In studies of recurrent events, joint modeling approaches are often needed to allow for potential dependent censoring by a terminal event such as death. Joint frailty models for recurrent events and death with an additional dependence parameter have been studied for cases in which individuals are observed from the start of the event processes. However, the samples are often selected at a later time, which results in delayed entry. Thus, only individuals who have not yet experienced the terminal event will be included in the study. We propose a method for estimating the joint frailty model from such left-truncated data. The frailty distribution among the selected survivors differs from the frailty distribution in the underlying population if the recurrence process and the terminal event are associated. The correctly adjusted marginal likelihood can be expressed as a ratio of two integrals over the frailty distribution, which may be approximated using Gaussian quadrature. The baseline rates are specified as piecewise constant functions, and the covariates are assumed to have multiplicative effects on the event rates. We assess the performance of the estimation procedure in a simulation study, and apply the method to estimate age-specific rates of recurrent urinary tract infections and mortality in an older population.

翻译：在复发事件研究中，通常需要采用联合建模方法来处理由终止事件（如死亡）导致的潜在相依删失。针对个体从事件过程起始时即被观察的情况，学者们已研究了包含额外相依参数的复发事件与死亡联合脆弱模型。然而，样本常常在后续时间点被选择，这导致延迟进入问题。因此，只有尚未经历终止事件的个体才会被纳入研究。我们提出了一种针对此类左截断数据的联合脆弱模型估计方法。若复发过程与终止事件存在关联，幸存个体筛选后的脆弱分布将与潜在总体中的脆弱分布存在差异。正确调整后的边际似然可表示为脆弱分布两个积分之比，该比值可通过高斯求积法进行近似。基线风险率被设定为分段常数函数，且假设协变量对事件发生率具有乘法效应。我们通过模拟研究评估了该估计程序的性能，并将该方法应用于估算老年人群中复发性尿路感染的年龄别发生率及死亡率。