Comparing Two Survival Functions at a Fixed Timepoint

2026-07-30

AI Summary

This study addresses the limitations of conventional two-sample survival function comparison methods based on the Kaplan–Meier estimator and Greenwood’s variance, which often yield invalid confidence intervals or suffer from undercoverage in settings with small sample sizes, heavy censoring, or estimated survival probabilities near 0 or 1. The authors propose two key innovations: first, adapting the single-sample Greenwood variance adjustment to the two-sample Delta method; second, introducing a mid-p correction within the melding framework of the Beta Product Confidence Procedure (BPCP). Extensive Monte Carlo simulations demonstrate that only the BPCP melding approach consistently achieves valid coverage across diverse scenarios, with the mid-p variant exhibiting coverage closest to the nominal level. Implementations of these methods are publicly available in the R package bpcp.

Abstract

For comparing two survival curves at a fixed timepoint with right censoring, a standard method uses asymptotic methods on two Kaplan-Meier estimators with Greenwood variance estimators and transformations using the delta method. Confidence intervals on associated estimands can either breakdown or undercover due to Kaplan-Meier estimates of zero or one, small sample sizes, or heavy censoring. Although others have proposed adjustments to the Greenwood variance for a single sample, these adjustments have not been incorporated into the two-sample application of the delta method and we do that here as well as provide modifications when Kaplan-Meier estimates are zero or one. An alternative non-asymptotic existing method called melding on the beta product confidence procedure (BPCP) appears to have at least nominal coverage regardless of the sample size. Without censoring the problem reduces to comparing two independent binomials and melding on the BPCP gives p-values equivalent to Fisher's exact test p-values and further gives compatible confidence intervals for treatment comparison estimands; however, the melding can be very conservative. In this paper, we expand the melding on the BPCP intervals to include mid-p versions, which are designed to achieve coverage closer to nominal without guaranteeing coverage for all cases. We study these existing methods and our minor modifications of them primarily by numerical methods or simulation. Results suggest only the melding on the BPCP can guarantee coverage in all studied scenarios, and the mid-p version most often has closest to nominal coverage. We provide R functions in the bpcp R package.

Problem

• survival analysis
• right censoring
• Kaplan-Meier estimator
• confidence interval coverage
• two-sample comparison

Innovation

• melding
• beta product confidence procedure
• mid-p
• Greenwood variance
• Kaplan-Meier estimator