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, samples are often selected at a later time, which results in delayed entry so that only individuals who have not yet experienced the terminal event will be included. In joint frailty models such left truncation has effects on the frailty distribution that need to be accounted for in both the recurrence process and the terminal event process, if the two are associated. We demonstrate, in a comprehensive simulation study, the effects that not adjusting for late entry can have and derive the correctly adjusted marginal likelihood, which can be expressed as a ratio of two integrals over the frailty distribution. We extend the estimation method of Liu and Huang (Stat Med 27:2665–2683, 2008. https://doi.org/10.1002/sim.3077) to include potential left truncation. Numerical integration is performed by Gaussian quadrature, the baseline intensities are specified as piecewise constant functions, potential covariates are assumed to have multiplicative effects on the intensities. We apply the method to estimate age-specific intensities of recurrent urinary tract infections and mortality in an older population.

在复发事件的研究中，通常需要联合建模方法，以允许通过死亡等终端事件进行潜在的依赖审查。对于从事件过程开始就观察个体的情况，已经研究了具有附加依赖参数的复发事件和死亡的联合脆弱模型。然而，样本通常是在较晚的时间选择的，这会导致延迟进入，因此只会包括尚未经历过终端事件的个人。在联合脆弱性模型中，这种左截断对脆弱性分布有影响，如果两者相关联，则需要在复发过程和终端事件过程中加以考虑。我们在综合模拟研究中证明，不调整延迟进入的影响可以产生并推导出正确调整的边际可能性，它可以表示为两个积分与脆弱分布的比率。我们扩展了 Liu 和 Huang (Stat Med 27:2665–2683, 2008. https://doi.org/10.1002/sim.3077) 的估计方法以包括潜在的左截断。数值积分由高斯求积法执行，基线强度指定为分段常数函数，假定潜在协变量对强度具有乘法效应。我们应用该方法来估计老年人群中复发性尿路感染的特定年龄强度和死亡率。我们扩展了 Liu 和 Huang (Stat Med 27:2665–2683, 2008. https://doi.org/10.1002/sim.3077) 的估计方法以包括潜在的左截断。数值积分由高斯求积法执行，基线强度指定为分段常数函数，假定潜在协变量对强度具有乘法效应。我们应用该方法来估计老年人群中复发性尿路感染的特定年龄强度和死亡率。我们扩展了 Liu 和 Huang (Stat Med 27:2665–2683, 2008. https://doi.org/10.1002/sim.3077) 的估计方法以包括潜在的左截断。数值积分由高斯求积法执行，基线强度指定为分段常数函数，假定潜在协变量对强度具有乘法效应。我们应用该方法来估计老年人群中复发性尿路感染的特定年龄强度和死亡率。