# Pseudo‐Observation Approach for Length‐Biased Cox Proportional Hazards Model

## TL;DR

This paper introduces a new method using pseudo-observations to estimate survival data in length-biased scenarios, improving accuracy and confidence intervals.

## Contribution

The paper proposes and evaluates pseudo-observation approaches for Cox models under length-biased right-censored data, showing improved confidence intervals.

## Key findings

- The pseudo-observation methods perform comparably to standard methods in terms of standard error.
- Proposed methods provide confidence intervals closer to the nominal level in large samples.
- One estimator is proven to be consistent and asymptotically normal.

## Abstract

Pseudo‐observations are used to estimate the expectation of a function of interest in a population when survival data are incomplete due to censoring or truncation. Length‐biased sampling is a special case of a left‐truncation model, in which the truncation variable follows a uniform distribution. This phenomenon is commonly encountered in various fields such as survival analysis and epidemiology, where the event of interest is related to the length or duration of an underlying process. In such settings, the probability of observing a data point is higher for longer lengths, leading to biased sampling. The goal of this paper is to apply pseudo‐observations to estimate the regression coefficients in the Cox proportional hazards model under length‐biased right‐censored (LBRC) data. We assess the accuracy and efficiency of two approaches that differ in their generation of pseudo‐observations, comparing them with two prominent standard methods in the presence of LBRC data. The results demonstrate that the two proposed pseudo‐observation methods are comparable to the standard methods in terms of standard error, with advantages in providing confidence intervals that are closer to the nominal level in large sample sizes and specific scenarios. Additionally, although length‐biased data are a special case of left‐truncated data, they must be addressed separately by utilizing the information that the left‐truncation variable follows a uniform distribution, as the simulation results show. We also establish the consistency and asymptotic normality of one of the proposed estimators. Finally, we applied the method to analyze a real dataset from LBRC.