A semi-parametric cox's regression model for zero-inflated left-censored time to event data

In some clinical, environmental, or economical studies, researchers are interested in a semi-continuous outcome variable which takes the value zero with a discrete probability and has a continuous distribution for the non-zero values. Due to the measuring mechanism, it is not always possible to fully observe some outcomes, and only an upper bound is recorded. We call this left-censored data and observe only the maximum of the outcome and an independent censoring variable, together with an indicator. In this article, we introduce a mixture semi-parametric regression model. We consider a parametric model to investigate the influence of covariates on the discrete probability of the value zero. For the non-zero part of the outcome, a semi-parametric Cox's regression model is used to study the conditional hazard function. The different parameters in this mixture model are estimated using a likelihood method. Hereby the infinite dimensional baseline hazard function is estimated by a step function. As results, we show the identifiability and the consistency of the estimators for the different parameters in the model. We study the finite sample behaviour of the estimators through a simulation study and illustrate this model on a practical data example.