Pseudo and conditional score approach to joint analysis of current count and current status data

We develop a joint analysis approach for recurrent and nonrecurrent event processes subject to case I interval censorship, which are also known in literature as current count and current status data, respectively. We use a shared frailty to link the recurrent and nonrecurrent event processes, while leaving the distribution of the frailty fully unspecified. Conditional on the frailty, the recurrent event is assumed to follow a nonhomogeneous Poisson process, and the mean function of the recurrent event and the survival function of the nonrecurrent event are assumed to follow some general form of semiparametric transformation models. Estimation of the models is based on the pseudo‐likelihood and the conditional score techniques. The resulting estimators for the regression parameters and the unspecified baseline functions are shown to be consistent with rates of square and cubic roots of the sample size, respectively. Asymptotic normality with closed‐form asymptotic variance is derived for the estimator of the regression parameters. We apply the proposed method to a fracture‐osteoporosis survey data to identify risk factors jointly for fracture and osteoporosis in elders, while accounting for association between the two events within a subject.