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 \href{https://github.com/gozhang/DRM-combined-survival}{DRM-combined-survival}.

翻译：密度比模型（DRM）是一种半参数模型，通过指数倾斜将多个样本的分布与一个非参数定义的参考分布联系起来，其形状差异由有限维参数控制。在观察性研究中收集到多种类型的部分观测（删失/截断）失效时间数据时，可利用DRM对合并数据进行统一分析。本文在DRM框架下扩展了针对删失长度偏倚/截断数据的方法论，并采用经验似然进行统计推断。我们开发了一种EM算法，用于计算基于DRM的模型参数与生存函数的最大经验似然估计量，并通过大量模拟实验评估其性能，涵盖正确模型设定、过度设定及错误设定等多种场景，涉及不同失效时间分布与删失比例。此外，我们通过分析加拿大蒙特利尔地区医院患者从入院到出院的时间数据，验证了所提方法的有效性。实现本方法的R代码已发布于GitHub平台：\\href{https://github.com/gozhang/DRM-combined-survival}{DRM-combined-survival}。