  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.
