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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.

No abstract

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.