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Causal Mediation Analysis with a Time-Dependent Mediator, Time-Dependent Confounders and a Time-to-Event Outcome: Revisiting the Difference Method

This paper evaluates the performance of the difference method for causal mediation analysis with time-dependent mediators, confounders, and time-to-event outcomes, finding that while it can provide unbiased estimates under specific assumptions (such as no treatment effect on time-dependent confounders) using Aalen or Cox models, it generally fails due to bias or collapsibility issues compared to the more robust parametric mediational g-formula.

Original authors: Robin Denz, Nina Timmesfeld

Published 2026-08-14

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CC BY 4.0

Original authors: Robin Denz, Nina Timmesfeld

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

Imagine you are a detective trying to solve a mystery: How does a specific action, like taking a medicine, actually change a person's future? Sometimes the medicine works directly, like a magic shield. But often, it works indirectly, like a coach who trains a player, and that player then wins the game. In the world of medical science, this is called mediation analysis. It's the art of splitting a total effect into two parts: the "direct" path and the "indirect" path.

Now, imagine the mystery gets harder. The "player" (a biological marker in the body) isn't just a snapshot; they change every day. The "coach" (the treatment) might also change the environment around them every day. And the "game" (the outcome, like survival) doesn't happen at a fixed time; it could happen tomorrow or ten years from now. This is the messy reality of time-to-event data with time-dependent variables. Scientists have built incredibly complex, super-powered computers to solve these puzzles, but they are so heavy and complicated that most doctors and researchers can't use them. So, many people try to use a "shortcut" called the difference method. It's like trying to guess the weight of a hidden object by weighing the whole box, then weighing the box without the object, and subtracting the two. It's simple, fast, and requires very little math. But does this shortcut work when the game is this complicated, or does it lead you to the wrong suspect?

This paper is a rigorous test drive of that shortcut. The authors, Robin Denz and Nina Timmesfeld, set up a massive digital laboratory to see if the "difference method" can survive the chaos of time-changing variables and survival outcomes. They didn't just guess; they built thousands of fake worlds (simulations) where they knew the exact truth. They pitted the simple difference method against a heavy-duty, complex tool called the parametric mediational g-formula, which acts as the gold standard.

Here is what they found in their simulations:

First, the shortcut is not a universal fix. If the treatment changes a time-dependent factor (like a side effect that appears later) which then messes up the relationship between the mediator and the outcome, the difference method fails spectacularly. It gets confused, mixing up direct effects with indirect ones, and produces biased results. In these messy scenarios, the shortcut is simply wrong.

Second, the shortcut works sometimes, but only under very specific conditions. If the outcome is rare (like a disease that happens to very few people) and the researchers use a specific type of math model called the Cox proportional hazards model, the difference method can give unbiased answers. Similarly, if they use an Aalen additive hazards model, it works well even if the outcome is common, provided the treatment doesn't directly cause the time-dependent confounders. However, if they try to use the shortcut with Accelerated Failure Time (AFT) models, it almost always produces a small but consistent error, no matter how clean the data is. This is because of a mathematical quirk called "non-collapsibility," where the math doesn't add up the way you'd expect.

Finally, the authors applied this test to a real-world example: a study on Ursodeoxycholic acid (UDCA), a drug used to treat a liver disease called primary biliary cholangitis. They wanted to see if the drug helped patients live longer by slowing down the damage to their liver tissue (the mediator). In this specific, small dataset, both the simple shortcut and the complex gold-standard tool gave very similar results. However, the authors warn that this similarity might be a fluke caused by the tiny sample size and the fact that the data was so clean. They emphasize that in the real world, where data is messy and assumptions are often violated, relying on the simple difference method is risky.

In short, the paper suggests that while the difference method is a tempting, easy-to-use tool, it is a bit of a "trap" for time-dependent survival data. It only works if you are extremely careful about your assumptions and the type of math you use. For most real-world scenarios, the authors recommend sticking to the complex, heavy-duty methods like the parametric g-formula, even if they are harder to use, because they are far more likely to tell you the truth. The shortcut might save you time, but in the high-stakes game of medical research, it might cost you the answer.

Technical Summary: Causal Mediation Analysis with a Time-Dependent Mediator, Time-Dependent Confounders and a Time-to-Event Outcome

1. Problem Statement

Mediation analysis is a standard tool for decomposing total treatment effects into direct and indirect components. However, applying these methods to settings involving time-to-event outcomes, time-dependent mediators, and time-dependent confounders presents significant challenges. Existing rigorous methods (e.g., parametric mediational g-formula, dynamic path analysis, multi-state modeling) are often statistically complex, computationally intensive, and lack user-friendly software implementations.

Consequently, researchers are often tempted to use the difference method, a simple approach where the indirect effect is estimated by subtracting the direct effect coefficient from the total effect coefficient in two separate regression models. While widely used for continuous outcomes and time-fixed mediators, the performance of the difference method in complex longitudinal survival settings is controversial. Previous studies suggest it may yield biased estimates due to non-collapsibility (in Cox models) or structural violations (e.g., time-dependent confounders caused by treatment). However, there has been no systematic evaluation of its empirical performance in realistic data-generating processes (DGPs) with time-dependent mediators for well-defined target estimands.

2. Methodology

2.1 Target Estimands and Identification

The authors define target estimands based on path-specific effects using counterfactual notation (Vansteelandt et al., 2019). This approach circumvents issues with post-treatment confounding by defining counterfactual histories for time-dependent confounders ( $L_t$ ) and mediators ( $M_t$ ) recursively.

Total Effect (TE): Contrast between $Y^{1,1}$ and $Y^{0,0}$ .
Direct Effect (DE): Contrast between $Y^{1,0}$ and $Y^{0,0}$ (mediator set to counterfactual under control).
Indirect Effect (IE): Contrast between $Y^{1,1}$ and $Y^{1,0}$ .
These effects are evaluated on three scales corresponding to popular survival models: log hazard ratio (Cox), hazard difference (Aalen), and log survival time (Accelerated Failure Time - AFT).

2.2 Methods Compared

The study compares the difference method against the parametric mediational g-formula (used as a benchmark).

Difference Method: Fits two models:
1. Total Effect Model: Outcome $\sim$ Treatment + Baseline Confounders.
2. Direct Effect Model: Outcome $\sim$ Treatment + Time-Dependent Mediator + Time-Dependent Confounders + Baseline Confounders.
  The indirect effect is estimated as the difference in the treatment coefficients ( $\hat{\theta}_A - \hat{\beta}_A$ ).
- Models used: Cox Proportional Hazards, Aalen Additive Hazards, and AFT (Weibull).
Parametric Mediational G-Formula: A simulation-based approach that estimates the expectation of counterfactual outcomes by modeling the conditional probabilities of the mediator, confounders, and outcome over time, then simulating data under different intervention scenarios ( $a, a'$ ) to estimate effects.

2.3 Simulation Study Design

A discrete-event simulation (Gillespie algorithm) was used to generate data with $n=2000$ individuals.

Variables: Binary baseline treatment ( $A$ ), baseline covariates ( $X, Z$ ), time-dependent confounder ( $L_t$ ), time-dependent mediator ( $M_t$ ), and time-dependent outcome ( $Y_t$ ).
Scenarios: Six scenarios were constructed to vary the causal structure:
1. Scenario 1: Randomized treatment; $L_t$ is a pure predictor of outcome (no confounding).
2. Scenario 2: $L_t$ is a time-dependent confounder of the mediator-outcome relationship.
3. Scenario 3: Feedback loop between $L_t$ and $M_t$ .
4. Scenario 4: $L_t$ is directly caused by treatment $A$ (violating the assumption that treatment does not affect time-dependent confounders).
5. Scenario 5: Non-randomized treatment with confounding of the treatment-mediator relationship.
6. Scenario 6: Non-randomized treatment with confounding of the treatment-outcome relationship.
Parameters: Scenarios included variations in baseline event probability (rare vs. common outcomes) and the presence/absence of indirect effects.
Performance Metrics: Bias and Mean Squared Error (MSE) of the estimated indirect effect.

2.4 Real-World Application

The methods were applied to data from a randomized controlled trial (Lindor et al., 1994) on Ursodeoxycholic acid (UDCA) for Primary Biliary Cholangitis (PBC).

Treatment: UDCA vs. Placebo.
Mediator: Histological progression (time-dependent).
Outcome: Liver transplant-free survival (time-to-event).
Confounders: Baseline disease stage, Mayo risk score, bilirubin; time-dependent varices.

3. Key Results

3.1 Simulation Findings

Aalen Model (Difference Method): Produced unbiased estimates in Scenarios 1–3 and 5–6, provided the model was correctly specified and the treatment did not directly cause the time-dependent confounder ( $L_t$ ). This held for both rare and common outcomes.
Cox Model (Difference Method): Produced unbiased estimates only when the outcome was rare and the treatment did not directly cause $L_t$ . With common outcomes, the method produced biased estimates due to non-collapsibility, regardless of the scenario.
AFT Model (Difference Method): Produced biased estimates in almost all scenarios, even when the outcome was rare and no time-dependent confounding existed. The bias stemmed from the non-collapsibility of the AFT coefficients; the marginal direct effect (target estimand) differed from the conditional coefficient estimated by the model.
Scenario 4 (Treatment $\to$ Confounder): In all scenarios where the treatment directly caused the time-dependent confounder ( $L_t$ ), all versions of the difference method produced biased estimates, regardless of the model used or whether $L_t$ was included in the direct effect model. This is because conditioning on $L_t$ removes part of the direct effect that flows through $A \to L_t \to Y$ .
Parametric G-Formula: Produced unbiased estimates in all scenarios and parameter combinations, serving as a robust benchmark.
Efficiency: When the difference method was unbiased, it generally exhibited lower Mean Squared Error (higher precision) than the g-formula, likely due to the simulation variance inherent in the g-formula's point estimate calculation in this study.

3.2 Real-World Application Results

In the PBC dataset:

Both the difference method (Cox/Aalen/AFT) and the g-formula yielded similar point estimates for total, direct, and indirect effects.
The total effect of UDCA showed a protective trend (Hazard Ratio $\approx$ 0.63), but confidence intervals were wide and included 1 due to the small sample size ( $n=170$ ).
Indirect effects via histological progression were close to 1 (null), with wide confidence intervals.
The authors note that the near-equivalence of results in this specific dataset may be coincidental or due to the small sample size masking structural violations (e.g., potential direct effects of UDCA on time-dependent confounders like varices).

4. Significance and Claims

The paper claims to fill a gap in the literature by systematically evaluating the difference method in the specific context of time-dependent mediators and confounders with time-to-event outcomes.

Conditional Utility: The difference method is not generally recommended for this setting. It is only a viable, simple alternative if strict assumptions hold:
1. The treatment does not have a direct causal effect on any time-dependent confounders.
2. For Cox models, the outcome must be rare.
3. For AFT models, the method is generally not recommended due to inherent bias from non-collapsibility, even under ideal conditions.
Robustness of G-Formula: The parametric mediational g-formula remains the preferred method for unbiased estimation across diverse structural scenarios, despite its computational complexity.
Practical Guidance: The authors emphasize that while the difference method is computationally efficient and easy to implement (e.g., two lines of code in R), its use requires careful assessment of the causal DAG. If the assumption that "treatment does not affect time-dependent confounders" cannot be defended, the difference method will yield biased results.

The paper concludes that while specialized methods are complex, they are necessary for valid inference in complex longitudinal survival data. The difference method should be used with extreme caution and only when its restrictive structural assumptions are plausible.