  The proportional hazards (PH) model is arguably one of the most popular
models used to analyze time to event data arising from clinical trials and
longitudinal studies, among many others. In many such studies, the event time
of interest is not directly observed but is known relative to periodic
examination times; i.e., practitioners observe either current status or
interval-censored data. The analysis of data of this structure is often fraught
with many difficulties. Further exacerbating this issue, in some such studies
the observed data also consists of instantaneous failures; i.e., the event
times for several study units coincide exactly with the time at which the study
begins. In light of these difficulties, this work focuses on developing a
mixture model, under the PH assumptions, which can be used to analyze
interval-censored data subject to instantaneous failures. To allow for modeling
flexibility, two methods of estimating the unknown cumulative baseline hazard
function are proposed; a fully parametric and a monotone spline representation
are considered. Through a novel data augmentation procedure involving latent
Poisson random variables, an expectation-maximization (EM) algorithm was
developed to complete model fitting. The resulting EM algorithm is easy to
implement and is computationally efficient. Moreover, through extensive
simulation studies the proposed approach is shown to provide both reliable
estimation and inference.
