Truncation by death in the sufficient cause framework

1. Problem Statement

The paper addresses the methodological challenge of truncation by death in epidemiologic studies. This occurs when the outcome of interest ($Y$) is undefined for individuals who die before the assessment time ($S=0$).

• The Core Issue: Standard analyses that condition on observed survival (e.g., comparing outcomes only among those alive at follow-up) do not yield a valid causal estimand. They compare distinct subpopulations under different treatment levels, introducing selection bias (collider bias) because the set of survivors differs between treatment groups.
• Current Approach: The Principal Stratification approach (based on the potential outcomes framework) is the standard solution. It defines the Survivor Average Causal Effect (SACE), which estimates the effect of treatment on the outcome for the subpopulation of "always-survivors" (individuals who would survive regardless of treatment assignment).
• The Gap: While the potential outcomes framework is well-established for this problem, the authors argue that the Sufficient Cause Framework (based on causal mechanisms and component causes) has been underutilized in this context. They aim to provide a mechanistic description of truncation by death to deepen the understanding of biases and the conditions under which causal effects are null.

2. Methodology

The authors employ the Sufficient Cause Framework to model the joint processes of survival and the outcome.

• Causal Structure:
  • Variables: Treatment ($X$), Survival ($S$), Outcome ($Y$), and a common cause of both survival and outcome ($U$, e.g., age).
  • Sufficient Causes for Survival ($S$): The authors enumerate nine types of sufficient causes for survival involving combinations of $X$, $\bar{X}$ (absence of treatment), $U$, $\bar{U}$, and background factors ($A_k$).
  • Sufficient Causes for Outcome ($Y$): Since $Y$ is undefined if $S=0$, all sufficient causes for $Y$ must include $S$ as a component cause. They enumerate nine types of sufficient causes for $Y$ involving $S$, $X$, $U$, and background factors ($B_k$).
• Assumptions:
  • Sufficient Cause Positive Monotonicity: The authors assume that the absence of treatment ($\bar{X}$) and the presence of the risk factor ($U$) are not necessary components of any sufficient cause for survival or the outcome. This simplifies the model to four types of sufficient causes for $S$ and four for $Y$.
  • Risk Status Types: Under these assumptions, the population is partitioned into $2^4 \times 2^4 = 256$ distinct risk status types, defined by the specific combination of background factors ($A_k, B_k$) to which an individual is susceptible.
• Analytical Approach:
  • The authors map these 256 risk status types to the principal strata (Always-Survivors, Never-Survivors, Protectable, etc.).
  • They derive mathematical expressions for the Crude Estimand (conditional on observed survival) and the SACE in terms of the population-level frequencies of these background factors ($A_k, B_k$) and the common cause $U$.
  • They analyze the conditions under which the SACE is null versus when the crude estimand is non-null.

3. Key Contributions

1. Mechanistic Interpretation of Bias: The paper demonstrates that the non-causal nature of the crude estimand (conditioning on observed survival) arises because it compares individuals with different risk status types. Specifically, certain risk types contribute data to the treatment group (where they survive) but not to the control group (where they die), or vice versa. This creates a comparison of distinct populations rather than a causal contrast.
2. Characterization of the SACE: The authors provide explicit formulas for the SACE and the stratum-specific SACE ($SACE_u$) based on the frequencies of background factors. This links the abstract potential outcomes concept to concrete causal mechanisms.
3. Conditions for Null Effects: The paper establishes precise conditions under which the SACE is zero.
   • $SACE_u = 0$ if, among always-survivors with a specific $U$, there are no individuals susceptible to sufficient causes where the treatment effect is non-null.
   • Crucially, the authors show that even if the SACE is null (no causal effect among always-survivors), the crude estimand may still be non-zero due to the differential composition of survivors across treatment arms.
4. Role of Effect Modification ($U$): The study highlights that membership in the "always-survivor" stratum is not static; it depends on the value of the common cause $U$. For example, an individual might be an "always-survivor" if young ($U=1$) but a "never-survivor" if old ($U=0$). This dynamic membership affects the weighting and interpretation of the SACE.

4. Key Results

• Risk Status Enumeration: The authors identified 256 possible risk status types.
  • Always-Survivors: There are 192 types when $U=1$ and 128 types when $U=0$.
  • Null SACE Conditions:
    • For $U=1$, the SACE is null if the population contains no individuals susceptible to specific combinations of $A$ and $B$ factors (specifically those where treatment changes the outcome).
    • For $U=0$, similar conditions apply but involve different subsets of background factors.
• Divergence of Estimands: The paper mathematically proves that $E[Y|S=1, X=1] - E[Y|S=1, X=0]$ (the crude difference) can be non-zero even when $SACE = 0$. This occurs because the set of survivors in the treated group includes risk types that do not exist (or die) in the control group, leading to a biased comparison of "survivors" rather than "potential outcomes."
• Formulas: The paper provides a comprehensive table (Table 2) expressing various estimands (including SACE, crude effects, and survival probabilities) as functions of the joint probabilities of background factors ($A_k, B_k$) and $U$.

5. Significance

• Theoretical Advancement: This work bridges the gap between the Potential Outcomes Framework (dominant in modern causal inference) and the Sufficient Cause Framework (historically important for understanding mechanisms). It shows that the sufficient cause framework can offer unique insights into why certain estimands fail and what specific mechanisms they capture.
• Clarification of Principal Stratification: By breaking down the SACE into its constituent risk status types, the authors provide a deeper understanding of the "always-survivor" population, moving beyond abstract definitions to mechanistic descriptions.
• Practical Implications: The findings reinforce that conditioning on post-treatment variables (like survival) is inherently problematic for causal inference. The detailed derivation of conditions for null effects helps researchers understand when a lack of observed effect might be due to the specific composition of the survivor population rather than a true lack of treatment efficacy.
• Future Directions: The authors suggest that this framework could be extended to other post-treatment problems (like treatment adherence) and quantitative outcomes (via hazard models), encouraging further integration of sufficient cause models into complex causal inference settings.

In summary, the paper provides a rigorous, mechanism-based re-examination of truncation by death, demonstrating how the sufficient cause framework elucidates the biases in standard analyses and clarifies the structural conditions required for valid causal estimation in the presence of death.