📊 statistics

Efficient and Debiased Learning of Average Hazard Under Non-Proportional Hazards

This paper proposes a semiparametric, doubly robust framework with cross-fitted machine learning to estimate the average hazard as a stable causal effect measure under non-proportional hazards, enabling valid $\sqrt{n}$ -consistent inference and demonstrating superior performance over traditional Cox-based summaries in both simulations and real-world comparative effectiveness research.

Original authors: Xiang Meng, Lu Tian, Kenneth Kehl, Hajime Uno

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xiang Meng, Lu Tian, Kenneth Kehl, Hajime Uno

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 trying to decide between two different diets (let's call them Diet A and Diet B) to see which one helps people live longer. You have a group of people, and you track how long it takes for them to get sick or pass away.

For decades, scientists have used a tool called the Hazard Ratio to make this comparison. Think of the Hazard Ratio like a speedometer that gives you a single number representing how much faster or slower people are "dying" in one group compared to the other.

The Problem: The Speedometer is Broken
The problem is that this "speedometer" assumes the risk of dying stays constant over time. But in real life, especially with modern cancer treatments like immunotherapy, things are messy:

The "Delayed Effect" Trap: Some drugs don't work immediately. They might take a year to kick in. The old speedometer gets confused by this delay.
The "Crossing" Trap: Sometimes, Diet A is better at first, but Diet B becomes better later. The old speedometer tries to average these two conflicting stories into one number, often resulting in a confusing or misleading answer.
The "Who's Watching" Trap: The old speedometer's reading changes depending on when people drop out of the study or when the study ends. If you stop the study early, you get a different answer than if you wait longer, even if the diets themselves haven't changed.

The Solution: The "Average Hazard" (AH)
This paper introduces a new, smarter way to measure treatment success called the Average Hazard (AH).

Instead of looking at a split-second speedometer reading, the authors suggest looking at the total mileage covered.

Here is the analogy:
Imagine you are driving two different cars (Treatment A and Treatment B) on a long road trip.

The Old Way (Hazard Ratio): You check the speedometer every second and try to average those speeds. If one car speeds up and slows down wildly, or if you only watch them for the first 10 minutes, your average speed calculation might be totally wrong.
The New Way (Average Hazard): You look at the total distance traveled divided by the total time spent driving.
- Formula: (Total Accidents) ÷ (Total Time Everyone was on the road).

This gives you a rate: "How many accidents happen per 100 hours of driving?"

If Treatment A has 5 accidents in 1,000 hours, the rate is 0.005.
If Treatment B has 2 accidents in 1,000 hours, the rate is 0.002.

This rate is stable. It doesn't matter if you stop the study early or late; it doesn't matter if the risk changes over time. It just tells you the true, overall "danger level" of the treatment for the whole group.

The Secret Sauce: "Debiased" Machine Learning
Calculating this new rate is hard because real-world data is messy. People have different ages, health conditions, and reasons for dropping out of the study.

The authors built a sophisticated "calculator" (a statistical framework) that uses Machine Learning to clean up this mess.

The "Double Robust" Shield: Imagine you are trying to predict the weather. You have two different weather models. If either one of them is right, your prediction will be accurate. This new method works the same way. It uses two different ways to guess the "noise" in the data (like who is likely to drop out). As long as one of those guesses is good, the final result is correct.
Cross-Fitting: To make sure the computer doesn't "cheat" by memorizing the data (overfitting), they split the data into groups. They train the computer on one group and test it on another, swapping them around. This ensures the result is honest and reliable.

Real-World Test: Melanoma Treatment
The authors tested this new method on real data from older adults with advanced melanoma (a type of skin cancer) who were taking different immunotherapy drugs.

The Result: The old method (Hazard Ratio) struggled because the drugs worked in complex, non-linear ways.
The New Method: The "Average Hazard" clearly showed that the combination of drugs (Nivolumab + Ipilimumab) and Nivolumab alone were both significantly better than Ipilimumab alone. The result was stable whether they looked at 2 years or 3 years of data.

Why This Matters
This paper gives doctors and researchers a better ruler to measure how well treatments work.

It's Honest: It doesn't get confused by changing risks over time.
It's Clear: It tells patients, "If you take this drug, you can expect X number of events per year," which is much easier to understand than a complex statistical ratio.
It's Flexible: It works even when we use powerful AI tools to analyze complex patient data, ensuring the results aren't just a trick of the math.

In short, they replaced a shaky, confusing speedometer with a sturdy, reliable odometer that tells the true story of survival.

1. Problem Statement

In time-to-event analysis (survival analysis), the Hazard Ratio (HR) derived from the Cox proportional hazards model is the standard metric for treatment effect. However, the Cox HR relies on the proportional hazards (PH) assumption, which is frequently violated in modern clinical settings (e.g., immuno-oncology with delayed effects, or crossing hazards).

When PH is violated:

The Cox HR loses a stable causal interpretation, becoming a weighted average of time-varying effects where weights depend on study-specific censoring and follow-up patterns.
The estimator is non-collapsible, meaning it can vary significantly between studies with identical biological effects but different administrative censoring.
It lacks an absolute reference, making clinical translation difficult.

While alternatives like Restricted Mean Survival Time (RMST) exist, they operate on the time scale. There is a need for a rate-based summary (events per person-time) that remains interpretable under non-proportional hazards and supports covariate adjustment in high-dimensional settings using machine learning (ML) while maintaining valid statistical inference ( $\sqrt{n}$ -consistency).

2. Methodology

The authors propose a semiparametric, doubly robust framework for estimating the Average Hazard (AH) and the Log Average Hazard Ratio (log-AHR).

A. Causal Definition and Identification

Estimand: The Average Hazard for treatment arm $a$ at horizon $\tau$ is defined as:
$\eta_a(\tau) = \frac{F_a(\tau)}{R_a(\tau)} = \frac{1 - S_a(\tau)}{\int_0^\tau S_a(t) dt}$
where $S_a(t)$ is the marginal counterfactual survival function, $F_a$ is cumulative incidence, and $R_a$ is the Restricted Mean Survival Time (RMST).
Interpretation: $\eta_a(\tau)$ represents the population-level person-time event rate (e.g., events per 100 person-years). Unlike the "marginalized conditional incidence rate" (which averages stratum-specific rates), AH weights strata by their contribution to total person-time, ensuring it reflects the true population rate.
Identification: Under standard causal assumptions (Consistency, Conditional Exchangeability, Positivity, Conditional Independent Censoring), the AH is identified from observed right-censored data.

B. Efficient Influence Function (EIF) Derivation

To enable efficient inference, the authors derive the Efficient Influence Function (EIF) for the log-AH ( $\theta_a = \log \eta_a$ ):

Step 1: They utilize the known EIF for the marginal survival curve $S_a(t)$ (from Westling et al., 2024), which combines an outcome regression component with Inverse Probability of Treatment Weighting (IPTW) and Inverse Probability of Censoring Weighting (IPCW) augmentation.
Step 2: Using the Functional Delta Method, they map the EIF of the survival curve to the EIF of the log-AH.
$\phi^*_{\theta_a}(O) = -\frac{1}{F_a(\tau)}\phi^*_{S_a(\tau)}(O) - \frac{1}{R_a(\tau)}\int_0^\tau \phi^*_{S_a(u)}(O) du$
This derivation proves that the log-AH is pathwise differentiable, a prerequisite for semiparametric efficiency.

C. Estimation Strategy: Cross-Fitted Debiased Estimator

The authors construct a cross-fitted, doubly robust estimator (AH-DML) to handle high-dimensional nuisance parameters (propensity scores, censoring models, and outcome models) estimated via flexible machine learning:

Cross-Fitting: Data is split into $K$ folds. Nuisance models are trained on $K-1$ folds and applied to the held-out fold to prevent overfitting bias.
One-Step Estimator: They construct a one-step estimator for the survival curve using the EIF, then apply the plug-in formula for AH.
Double Robustness: The estimator remains consistent if either:
- The outcome model (survival function) is correctly specified, OR
- Both the treatment assignment (propensity score) and censoring models are correctly specified.
Variance Estimation: The asymptotic variance is estimated using the empirical variance of the estimated EIF, allowing for the construction of valid Wald-type confidence intervals.

3. Key Contributions

Formalization of AH as a Causal Estimand: The paper rigorously defines AH within the potential outcomes framework, distinguishing it from non-collapsible alternatives and proving its identification under right-censoring.
Theoretical Derivation: It establishes the pathwise differentiability of the AH and derives its Efficient Influence Function, providing the theoretical foundation for efficient, nonparametric inference.
Algorithmic Framework: It introduces a cross-fitted, doubly robust estimator that leverages modern machine learning for nuisance estimation while guaranteeing $\sqrt{n}$ -consistency and asymptotic normality under mild product-rate conditions (e.g., nuisance estimators converging faster than $n^{-1/4}$ ).
Robustness to Non-Proportionality: The method provides stable inference even when hazards cross or treatment effects are delayed, scenarios where Cox HR fails.

4. Results

Simulation Studies

Settings: The authors tested the estimator under Proportional Hazards (PH), Non-Proportional Hazards (Non-PH), and Crossing Hazards (Cross-A) scenarios with varying sample sizes ( $n=500$ to $1500$).
Comparisons: They compared their AH-DML estimator against a standard G-computation (outcome regression only) and a Cox model with covariate adjustment.
Findings:
- Bias: AH-DML achieved small bias across all settings.
- Coverage: It maintained near-nominal (95%) confidence interval coverage, whereas Cox-based summaries showed unstable coverage in Non-PH and Crossing Hazards regimes.
- Double Robustness: In stress tests where the censoring model was misspecified, AH-DML maintained performance as long as the outcome model was correct, validating the double robustness property.

Real-World Data Analysis (SEER-Medicare)

Context: Comparative effectiveness of immunotherapy regimens for advanced melanoma in older adults (66+ years).
Comparison: Nivolumab + Ipilimumab vs. Ipilimumab monotherapy; Nivolumab vs. Ipilimumab.
Findings:
- Both Nivolumab-containing regimens showed a significantly lower Average Hazard of death compared to Ipilimumab monotherapy.
- Stability: The estimated AH ratios were stable across time horizons (24 and 36 months), with ratios around 0.59–0.60 (indicating ~40% reduction in event rate).
- Contrast with Cox: The study highlights that AH provides a model-free, interpretable rate summary that avoids the instability of HRs in the presence of delayed treatment effects common in immunotherapy.

5. Significance

This paper bridges a critical gap in causal survival analysis by providing a rate-based alternative to the Hazard Ratio that is robust to non-proportional hazards.

Clinical Utility: It offers clinicians and decision-makers an interpretable metric (events per person-time) that does not rely on the often-violated PH assumption.
Methodological Advancement: It demonstrates how to combine machine learning (for flexible nuisance estimation) with semiparametric theory (for valid inference) in a doubly robust manner for complex time-to-event estimands.
Applicability: The framework is particularly valuable for modern oncology trials (e.g., immuno-oncology) and real-world evidence studies where treatment effects evolve over time and standard Cox models may yield misleading conclusions.