[Paper Review] Inference for the Mann-Whitney Effect for Right-Censored and Tied Data

Dennis Dobler, Markus Pauly|arXiv (Cornell University)|May 16, 2016

Statistical Methods and Inference52 references3 citations

TL;DR

This paper develops and compares bootstrap- and permutation-based inference methods for the Mann-Whitney effect in two-sample right-censored survival data with tied event times. It proposes normalized Kaplan-Meier estimators for the nonparametric effect measure $ p = P(T_1 > T_2) + \frac{1}{2}P(T_1 = T_2) $, showing that the permutation procedure is asymptotically robust and finitely exact under exchangeability, outperforming standard normal and bootstrap approaches in small samples.

ABSTRACT

The Mann-Whitney effect is an intuitive measure for discriminating two survival distributions. Here we analyze various inference techniques for this parameter in a two-sample survival setting with independent right-censoring, where the survival times are even allowed to be discretely distributed. This allows for ties in the data and requires the introduction of normalized versions of Kaplan-Meier estimators from which adequate point estimates are deduced. From an asymptotic analysis of the latter, asymptotically exact inference procedures based on standard normal, bootstrap- and permutation-quantiles are developed and compared in simulations. Here, the asymptotically robust and, in case of equal survival and censoring distributions, even finitely exact permutation procedure turned out to be the best. Finally, all procedures are illustrated using a real data set.

Motivation & Objective

• To develop valid statistical inference for the Mann-Whitney effect in two-sample survival data with independent right-censoring and tied event times.
• To address the limitation of existing methods that exclude ties or assume continuous distributions.
• To compare asymptotic, bootstrap, and permutation-based inference procedures for the Mann-Whitney effect $ p $ and its odds $ w = p/(1-p) $.
• To establish the finite-sample validity and robustness of permutation-based inference under exchangeability.
• To provide confidence intervals and hypothesis tests for $ H_0^p: p = 1/2 $, extending the nonparametric Behrens-Fisher problem to censored data.

Proposed method

• Uses normalized versions of the Kaplan-Meier estimator to estimate the Mann-Whitney effect $ p $, accounting for ties and right-censoring.
• Applies the functional $ \delta $-method to derive asymptotic distributions of the estimators under regularity conditions.
• Employs Efron’s bootstrap and permutation resampling to improve small-sample performance of test statistics.
• Derives asymptotic distributions of bootstrap and permutation estimators using Donsker-type theorems for empirical processes.
• Uses the Wilcoxon-type statistic $ \phi $ as a basis for the test, with its asymptotic variance estimated via bootstrap and permutation variance estimators.
• Establishes finite exactness of the permutation procedure under exchangeable data using Hadamard differentiability and weak convergence of permutation empirical processes.

Experimental results

Research questions

• RQ1 How can the Mann-Whitney effect be consistently estimated in the presence of right-censoring and tied survival times?
• RQ2 Which inference procedure—standard normal, bootstrap, or permutation—provides the most accurate type I error control in small samples with censored and tied data?
• RQ3 Is the permutation-based test asymptotically robust and finitely exact under exchangeability in the censored, tied data setting?
• RQ4 How do the confidence intervals for $ p $ and $ w $ behave under various censoring and tie patterns?
• RQ5 Can the proposed methods be extended to test the nonparametric Behrens-Fisher hypothesis $ H_0^p: p = 1/2 $ in a nonparametric, non-continuous survival model?

Key findings

• The permutation-based inference procedure is asymptotically robust and finitely exact under exchangeability, outperforming standard normal and bootstrap methods in small samples.
• The normalized Kaplan-Meier estimator provides consistent and asymptotically normal estimates of the Mann-Whitney effect $ p $, even under discrete and censored data.
• Bootstrap and permutation variance estimators are consistent under regularity conditions, with the permutation variance estimator achieving finite-sample exactness under exchangeability.
• The permutation test maintains correct size control under the null hypothesis $ H_0^p: p = 1/2 $, even with heavy censoring and ties.
• The bootstrap-based inference shows improved performance over asymptotic normal approximation, especially in small samples with heteroscedasticity.
• The proposed methods are applicable to real-world survival data, as demonstrated on a real dataset, and are implemented via an R package for practical use.