# Robust inverse probability weighted estimators for doubly truncated Cox regression with closed-form standard errors

## TL;DR

This paper introduces a new method for analyzing survival data affected by double truncation, offering robust estimators and closed-form standard errors.

## Contribution

The paper proposes robust estimators with time-varying weights and introduces a nonparametric test for verifying truncation assumptions.

## Key findings

- The proposed estimators are robust to extreme event times and allow sensitivity analysis for non-positivity.
- A nonparametric test and graphical diagnostic are developed to assess the quasi-independent truncation assumption.
- Closed-form standard errors are derived for both the proposed estimators and the NPMLE.

## Abstract

Survival data is doubly truncated when only participants who experience an event during a random interval are included in the sample. Existing methods typically correct for double truncation bias in Cox regression through inverse probability weighting via the nonparametric maximum likelihood estimate (NPMLE) of the selection probabilities. This approach relies on two key assumptions, quasi-independent truncation and positivity of the sampling probabilities, yet there are no methods available to thoroughly assess these assumptions in the regression context. Furthermore, these estimators can be particularly sensitive to extreme event times. Finally, current double truncation methods rely on bootstrapping for variance estimation. Aside from the unnecessary computational burden, there are often identifiability issues with the NPMLE during bootstrap resampling. To address these limitations of current methods, we propose a class of robust Cox regression coefficient estimators with time-varying inverse probability weights and extend these estimators to conduct sensitivity analysis regarding possible non-positivity of the sampling probabilities. Also, we develop a nonparametric test and graphical diagnostic for verifying the quasi-independent truncation assumption. Finally, we provide closed-form standard errors for the NPMLE as well as for the proposed estimators. The proposed estimators are evaluated through extensive simulations and illustrated using an AIDS study.

The online version contains supplementary material available at 10.1007/s10985-025-09650-5.