is broken.

2. The "Always-Survivor" Group (Principal Stratification)

The Analogy: Imagine you only look at the runners who were so tough they would have finished the race no matter which team they were on. You ignore everyone who dropped out.
The Result: This gives a very fair comparison of the treatment's effect on the people who survived. However, it's like looking at a tiny slice of the pie. It tells you how the treatment helps the survivors, but it doesn't tell you if the treatment actually helped people live longer. It's a "one-dimensional" view.

3. The "Composite" Score (Combining Death and Health)

The Analogy: Imagine you create a new rule: "Being alive is the most important thing. If you are alive, your speed matters. If you are dead, you get the lowest possible score, no matter how fast you were running."
The Result: This combines survival and health into one big score.

Pairwise Comparison: This asks, "If I pick a random person from Team A and a random person from Team B, who is doing better?" It counts how often Team A wins. It's simple and clear.
Survival-Incorporated Median: This looks at the "middle" person in the group. If the middle person is alive and doing well, the treatment is good.
The Catch: To make this work, you have to decide exactly how much "being alive" is worth compared to "running fast." The paper argues that the "Pairwise Comparison" is the most honest way to do this because it doesn't require inventing complex math to combine life and speed.

4. The "Restricted Mean Survival Time" (RMST)

The Analogy: Instead of looking at speed, you just count how many days each person stayed on the track.
The Result: This is a very clear, standard way to measure if a treatment helps people live longer. The paper suggests using this alongside the "Always-Survivor" method to get the full picture.

The Solution: A Bayesian Toolkit

The authors didn't just talk about these ideas; they built a mathematical toolkit (using a method called Bayesian estimation) to calculate these scores accurately.

Think of their toolkit as a smart detective.

When a patient dies, the detective doesn't just throw away their data.
Instead, the detective uses clues (like the patient's age, initial health, and how fast they were declining) to guess what would have happened if they had stayed alive.
The detective admits, "I'm not 100% sure, so I'll give you a range of possibilities rather than a single number." This honesty about uncertainty is the key strength of their method.

What They Found (The Simulation)

They tested their toolkit using computer simulations (fake data) with four different scenarios:

No effect: Neither treatment works.
Beneficial: Both treatments help people live longer and feel better.
Mixed: One treatment helps people feel better but might shorten their lives (or vice versa).
Censoring: People drop out for reasons other than death.

The Verdict:

The RMST (survival time) and SACE (always-survivor effect) worked best together. They gave a complete picture: "Treatment A helps people live longer, and for those who survive, it helps them move better."
The Pairwise Comparison was great at showing the trade-off. In the "Mixed" scenario, it showed that early on, the treatment looked good (people felt better), but as more people died, the score dropped, revealing the downside.
The Survival-Incorporated Median sometimes got stuck. If too many people died in a specific group, the "middle" person became hard to define, making the result unclear.

The Real-World Test: The ALS Trial

Finally, they applied their toolkit to real data from a trial for a drug called Olesoxime used to treat ALS.

The old way of looking at the data (the "Naive" method) showed a tiny, temporary benefit that disappeared later.
Their new method showed a slightly different story: The treatment seemed to have a beneficial effect on both survival and function, but the evidence wasn't strong enough to be 100% certain (the "confidence intervals" included zero).
Crucially, their method showed why the results were uncertain: because the number of deaths made it hard to know exactly what the treatment was doing to the patients who survived.

The Main Takeaway

The paper argues that when patients die during a study, you cannot just look at the survivors. You have to accept that the problem is multifaceted (it has many faces).

To get the truth, you should use two tools at once:

One tool to measure how long people lived (RMST).
One tool to measure how well the survivors were doing (SACE).

Trying to mash them into a single number often hides the truth. By using their new Bayesian methods, scientists can be more honest about what they know, what they don't know, and how the treatment truly affects both life and quality of life.

1. Problem Statement

In clinical trials for severe, life-threatening diseases (e.g., Amyotrophic Lateral Sclerosis - ALS), longitudinal patient-reported outcomes (such as quality of life or functional scores) are often truncated by death. Once a patient dies, their longitudinal outcome becomes undefined. This creates a fundamental challenge in causal inference:

Lack of Ordering: There is no natural way to order a "living patient with a specific score" against a "deceased patient."
Lack of Distance: It is impossible to define a meaningful metric (distance) between a change in functional score and time of death.
Current Limitations: Existing methods often rely on naive comparisons of survivors (introducing selection bias), composite endpoints with unclear causal interpretations (e.g., Win Ratio), or hypothetical scenarios that assume death can be prevented without affecting the disease trajectory (often clinically implausible).

The paper argues that without a unified framework defining the causal estimand (the specific quantity being estimated) and the assumptions required to identify it, the interpretation of treatment effects in these settings remains ambiguous.

2. Methodology

A. Causal Framework and Estimands

The authors propose a framework based on the Potential Outcomes paradigm and Principal Stratification. They categorize patients into four principal strata based on their potential survival under treatment ( $W=1$ ) and control ( $W=0$ ):

Always-survivors ($LL$): Survive under both.
Protected ($DL$): Die under control, survive under treatment.
Harmed ($LD$): Survive under control, die under treatment.
Never-survivors ($DD$): Die under both.

They review and formalize four distinct causal estimands:

Stratified Average Causal Effect (SACE): The average treatment effect restricted to the Always-survivor stratum ($LL$). This preserves the natural metric of the longitudinal outcome but requires identifying the unobservable stratum.
Restricted Mean Survival Time (RMST): A survival-based estimand representing the average survival time up to a specific horizon.
Pairwise Comparison (PC): A composite estimand treating death as the worst possible outcome. It calculates the probability that a randomly selected patient has a "better" composite outcome (survival + score) under treatment than under control. It relies on ordering but avoids defining a finite metric for all pairs.
Survival-Incorporated Median (SIM): The median of the distribution of composite outcomes. This attempts to provide a "magnitude" of effect but may be undefined (infinite) if a large proportion of patients die.

B. Bayesian Estimation Approach

The authors develop a Bayesian model-based framework (using the PyMC library) to estimate these quantities. Key modeling choices include:

Survival Model: A Bayesian piecewise-constant proportional hazards model to estimate survival probabilities ( $S_{i,t}$ ) and identify the principal strata.
Longitudinal Model: A linear mixed-effects model for the longitudinal outcome, conditional on the patient being alive.
Principal Ignorability: A critical assumption stating that, conditional on baseline covariates ( $X$ ), the potential survival status is independent of the potential longitudinal outcome. This allows the imputation of missing longitudinal data for those who died.
Two-Step Procedure: The survival model is estimated first to generate probabilities of belonging to the "always-survivor" stratum. These probabilities are then used as weights in the likelihood function for the longitudinal model.
Handling Uncertainty: The estimators explicitly account for the uncertainty in stratum membership (e.g., a patient observed to survive might theoretically have died under the alternative treatment).

3. Key Contributions

Unified Framework: The paper provides a comprehensive causal framework that explicitly links the definition of estimands to the underlying assumptions (e.g., Principal Ignorability, Monotonicity) required for their identification.
Bayesian Estimators: It introduces novel Bayesian estimators for SACE, PC, SIM, and RMST that rigorously handle the uncertainty of principal strata membership, avoiding the "hard" classification often used in frequentist approaches.
Clarification of Trade-offs: The authors demonstrate that no single estimand is perfect.
- SACE + RMST provides a complete, multidimensional picture (separating survival and functional effects) but yields multiple numbers.
- PC provides a single scalar probability of benefit but relies on the strong assumption that death is strictly worse than any functional score.
- SIM is intuitive but can be undefined in high-mortality settings.
Software Implementation: The authors provide open-source code (Truncation By Death Estimates repository) to facilitate the application of these methods.

4. Results

Simulation Study

The authors conducted a simulation study across four scenarios (no effect, beneficial, mixed effects, and censoring) to validate the estimators:

RMST: Performed robustly across all scenarios with accurate coverage.
SACE: Showed increasing uncertainty (wider credibility intervals) as the number of deaths increased, reflecting the difficulty of identifying the "always-survivor" stratum. However, when combined with RMST, it provided a complete picture of treatment effects in "mixed" scenarios (where treatment improves function but harms survival, or vice versa).
PC: Successfully captured the trade-off between survival and function. In mixed scenarios, the PC estimate shifted from >0.5 (beneficial) to <0.5 (detrimental) as mortality increased under treatment.
SIM: Often undefined in scenarios with high mortality rates (due to infinite values in the distribution), highlighting its limitations in severe disease contexts.

Application to ALS Data (Olesoxime Trial)

The methods were applied to a randomized controlled trial of Olesoxime in ALS patients ( $N=512$ ).

Findings: The naive analysis (comparing only survivors) showed a modest, non-sustained benefit.
Bayesian Results:
- SACE & RMST: Suggested a beneficial treatment effect, though 95% credibility intervals included the null.
- PC & SIM: Showed more pronounced effects, particularly at 12 months. The PC estimate was $0.53$ (95% CI: $0.53, 0.60$), indicating a probability of a better composite outcome favoring treatment.
Model Performance: The reconstruction of missing data showed adequate calibration (Integrated Brier Score) but moderate discrimination, consistent with the difficulty of predicting survival in ALS.

5. Significance and Conclusion

This paper addresses a critical gap in the analysis of clinical trials for fatal diseases. By moving away from purely model-based approaches (like joint modeling) toward causal estimand-based approaches, the authors ensure that the statistical question being answered aligns with the clinical question.

Recommendation: The authors recommend the joint use of SACE and RMST. This approach disentangles the survival component from the longitudinal functional component, acknowledging the inherently multifactorial nature of treatment effects in fatal diseases.
Impact: The framework helps regulators (like the FDA and EMA) and researchers interpret complex trial data where death truncates the primary endpoint, preventing misleading conclusions derived from naive survivor analyses or overly simplistic composite scores.
Future Work: The framework is noted as applicable to other intercurrent events (e.g., treatment non-adherence) where defining ordering and distance is challenging.