📊 statistics

Estimating conditional Mann-Whitney effects using pseudo-observation-based regression

This paper introduces a distribution-free regression framework using pseudo-observations to estimate conditional Mann-Whitney effects for both ordinal and right-censored outcomes, providing consistent coefficient estimates and bootstrap-based hypothesis tests that are validated through simulations and applied to breast cancer survival data.

Original authors: Dennis Dobler, Alina Schenk, Matthias Schmid

Published 2026-04-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Dennis Dobler, Alina Schenk, Matthias Schmid

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: A New Way to Compare Treatments

Imagine you are a doctor trying to decide if a new medicine (Treatment A) is better than the old standard (Treatment B). Traditionally, statisticians have used tools like the Cox model to answer this. Think of the Cox model as a very strict referee who only accepts the game if the players follow a specific rule: the "hazard ratio" must stay constant over time. If the players change their speed or strategy halfway through the game, this referee gets confused and might give the wrong answer.

The authors of this paper, Dennis Dobler, Alina Schenk, and Matthias Schmid, wanted to build a new referee that doesn't care about those strict rules. They created a method based on something called the Mann-Whitney effect.

The Core Concept: The "Who Wins?" Probability

Instead of asking "How much faster does Treatment A make you recover?", they ask a simpler, more intuitive question:

"If I pick one random patient from the Treatment A group and one random patient from the Treatment B group, what is the probability that the Treatment A patient will survive longer?"

If the answer is 50%, the treatments are equal.
If the answer is 60%, Treatment A is generally better.
If the answer is 40%, Treatment B is generally better.

This is the Mann-Whitney effect. It's a "relative effect." It's easy to understand because it's just a probability, like a coin flip.

The Problem: What About Patient Differences?

In real life, patients aren't identical. Some are young, some are old; some have mild tumors, some have aggressive ones.

The Old Way: You might try to calculate the "average" probability for everyone. But this hides the truth. Maybe Treatment A is amazing for young people but terrible for old people. An average would miss this completely.
The New Goal: The authors wanted a way to say: "For a 50-year-old with a specific type of tumor, what is the probability they will do better on Treatment A?"

They wanted to build a regression model (a mathematical formula) that takes patient details (covariates) and predicts that "Who Wins?" probability.

The Solution: The "Pseudo-Observation" Trick

Here is the tricky part: You can't easily calculate the "Who Wins?" probability for a single person because it requires comparing them to everyone else. It's like trying to calculate your average speed in a race by only looking at your own stopwatch; you need to know how everyone else ran to know if you were fast.

To solve this, the authors used a clever statistical trick called Pseudo-Observations.

The Analogy: The "What If I Wasn't Here?" Game
Imagine you are in a room with 100 people. You want to know how your presence affects the average height of the room.

You calculate the average height of the whole room.
Then, you pretend you aren't there. You calculate the average height of the remaining 99 people.
You compare the two numbers. The difference tells you how much you contributed to the average.

The authors do this for every single patient in the study. They calculate the "Who Wins?" probability for the whole group, then remove one patient, recalculate, and see how much that one patient changed the result. This creates a "pseudo-observation" for every patient—a number that represents their specific contribution to the treatment effect.

Once they have these numbers for everyone, they can plug them into a standard regression model (like a linear equation) to see how age, tumor size, or gender changes the probability of winning.

The "Jackknife" and "Bootstrap" Tools

To make sure their new referee is accurate, they used two statistical tools:

The Jackknife: This is the "What If I Wasn't Here?" method described above. It helps them estimate the uncertainty of their calculations.
The Bootstrap: Imagine taking a bag of marbles (your data), pulling one out, writing it down, putting it back, and doing this 1,000 times to create 1,000 new "fake" bags. By seeing how the results change across these 1,000 fake bags, they can be very confident about how accurate their real results are.

Putting It to the Test: The SUCCESS-A Trial

The authors tested their new method on real data from a massive breast cancer study called SUCCESS-A.

The Old View: The original study used the strict Cox model and found no overall difference between the new drug (gemcitabine) and the standard drug.
The New View: Using their flexible "Who Wins?" model, they found that the answer wasn't "no difference." Instead, the difference depended on the patient!
- For patients with aggressive, advanced tumors, the new drug was significantly better.
- For patients with less aggressive tumors, the standard drug was actually slightly better.

The old model averaged these two groups together and saw a flat line (no effect). The new model saw the hills and valleys, identifying exactly which patients would benefit.

Why This Matters

No Strict Rules: Unlike the old methods, this new approach doesn't require the "hazard ratio" to stay constant. It works even if the treatment effect changes over time.
Easy to Understand: Instead of confusing numbers like "Hazard Ratios," doctors get probabilities (e.g., "There is a 65% chance this patient will do better on Drug A").
Personalized Medicine: It helps identify subgroups. It tells us that a "one-size-fits-all" approach might be wrong, and we need to treat patients based on their specific characteristics.

Summary

The authors built a new statistical tool that acts like a flexible, rule-breaking referee. Instead of asking "How much faster is the treatment?", it asks "What are the odds this patient wins?" By using a clever "What If I wasn't here?" trick (pseudo-observations), they can predict these odds for specific types of patients. When they applied this to breast cancer data, they discovered that a drug thought to be ineffective was actually a lifesaver for a specific group of patients—a nuance the old methods missed entirely.

1. Problem Statement

The paper addresses the challenge of analyzing the Mann-Whitney effect (also known as the relative treatment effect or probabilistic index), defined as $\theta = P(T_1 > T_2)$ , where $T_1$ and $T_2$ are event times from two independent groups. While the unconditional Mann-Whitney effect is a standard non-parametric measure for two-sample comparisons, existing methods often lack the ability to:

Adjust for covariates: Incorporate patient-specific characteristics ( $Z$ ) to estimate a conditional effect $\theta(z_1, z_2) = P(T_1 > T_2 | Z_1=z_1, Z_2=z_2)$ .
Handle censored data: Robustly manage randomly right-censored time-to-event data without relying on restrictive parametric assumptions (e.g., proportional hazards).
Interpretability: Provide results on a probability scale rather than a log-hazard scale, which is often difficult to interpret clinically.

The authors aim to develop a distribution-free regression framework that links the Mann-Whitney effect to a linear combination of covariates, applicable to both fully observed and right-censored data.

2. Methodology

The proposed approach combines pseudo-observations with generalized estimating equations (GEE).

A. The Model

The authors define a regression model for the conditional Mann-Whitney effect:
$\theta(z_1, z_2) = \mu(\beta_0 + \beta_1^\top z_1 + \beta_2^\top z_2)$
where:

$\mu(\cdot)$ is a strictly monotone inverse link function (e.g., identity or logistic).
$\beta$ is the parameter vector.
$z_1$ and $z_2$ are covariate vectors for individuals in group 1 and group 2, respectively.
The model allows for different sets of covariates in each group, though the special case $z_1 = z_2 = z$ allows for causal inference interpretation under standard assumptions.

B. Pseudo-Observation Construction

To fit the model, the authors adapt the two-sample jackknife technique to generate pseudo-observations ( $\tilde{\theta}_{i_1 i_2}$ ).

Fully Observable Case: The pseudo-observation for a pair $(i_1, i_2)$ is derived by removing the $i_1$ -th observation from group 1 and the $i_2$ -th from group 2, recalculating the estimator, and combining these leave-one-out estimates. In the uncensored case, this simplifies to the indicator function $1\{T_{1i_1} > T_{2i_2}\}$ .
Censored Case: The standard estimator $\hat{\theta}$ is replaced by an integral involving Kaplan-Meier survival estimators ( $\hat{S}_1, \hat{S}_2$ ). The pseudo-observations are constructed using the same jackknife logic but based on the Kaplan-Meier estimators derived from censored data.
Tie Correction: A continuity correction is applied for tied event times, adding $0.5 \cdot P(T_1 = T_2)$ to the probability estimate.

C. Parameter Estimation

The regression parameters $\beta$ are estimated by solving a Generalized Estimating Equation (GEE):
$U(\beta) = \sum_{i_1} \sum_{i_2} A_{i_1 i_2} (\tilde{\theta}_{i_1 i_2} - \mu_{i_1 i_2}) = 0$
where $A_{i_1 i_2}$ is the derivative of the link function with respect to $\beta$ , and $\mu_{i_1 i_2}$ is the model-predicted value.

D. Asymptotic Theory and Inference

Consistency and Normality: The authors prove a Central Limit Theorem (CLT) showing that the estimator $\hat{\beta}$ is consistent and asymptotically normal under mild regularity conditions (Assumption 1), even in the presence of right-censoring.
Hypothesis Testing: Since the asymptotic variance structure is complex (involving second derivatives of the functional), the authors propose bootstrap-based hypothesis tests (specifically using the warp-speed bootstrap for efficiency) to infer the significance of covariate effects. Four test statistics are evaluated:
1. Empirical standard deviation.
2. Inter-quartile range (IQR).
3. Median absolute deviation (MAD).
4. Percentile method.

3. Key Contributions

Novel Regression Framework: This is the first paper to propose a regression model for the Mann-Whitney effect that handles randomly right-censored and tied data using a distribution-free approach.
Pseudo-Oobservation Extension: It extends the jackknife pseudo-observation methodology from one-sample problems to the two-sample setting, providing a rigorous theoretical foundation for the asymptotic properties of the estimator.
Flexible Covariate Handling: The model allows for different covariate sets in the two groups, making it suitable for comparing data from different sources (e.g., historical controls vs. new trials).
Interpretability: Unlike Cox models which estimate log-hazard ratios, this method estimates effects on the probability scale ( $P(T_1 > T_2)$ ), offering direct clinical interpretation regarding the likelihood of one treatment outperforming another.
Causal Interpretation: In the special case where covariates are identical ( $z_1=z_2$ ), the model implicitly defines a causal estimand for the probability of treatment benefit.

4. Results

Simulation Study

Setup: Extensive simulations were conducted with Weibull-distributed event times, varying sample sizes, censoring proportions (up to 75%), and covariate dimensions.
Type-I Error: The bootstrap tests based on empirical standard deviation and empirical quantiles controlled the Type-I error rate well (close to the nominal 5%), even under high censoring. Tests based on IQR and MAD were found to be slightly liberal (over-rejecting).
Power: The proposed tests demonstrated satisfactory power, comparable to the z-test from a Cox regression model, provided censoring did not exceed 50%.
Comparison to Cox: The Cox model's z-test was often too liberal in small samples. The proposed method offered a robust alternative without assuming proportional hazards.

Real Data Application (SUCCESS-A Trial)

The method was applied to the SUCCESS-A breast cancer trial (3,754 patients) to analyze Disease-Free Survival (DFS).

Findings: While the original analysis found no overall treatment benefit, the proposed regression identified significant heterogeneity in treatment effects:
- Benefit from Intervention (Gemcitabine): Patients with advanced tumor stages (pT2–pT4), poor differentiation (G3), and triple-negative molecular subtypes showed a higher probability of benefit from the intervention.
- Benefit from Control: Patients with early-stage tumors, hormone receptor-positive (ER+/PR+) status, and non-affected lymph nodes showed a higher probability of benefit from the standard chemotherapy alone.
Comparison to Cox: The proposed model identified different effect directions for certain covariates (e.g., tumor stage) compared to the Cox model. The authors attribute this to the different estimands: the proposed model uses a truncated effect over a finite time horizon ( $\tau = 5.5$ years), whereas the Cox model estimates an untruncated effect assuming proportional hazards.
Clinical Utility: The model successfully classified individual patients into "intervention-benefit" and "control-benefit" groups with 95% confidence intervals, highlighting the potential for personalized treatment decisions.

5. Significance and Conclusion

This paper provides a powerful, distribution-free alternative to the Cox proportional hazards model for analyzing time-to-event data. Its significance lies in:

Relaxing Assumptions: It removes the need for the proportional hazards assumption, which is frequently violated in clinical trials.
Direct Probability Interpretation: It translates complex survival data into intuitive probabilities ( $P(T_1 > T_2)$ ), facilitating better communication with clinicians and patients.
Subgroup Analysis: It offers a unified framework for identifying predictive covariates and patient subgroups that benefit most from specific treatments, addressing the growing need for precision medicine in oncology.
Robustness: The method remains valid even with high rates of censoring and tied event times, common issues in real-world clinical data.

The authors conclude that the pseudo-observation-based regression for the Mann-Whitney effect is a robust, flexible, and interpretable tool for modern survival analysis, particularly when the goal is to estimate conditional probabilities of treatment superiority.