Incorporating delayed entry into the joint frailty model for recurrent events and a terminal event.

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.

Information measures and design issues in the study of mortality deceleration: findings for the gamma-Gompertz model.

Mortality deceleration, or the slowing down of death rates at old ages, has been repeatedly investigated, but empirical studies of this phenomenon have produced mixed results. The scarcity of observations at the oldest ages complicates the statistical assessment of mortality deceleration, even in the parsimonious parametric framework of the gamma-Gompertz model considered here. The need for thorough verification of the ages at death can further limit the available data. As logistical constraints may only allow to validate survivors beyond a certain (high) age, samples may be restricted to a certain age range. If we can quantify the effects of the sample size and the age range on the assessment of mortality deceleration, we can make recommendations for study design. For that purpose, we propose applying the concept of the Fisher information and ideas from the theory of optimal design. We compute the Fisher information matrix in the gamma-Gompertz model, and derive information measures for comparing the performance of different study designs. We then discuss interpretations of these measures. The special case in which the frailty variance takes the value of zero and lies on the boundary of the parameter space is given particular attention. The changes in information related to varying sample sizes or age ranges are investigated for specific scenarios. The Fisher information also allows us to study the power of a likelihood ratio test to detect mortality deceleration depending on the study design. We illustrate these methods with a study of mortality among late nineteenth-century French-Canadian birth cohorts.

Joint modeling of interval counts of recurrent events and death.

When a recurrent event process is ended by death, this may imply dependent censoring if the two processes are associated. Such dependent censoring would have to be modeled to obtain a valid inference. Moreover, the dependence between the recurrence process and the terminal event may be the primary topic of interest. Joint frailty models for recurrent events and death, which include a separate dependence parameter, have been proposed for exactly observed recurrence times. However, in many situations, only the number of events experienced during consecutive time intervals are available. We propose a method for estimating a joint frailty model based on such interval counts and observed or independently censored terminal events. The baseline rates of the two processes are modeled by piecewise constant functions, and Gaussian quadrature is used to approximate the marginal likelihood. Covariates can be included in a proportional rates setting. The observation intervals for the recurrent event counts can differ between individuals. Furthermore, we adapt a score test for the association between recurrent events and death to the setting in which only individual interval counts are observed. We study the performance of both approaches via simulation studies, and exemplify the methodology in a biodemographic study of the dependence between budding rates and mortality in the species Eleutheria dichotoma.