Left-truncated data are often encountered in epidemiological cohort studies, where individuals are recruited according to a certain cross-sectional sampling criterion. Length-biased data, a special case of left-truncated data, assume that the incidence of the initial event follows a homogeneous Poisson process. In this article, we consider an analysis of length-biased and interval-censored data with a nonsusceptible fraction. We first point out the importance of a well-defined target population, which depends on the prior knowledge for the support of the failure times of susceptible individuals. Given the target population, we proceed with a length-biased sampling and draw valid inferences from a length-biased sample. When there is no covariate, we show that it suffices to consider a discrete version of the survival function for the susceptible individuals with jump points at the left endpoints of the censoring intervals when maximizing the full likelihood function, and propose an EM algorithm to obtain the nonparametric maximum likelihood estimates of nonsusceptible rate and the survival function of the susceptible individuals. We also develop a novel graphical method for assessing the stationarity assumption. When covariates are present, we consider the Cox proportional hazards model for the survival time of the susceptible individuals and the logistic regression model for the probability of being susceptible. We construct the full likelihood function and obtain the nonparametric maximum likelihood estimates of the regression parameters by employing the EM algorithm. The large sample properties of the estimates are established. The performance of the method is assessed by simulations. The proposed model and method are applied to data from an early-onset diabetes mellitus study.

在流行病学队列研究中经常遇到左截断数据，其中个人是根据一定的横断面抽样标准招募的。偏长数据是左截断数据的一种特殊情况，假设初始事件的发生率遵循齐次泊松过程。在本文中，我们考虑对具有不敏感分数的长度偏倚和区间删失数据进行分析。我们首先指出明确目标人群的重要性，这取决于支持易感个体故障时间的先验知识。给定目标人群，我们继续进行有偏长度的抽样，并从有偏长度的样本中得出有效的推论。当没有协变量时，我们表明，在最大化全似然函数时，考虑易感个体的生存函数的离散版本就足够了，跳跃点位于审查区间的左端点，并提出了一种 EM 算法来获得非易感个体的非参数最大似然估计易感个体的生存率和生存功能。我们还开发了一种新的图形方法来评估平稳性假设。当存在协变量时，我们将 Cox 比例风险模型用于易感个体的生存时间，将逻辑回归模型用于易感个体的概率。我们构建了全似然函数，并通过采用EM算法获得了回归参数的非参数最大似然估计。估计的大样本特性已建立。通过模拟评估该方法的性能。所提出的模型和方法适用于早发性糖尿病研究的数据。