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Causal Fairness for Survival Analysis

This paper introduces a non-parametric causal framework for fairness in survival analysis that decomposes temporal disparities into direct, indirect, and spurious pathways to explain their origins and evolution, demonstrated through an analysis of racial disparities in ICU outcomes.

Original authors: Drago Plecko

Published 2026-05-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Drago Plecko

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 Picture: Why "Fairness" Needs a Time Machine

Imagine you are running a hospital, and you want to know if your treatment is fair to everyone. You look at your data and see that one group of patients (let's call them Group A) seems to survive longer than another group (Group B).

In the past, data scientists would stop there. They would say, "Okay, Group A lives longer. Maybe we need to fix the treatment for Group B." They looked at the snapshot of the result.

But this paper argues that looking at the snapshot isn't enough. It's like seeing a car crash and only asking, "Who was driving?" without asking, "Was the road icy? Was the car broken? Did the driver get a flat tire on the way?"

The author, Drago Plečko, wants to build a "Time Machine for Fairness." Instead of just looking at the final result (who died and when), this method looks at the journey to see why the groups ended up differently. It asks: Did the difference happen because of the group's identity directly? Did it happen because of the medical conditions they arrived with? Or did it happen because of hidden factors like where they live or their income?

The Problem with Old Methods: The "Black Box" of Statistics

Current methods for checking fairness in time-based data (like "time until death" or "time until a machine breaks") are like a black box. You put data in, and you get a number out that says, "There is a 5% difference between Group A and Group B."

The problem is that this number doesn't tell you why.

Is the 5% difference because the hospital treats Group B worse? (Direct unfairness).
Is it because Group B arrives sicker? (Indirect unfairness).
Is it because Group B is generally younger or older, which naturally changes survival rates? (Spurious/Confounded unfairness).

Old statistical methods can't separate these three reasons. They just give you a blurry picture.

The Solution: A Causal Map (The "Recipe" for Disparities)

This paper introduces a new framework that acts like a detailed recipe for understanding disparities. It breaks the "survival gap" down into three distinct ingredients:

The Direct Path (The "Head-On" Effect): This is the effect of the group identity itself. Imagine two runners starting a race. If the referee (the system) trips one runner but not the other, that's a direct unfair effect. In the hospital example, this would be if the treatment protocol itself treats one race differently than another, regardless of their health.
The Indirect Path (The "Detour" Effect): This is where the group identity leads to a different starting condition, which then leads to the result. Imagine Group B has to walk a longer, rougher path to get to the starting line because of their neighborhood. They arrive tired and out of breath. The race result is worse, but not because the referee tripped them; it's because of the detour they were forced to take. In the paper, this represents how race might influence illness severity or access to care before the ICU admission.
The Spurious Path (The "Coincidence" Effect): This is a fake connection. Imagine Group B happens to be younger on average. Younger people naturally run faster. If Group B wins, it's not because of the race rules or the path; it's just because they are younger. This is a "spurious" correlation. The paper's method helps us see that this isn't "unfairness" in the system, but a natural demographic difference.

The Three Scenarios: Handling the "Missing Pieces"

In real life, data is messy. Sometimes we don't know exactly when a patient died because they left the study early (this is called "censoring"). The paper handles three different ways this messiness can happen:

The "Random Exit" (Non-Informative Censoring): Imagine a patient leaves the study just because the study ended, or they moved away. This is random. The paper's method works perfectly here, using standard tools to untangle the three paths mentioned above.
The "Competing Risks" (Multiple Bad Outcomes): Imagine a patient could die from a heart attack OR a stroke. Both are bad, but they are different events. The paper adapts its method to track fairness for each specific type of bad event, ensuring we don't mix them up.
The "Suspicious Exit" (Informative Censoring): This is the tricky one. Imagine patients leave the study early because they are getting sicker, and the researchers stop tracking them. The reason they left is connected to the outcome. The paper uses a mathematical tool called a "Copula" (think of it as a flexible rubber sheet that stretches to fit the data) to guess what would have happened if we could see the whole picture. It doesn't give one single answer but provides a range of possibilities (bounds) based on how strong that "suspicious" connection might be.

The Real-World Test: ICU Patients in Australia

To prove this works, the author applied it to real data from the Intensive Care Unit (ICU) in Australia and New Zealand. They looked at nearly 250,000 patients, comparing Indigenous and non-Indigenous patients.

What they found:
If you just looked at the total survival difference, the numbers were confusing. Sometimes Indigenous patients seemed to do worse, sometimes better, and the average difference was almost zero. It looked like "no big deal."

But when they used the "Time Machine" (the causal decomposition):

The Direct Path: Showed that Indigenous patients actually had a slight survival advantage if you ignored everything else.
The Indirect Path: Showed a huge disadvantage. Indigenous patients arrived with much sicker chronic conditions and more severe illnesses. This "detour" was dragging their survival down significantly.
The Spurious Path: Showed an advantage. Indigenous patients were, on average, younger when admitted, which naturally helped them survive longer.

The Lesson: The "zero difference" in the total numbers was a lie. It was hiding two massive, opposing forces: a huge disadvantage caused by health inequalities (Indirect) and a natural advantage caused by age (Spurious). Without this new method, a hospital administrator might have thought, "Everything is fair," and missed the fact that Indigenous patients were arriving at the ICU in much worse shape.

Summary

This paper provides a new toolkit for data scientists and doctors. Instead of just asking, "Is there a gap?", it asks, "Where does the gap come from?"

It separates the unfair treatment (Direct), the unfair starting conditions (Indirect), and the natural differences (Spurious). By doing this, it helps us understand that "fairness" isn't just about the final score; it's about the entire journey leading up to it. This allows us to fix the specific part of the system that is broken, rather than guessing.

Technical Summary: Causal Fairness for Survival Analysis

Problem Statement

In high-stakes domains such as healthcare, employment, and criminal justice, machine learning systems increasingly rely on large-scale datasets to inform decisions. While the field of Fair Machine Learning (Fair ML) has made significant strides in bias detection and fair prediction, these efforts largely focus on static settings (single time-point snapshots). Conversely, fairness in temporal contexts, specifically Survival Analysis or Time-to-Event (TTE) analysis, remains underexplored.

Existing approaches to fair survival analysis predominantly adopt statistical fairness definitions. The paper argues that even with unlimited data, statistical definitions cannot disentangle the causal mechanisms generating disparities. They fail to distinguish between direct effects of a protected attribute, indirect effects mediated through other variables, and spurious (confounded) effects. This limitation prevents a human-understandable explanation of why disparities arise and how they evolve over time.

Methodology

The authors propose a non-parametric causal framework for fairness in TTE analysis. The approach is grounded in Structural Causal Models (SCMs) and proceeds through four distinct steps:

Formalization of Assumptions: The authors adapt the Standard Fairness Model (SFM) to survival analysis. This graphical model includes:
- $X$ : Protected attribute (e.g., race).
- $Z$ : Confounders (not causally influenced by $X$ ).
- $W$ : Mediators (potentially influenced by $X$ ).
- $T$ : Survival time (unobserved).
- $C$ : Censoring time.
- $(M, \delta)$ : Observed data, where $M = \min(T, C)$ and $\delta$ indicates if the event occurred.
  The framework considers three censoring regimes:
- Non-Informative Censoring (NIC): $T \perp \perp C \mid X, Z, W$ .
- Competing Risks (CR): Multiple event types ( $T_1, \dots, T_K$ ) where all are of interest.
- Informative Censoring (IC): $T$ and $C$ share unobserved common causes, requiring sensitivity analysis.
Recovery of Conditional Survival: The method first recovers the conditional survival function $P(T \mid X, Z, W)$ (or Cumulative Incidence Functions in CR settings) from the observed censored data $(M, \delta)$ .
Causal Reduction Theorem: A core theoretical contribution is the Causal Reduction Theorem (Theorem 1). It proves that any functional $\Phi$ of the conditional survival distribution (e.g., survival probability at time $t$ ) is a deterministic function of $(X, Z, W)$ . This allows the problem to be reframed: instead of analyzing the complex survival curve directly, one can analyze the causal effects of $X$ on the derived variable $\Phi$ . This enables the decomposition of disparities into:
- Direct Effect (DE): $X \to T$ (or $X \to \Phi$ ).
- Indirect Effect (IE): $X \to W \to T$ .
- Spurious Effect (SE): $X \leftrightarrow Z \to T$ .
Efficient Estimation: The paper provides estimation strategies for these effects:
- Model-Based Estimation: Utilizing Random Survival Forests (RSF) to estimate conditional hazards and propensities.
- Doubly-Robust Estimation: The authors derive a new influence function (Theorem 2) for censored data, enabling consistent estimation even if one of the nuisance models (survival or censoring) is misspecified.
- Informative Censoring Handling: For IC settings, the paper employs a Copula-Graphic Estimator (CGE) based on Archimedean copulas. Since the joint distribution is not fully identifiable without strong assumptions, the authors derive sharp bounds (Theorem 3) for the survival function, which collapse to point estimates as the time grid refines.

Key Contributions

Formalization of Causal Fairness in TTE: The paper formalizes causal fairness specifically for survival analysis, proving the Causal Reduction Theorem to enable pathway decomposition of disparities over time.
Identification Results: It derives identification results for causal effects under three distinct regimes: non-informative censoring, competing risks, and informative censoring.
New Technical Estimators:
- A doubly-robust estimator for counterfactual effects in censored data (Theorem 2).
- Sharp bounds for copula-graphic estimation under non-disjoint increments (Theorem 3), addressing the challenge of informative censoring without assuming independence.
Empirical Application: The framework is applied to a real-world dataset of ICU admissions to analyze racial disparities in post-ICU mortality and readmission.

Results

The methodology was applied to the ANZICS Adult Patient Database (249,958 patients, including 10,436 Indigenous patients) to study racial disparities in:

Time-to-Death (Non-Informative Censoring):
- Standard statistical analysis (Total Variation measure) showed a near-zero disparity after 80 days, suggesting no difference.
- Causal Decomposition revealed that this null result masked substantial, opposing pathway-specific effects:
  - Direct Effect: Minority patients showed improved survival along the direct path.
  - Indirect Effect: Minority patients showed reduced survival due to higher burdens of chronic/acute illness (mediated by $W$ ).
  - Spurious Effect: Minority patients showed improved survival due to lower average age at admission (confounded by $Z$ ).
- The decomposition demonstrated that standard statistical fairness metrics can obscure the underlying causal mechanisms driving disparities.
Time-to-Readmission (Informative Censoring):
- Analyzed under the assumption that death acts as informative censoring for readmission.
- Using sensitivity analysis with varying dependence parameters ( $\tau$ ), the study found that minority patients faced a higher burden of readmission, driven largely by indirect effects (more complex diagnoses and worse chronic health) and direct effects, despite a lower spurious burden due to age.

Significance and Claims

The paper claims that its approach provides a principled, human-understandable explanation of survival disparities that standard statistical fairness analyses miss. By moving beyond static snapshots and statistical definitions, the framework allows researchers to:

Disentangle whether disparities are caused by the protected attribute itself (direct), by systemic factors affecting mediators (indirect), or by confounding variables (spurious).
Observe how these disparities evolve over time, rather than treating them as a single aggregate number.
Handle complex real-world data scenarios, including competing risks and informative censoring, which are common in healthcare but often ignored in fair ML literature.

The authors emphasize that while their framework relies on specific causal assumptions (encoded in the SFM), these assumptions allow for the identification of causal mechanisms that are otherwise unidentifiable with statistical methods alone. They acknowledge limitations, such as the reliance on the Archimedean copula assumption for informative censoring and the assumption of time-invariant covariates, noting these as directions for future work.