Fitting Cox models with doubly censored data using spline-based marginal likelihood

In some applications, the variable of interest is the time from a first event (starting point) to a second event, while both times are interval censored.We propose fitting Cox proportional hazards model to this type of data using a spline-based sieve maximum marginal likelihood, where the time to first event is integrated out in the empirical likelihood function of the time of interest. This greatly reduces the complexity of the likelihood function compared with the full semiparametric likelihood. The dependence of the time of interest on time to the first event (origin)is induced by including time to the first event as a covariate in the Cox model for the time of interest. Asymptotic theory for the estimator is established and a simulation study is conducted to assess its finite performance. It is also applied to the analysis of a data set in HIV infection.