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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Bickel P, Klaassen C, Ritov Y, Wellner JA (1993) Efficient and adaptive estimation for semiparametric models. Johns Kopkins University Press, Baltimore
Chen MH, Ibrahim JG, Sinha D (1999) A new bayesian model for survival data with a survival fraction. J Am Stat Assoc 94:909–919
Chen X, Fan Y, Tsyrennikov V (2006) Efficient estimation of semiparametric multivariate copula models. J Am Stat Assoc 475:1228–1240
Cheng G, Zhou L, Chen X, Huang JZ (2014) Efficient estimation of semiparametric copula models for bivariate survival data. J Multivar Anal 123:330–344
Farrington CP (2000) Residuals for proportional hazards models with interval-censored survival data. Biometrics 56:473–482
Geman A, Hwang CR (1982) Nonparametric maximum likelihood estimation by the method of sieves. Ann Stat 10:401–414
Hu T, Xiang L (2013) Efficient estimation for semiparametric cure models with interval-censored data. J Multivar Anal 121:139–151
Hu T, Xiang L (2016) Partially linear transformation cure models for interval-censored data. Comput Stat Data Anal 93:257–269
Huang J, Zhang Y, Hua L (2008) A least-squares approach to consistent information estimation in semiparametric models. Technical report, Department of Biostatistics, University of Iowa
Jamshidian M (2004) On algorithms for restricted maximum likelihood estimation. Comput Stat Data Anal 45:137–157
Joly P, Commenges D, Letenneur L (1998) A penalized likelihood approach for arbitrarily censored and truncated data: application to age-specific incidence of dementia. Biometrics 54:185–194
Kim JS (2003a) Efficient estimation for the proportional hazards model with left-truncated and “case 1” interval-censored data. Stat Sin 13:519–537
Kim JS (2003b) Maximum likelihood estimation for the proportional hazards model with partly interval-censored data. J Roy Stat Soc B 65:489–502
Lam KF, Xue H (2005) A semiparametric regression cure model with current status data. Biometrika 92:573–586
Liu H, Shen Y (2009) A semiparametric regression cure model for inverval-censored data. J Am Stat Assoc 104:1168–1178
Ma S (2010) Mixed case interval censored data with a cured subgroup. Stat Sin 20:1165–1181
Peng Y, Taylor JMG (2017) Residual-based model diagnosis methods for mixture cure models. Biostatistics 73:495–505
Ramsay JO (1988) Monotone regression splines in action. Stat Sci 3:425–441
Schumaker L (1981) Spline function: basic theory. John Wiley, New York
Shen X (1997) On methods of sieves and penalization. Ann Stat 25:2555–2591
Sun J (1997) The statistical analysis of interval-censored failure time data, vol 2006. Springer, New York
Sy JP, Taylor JMG (2000) Estimation in a cox proportional hazards cure model. Biometrics 56:227–236
Tsodikov A (1998) A proportional hazards model taking account of long-term survivors. Biometrics 54:1508–1516
van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, Cambridge
Wellner JA, Zhang Y (2007) Likelihood-based semiparametric estimation methods for panel count data with covariates. Ann Stat 35:2106–2142
Wu Y, Zhang Y (2012) Partially monotone tensor spline estimation of the joint distribution function with bivariate current status data. Ann Stat 40:1609–1636
Zeng D, Yin G, Ibrahim JG (2006) Semiparametric transformation models for survival data with a cure fraction. J Am Stat Assoc 101:670–684
Zhang Y, Hua L, Huang J (2010) A spline-based semiparametric maximum likelihood estimation for the cox model with interval-censored data. Scand J Stat 37:338–354
The research of Yuan Wu was supported in part by award number P01CA142538 from the National Cancer Institute. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of Health.
Electronic supplementary material
Below is the link to the electronic supplementary material.
About this article
Cite this article
Wu, Y., Chambers, C.D. & Xu, R. Semiparametric sieve maximum likelihood estimation under cure model with partly interval censored and left truncated data for application to spontaneous abortion. Lifetime Data Anal 25, 507–528 (2019). https://doi.org/10.1007/s10985-018-9445-4
- Empirical process
- Generalized gradient projection algorithm
- Spline function