  This work was motivated by observational studies in pregnancy with
spontaneous abortion (SAB) as outcome. Clearly some women experience the SAB
event but the rest do not. In addition, the data are left truncated due to the
way pregnant women are recruited into these studies. For those women who do
experience SAB, their exact event times are sometimes unknown. Finally, a small
percentage of the women are lost to follow-up during their pregnancy. All these
give rise to data that are left truncated, partly interval and right-censored,
and with a clearly defined cured portion. We consider the non-mixture Cox
regression cure rate model and adopt the semiparametric spline-based sieve
maximum likelihood approach to analyze such data. Using modern empirical
process theory we show that both the parametric and the nonparametric parts of
the sieve estimator are consistent, and we establish the asymptotic normality
for both parts. Simulation studies are conducted to establish the finite sample
performance. Finally, we apply our method to a database of observational
studies on spontaneous abortion.
