Semiparametric linear regression with an interval-censored covariate in the atherosclerosis risk in communities study

In longitudinal studies, investigators are often interested in understanding how the time since the occurrence of an intermediate event affects a future outcome. The intermediate event is often asymptomatic such that its occurrence is only known to lie in a time interval induced by periodic examinations. We propose a linear regression model that relates the time since the occurrence of the intermediate event to a continuous response at a future time point through a rectified linear unit activation function while formulating the distribution of the time to the occurrence of the intermediate event through the Cox proportional hazards model. We consider nonparametric maximum likelihood estimation with an arbitrary sequence of examination times for each subject. We present an EM algorithm that converges stably for arbitrary datasets. The resulting estimators of regression parameters are consistent, asymptotically normal, and asymptotically efficient. We assess the performance of the proposed methods through extensive simulation studies and provide an application to the Atherosclerosis Risk in Communities Study.