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.
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