OptBand: optimization-based confidence bands for functions to characterize time-to-event distributions

S. Tracy,H. Uno,T. Chen

doi:10.1007/s10985-021-09522-8

Abstract

Classical simultaneous confidence bands for survival functions (i.e., Hall–Wellner, equal precision, and empirical likelihood bands) are derived from transformations of the asymptotic Brownian nature of the Nelson–Aalen or Kaplan–Meier estimators. Due to the properties of Brownian motion, a theoretical derivation of the highest confidence density region cannot be obtained in closed form. Instead, we provide confidence bands derived from a related optimization problem with local time processes. These bands can be applied to the one-sample problem regarding both cumulative hazard and survival functions. In addition, we present a solution to the two-sample problem for testing differences in cumulative hazard functions. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies. The proposed bands are applied to clinical trial data to assess survival times for primary biliary cirrhosis patients treated with D-penicillamine.

Highlights

For time-to-event outcomes in clinical studies, inference for the cumulative distribution and survival functions are of interest
Pointwise confidence bands (CB) are narrower than simultaneous CBs, but the former do not attain the nominal coverage level (1 − α) on the specified interval [tL, tU ]
Two classical simultaneous CBs based on the asymptotics of the KM estimator were originally developed by Hall and Wellner (1984) and Nair (1984); they are popularly referred to as the Hall–Wellner (HW) and equal precision (EP) bands, respectively

Summary

Introduction

For time-to-event outcomes in clinical studies, inference for the cumulative distribution and survival functions are of interest. Hollander et al (1997) derived simultaneous CBs for survival and cumulative hazard functions based on the empirical likelihood (EL) approach for confidence intervals introduced by Thomas and Grunkemeier (1975). This technique has been applied to quantile functions by Li et al (1996), but was not applied to the two-sample problem. We propose a novel procedure to construct analytical simultaneous CBs {L(t), U(t)} for φ(t) which approximately target the HCDR. That is, such bands aim to minimize tU.

Functional approximation of the critical value parameter

One-sample problem

Two-sample problem

Simulation

Primary biliary cirrhosis data analysis

Findings

Discussion