Lifetime Data Analysis

, Volume 25, Issue 3, pp 507–528 | Cite as

Semiparametric sieve maximum likelihood estimation under cure model with partly interval censored and left truncated data for application to spontaneous abortion

  • Yuan WuEmail author
  • Christina D. Chambers
  • Ronghui Xu


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.


Empirical process Generalized gradient projection algorithm Spline function 



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.

Supplementary material

10985_2018_9445_MOESM1_ESM.pdf (302 kb)
Supplementary material 1 (pdf 301 KB)


  1. Bickel P, Klaassen C, Ritov Y, Wellner JA (1993) Efficient and adaptive estimation for semiparametric models. Johns Kopkins University Press, BaltimorezbMATHGoogle Scholar
  2. Chen MH, Ibrahim JG, Sinha D (1999) A new bayesian model for survival data with a survival fraction. J Am Stat Assoc 94:909–919CrossRefzbMATHGoogle Scholar
  3. Chen X, Fan Y, Tsyrennikov V (2006) Efficient estimation of semiparametric multivariate copula models. J Am Stat Assoc 475:1228–1240MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cheng G, Zhou L, Chen X, Huang JZ (2014) Efficient estimation of semiparametric copula models for bivariate survival data. J Multivar Anal 123:330–344MathSciNetCrossRefzbMATHGoogle Scholar
  5. Farrington CP (2000) Residuals for proportional hazards models with interval-censored survival data. Biometrics 56:473–482CrossRefzbMATHGoogle Scholar
  6. Geman A, Hwang CR (1982) Nonparametric maximum likelihood estimation by the method of sieves. Ann Stat 10:401–414MathSciNetCrossRefzbMATHGoogle Scholar
  7. Hu T, Xiang L (2013) Efficient estimation for semiparametric cure models with interval-censored data. J Multivar Anal 121:139–151MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hu T, Xiang L (2016) Partially linear transformation cure models for interval-censored data. Comput Stat Data Anal 93:257–269MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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 IowaGoogle Scholar
  10. Jamshidian M (2004) On algorithms for restricted maximum likelihood estimation. Comput Stat Data Anal 45:137–157MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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 54:185–194CrossRefzbMATHGoogle Scholar
  12. Kim JS (2003a) Efficient estimation for the proportional hazards model with left-truncated and “case 1” interval-censored data. Stat Sin 13:519–537MathSciNetzbMATHGoogle Scholar
  13. Kim JS (2003b) Maximum likelihood estimation for the proportional hazards model with partly interval-censored data. J Roy Stat Soc B 65:489–502MathSciNetCrossRefzbMATHGoogle Scholar
  14. Lam KF, Xue H (2005) A semiparametric regression cure model with current status data. Biometrika 92:573–586MathSciNetCrossRefzbMATHGoogle Scholar
  15. Liu H, Shen Y (2009) A semiparametric regression cure model for inverval-censored data. J Am Stat Assoc 104:1168–1178CrossRefzbMATHGoogle Scholar
  16. Ma S (2010) Mixed case interval censored data with a cured subgroup. Stat Sin 20:1165–1181MathSciNetzbMATHGoogle Scholar
  17. Peng Y, Taylor JMG (2017) Residual-based model diagnosis methods for mixture cure models. Biostatistics 73:495–505MathSciNetzbMATHGoogle Scholar
  18. Ramsay JO (1988) Monotone regression splines in action. Stat Sci 3:425–441CrossRefGoogle Scholar
  19. Schumaker L (1981) Spline function: basic theory. John Wiley, New YorkzbMATHGoogle Scholar
  20. Shen X (1997) On methods of sieves and penalization. Ann Stat 25:2555–2591MathSciNetCrossRefzbMATHGoogle Scholar
  21. Sun J (1997) The statistical analysis of interval-censored failure time data, vol 2006. Springer, New YorkGoogle Scholar
  22. Sy JP, Taylor JMG (2000) Estimation in a cox proportional hazards cure model. Biometrics 56:227–236MathSciNetCrossRefzbMATHGoogle Scholar
  23. Tsodikov A (1998) A proportional hazards model taking account of long-term survivors. Biometrics 54:1508–1516CrossRefzbMATHGoogle Scholar
  24. van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  25. Wellner JA, Zhang Y (2007) Likelihood-based semiparametric estimation methods for panel count data with covariates. Ann Stat 35:2106–2142MathSciNetCrossRefzbMATHGoogle Scholar
  26. Wu Y, Zhang Y (2012) Partially monotone tensor spline estimation of the joint distribution function with bivariate current status data. Ann Stat 40:1609–1636MathSciNetCrossRefzbMATHGoogle Scholar
  27. Zeng D, Yin G, Ibrahim JG (2006) Semiparametric transformation models for survival data with a cure fraction. J Am Stat Assoc 101:670–684MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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 37:338–354MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Biostatistics and BioinformaticsDuke UniversityDurhamUSA
  2. 2.Department of PediatricsUniversity of CaliforniaSan DiegoUSA
  3. 3.Department of Family Medicine and Public HealthUniversity of CaliforniaSan DiegoUSA
  4. 4.Department of MathematicsUniversity of CaliforniaSan DiegoUSA

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