# Density ratio model for multiple types of survival data with empirical likelihood

arXiv: https://arxiv.org/abs/2511.09398 · HTML: https://mlsanity.com/paper/2511.09398/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Nov 2025.

## Built on by (most cited first, 1 of 1)

- [Anytime PAC-Bayes for Constrained Density-Ratio Networks under Covariate Shift](https://mlsanity.com/paper/2605.17212/) (2026)

## Abstract (arXiv)

The density ratio model (DRM) is a semiparametric model that relates the distributions from multiple samples to a nonparametrically defined reference distribution via exponential tilting, with finite-dimensional parameters governing their differences in shape. When multiple types of partially observed (censored/truncated) failure time data are collected in an observational study, the DRM can be utilized to conduct a single unified analysis of the combined data. In this paper, we extend the methodology for censored length-biased/truncated data to the DRM framework and formulate the inference using empirical likelihood. We develop an EM algorithm to compute the DRM-based maximum empirical likelihood estimators of the model parameters and survival function, and assess its performance through extensive simulations under correct model specification, overspecification, and misspecification, across a range of failure-time distributions and censoring proportions. We also illustrate the efficacy of our method by analyzing the duration of time spent from admission to discharge in a Montreal-area hospital in Canada. The R code that implements our method is available on GitHub at hrefhttps://github.com/gozhang/DRM-combined-survivalDRM-combined-survival.