Lifetime Data Analysis

, Volume 21, Issue 4, pp 594–625 | Cite as

The current duration design for estimating the time to pregnancy distribution: a nonparametric Bayesian perspective

  • Dario Gasbarra
  • Elja Arjas
  • Aki Vehtari
  • Rémy Slama
  • Niels Keiding


This paper was inspired by the studies of Niels Keiding and co-authors on estimating the waiting time-to-pregnancy (TTP) distribution, and in particular on using the current duration design in that context. In this design, a cross-sectional sample of women is collected from those who are currently attempting to become pregnant, and then by recording from each the time she has been attempting. Our aim here is to study the identifiability and the estimation of the waiting time distribution on the basis of current duration data. The main difficulty in this stems from the fact that very short waiting times are only rarely selected into the sample of current durations, and this renders their estimation unstable. We introduce here a Bayesian method for this estimation problem, prove its asymptotic consistency, and compare the method to some variants of the non-parametric maximum likelihood estimators, which have been used previously in this context. The properties of the Bayesian estimation method are studied also empirically, using both simulated data and TTP data on current durations collected by Slama et al. (Hum Reprod 27(5):1489–1498, 2012).


McMC Posterior consistency Data augmentation Logistic process prior Generalized gamma convolution process 


  1. Arjas E, Gasbarra D (1994) Nonparametric Bayesian inference from right censored survival data, using the Gibbs sampler. Stat Sin 4:505–524MATHMathSciNetGoogle Scholar
  2. Baird DD, Wilcox AJ, Weinberg CR (1986) Use of time to pregnancy to study environmental exposures. Am J Epidemiol 124(3):470–480Google Scholar
  3. Brillinger DR (1986) The natural variability of vital rates and associated statistics. Biometrics 42(4):693–734MATHMathSciNetCrossRefGoogle Scholar
  4. Cox DR, Miller HD (1965) The theory of stochastic processes. Chapman, LondonMATHGoogle Scholar
  5. Diaconis P, Freedman D (1986) On the consistency of Bayes estimates (with discussion). Ann Stat 14:1–67MATHMathSciNetCrossRefGoogle Scholar
  6. Denby L, Vardi Y (1986) The survival curve with decreasing density. Technometrics 28(4):359–367MATHMathSciNetCrossRefGoogle Scholar
  7. Ghosh JK, Ramamoorthi RV (2003) Bayesian nonparametrics. Springer, New YorkMATHGoogle Scholar
  8. Grenander U (1956) On the theory of mortality measurement II. Skand Aktuarietidskr 39:125–153MathSciNetGoogle Scholar
  9. Hansen MB, Lauritzen SL (2002) Nonparametric Bayes inference for concave distribution functions. Stat Neerl 56(1):110–127MATHMathSciNetCrossRefGoogle Scholar
  10. Hjort NL (1990) Nonparametric Bayes estimators based on beta processes in models for life history data. Ann Stat 18(3):1259–1294MATHMathSciNetCrossRefGoogle Scholar
  11. Hjort NL, Holmes C, Müller P, Anderson MD, Walker SG (eds) (2010) Bayesian nonparametrics. Cambridge University Press, CambridgeMATHGoogle Scholar
  12. Ishikawa Y (2013) Stochastic calculus of variations for jump processes. De Gruyter, BerlinMATHCrossRefGoogle Scholar
  13. James LF, Roynette B, Yor M (2008) Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab Surv 5:346–415MATHMathSciNetCrossRefGoogle Scholar
  14. Kallenberg O (2002) Foundations of modern probability, 2nd edn. Springer, New YorkMATHCrossRefGoogle Scholar
  15. Keiding N (1991) Age-specific incidence and prevalence: a statistical perspective (with discussion). J R Stat Soc A 154:371–412MATHMathSciNetCrossRefGoogle Scholar
  16. Keiding N, Kvist K, Hartvig H, Tvede M, Juul S (2002) Estimating time to pregnancy from current durations in a cross-sectional sample. Biostatistics 3:565–578MATHCrossRefGoogle Scholar
  17. Keiding N, Hansen OHH, Srensen DN, Slama R (2012) The current duration approach to estimating time to pregnancy. Scand J Stat 39:185–204MATHMathSciNetCrossRefGoogle Scholar
  18. Lenk PJ (1988) The logistic normal distribution for Bayesian nonparametric predictive densities. JASA 83(402):509–516MATHMathSciNetCrossRefGoogle Scholar
  19. Lenk PJ (1991) Towards practicable Bayesian nonparametric density estimator. Biometrika 78(3):531–543MATHMathSciNetCrossRefGoogle Scholar
  20. McLaughlin KA, Green JG, Gruber MJ, Sampson NA, Zaslavsky AM, Kessler RC (2010) Childhood adversities and adult psychiatric disorders in the National Comorbidity Survey Replication II. Arch Gen Psychiatry 67:124–132CrossRefGoogle Scholar
  21. Phadia EG (2013) Prior processes and their applications, nonparametric Bayesian estimation. Springer, New YorkMATHCrossRefGoogle Scholar
  22. Robert C, Casella G (2004) Monte Carlo statistical methods, 2nd edn. Springer, New YorkMATHCrossRefGoogle Scholar
  23. Schwartz L (1965) On Bayes procedures. Z Wahrsch Verw Gebiete 4:10–26MATHCrossRefGoogle Scholar
  24. Sethuraman J (1994) A constructive definition of the Dirichlet process prior. Stat Sin 2:639–665MathSciNetGoogle Scholar
  25. Simon Th (2000) Support theorem for jump processes. Stoch Process Appl 89(1):1–30MATHCrossRefGoogle Scholar
  26. Slama R, Hansen OK, Ducot B, Bohet A, Sorensen D, Giorgis Allemand L, Eijkemans MJ, Rosetta L, Thalabard JC, Keiding N, Bouyer J (2012) Estimation of the frequency of involuntary infertility on a nation-wide basis. Hum Reprod 27(5):1489–1498CrossRefGoogle Scholar
  27. Tokdar ST, Ghosh JK (2007) Posterior consistency of logistic Gaussian process priors in density estimation. J Stat Plan Inference 137:34–42MATHMathSciNetCrossRefGoogle Scholar
  28. Van Es B, Klaassen CAJ, Oudshoorn K (2000) Survival analysis under cross sectional sampling: length bias and multiplicative censoring. J Stat Plan Inference 91:295–312MATHCrossRefGoogle Scholar
  29. Walker SG, Hjort NL (2001) On Bayesian consistency. J R Stat Soc Ser B 63:811–821MATHMathSciNetCrossRefGoogle Scholar
  30. Walker SG (2004) New approaches to Bayesian consistency. Ann Stat 32(5):2028–2043MATHCrossRefGoogle Scholar
  31. Walker SG, Lijoi A, Prünster I (2005) Data tracking and the understanding of Bayesian consistency. Biometrika 92(4):765–778MATHMathSciNetCrossRefGoogle Scholar
  32. Weinberg CR, Gladen BC (1986) The beta-geometric distribution applied to comparative fecundability studies. Biometrics 42:547–560CrossRefGoogle Scholar
  33. Williamson RE (1956) Multiply monotone functions and their Laplace transforms. Duke Math J 23:189–207MATHMathSciNetCrossRefGoogle Scholar
  34. Woodroofe M, Sun J (1993) A penalized maximum likelihood estimate of \(f(0+)\) when \(f\) is non-increasing. Stat Sin 3:501–515MATHMathSciNetGoogle Scholar
  35. Yamaguchi K (2003) Accelerated failure-time mover-stayer regression models for the analysis of last-episode data. Sociol Methodol 33:81–110CrossRefGoogle Scholar
  36. Zelen M (2004) Forward and backward recurrence times and length biased sampling: age specific models. Lifetime Data Anal 10(4):325–334MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Dario Gasbarra
    • 1
  • Elja Arjas
    • 1
  • Aki Vehtari
    • 2
  • Rémy Slama
    • 3
  • Niels Keiding
    • 4
  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  2. 2.Department of Computer ScienceAalto UniversityEspooFinland
  3. 3.French Institute of Health and Medical Research, Team of Environmental Epidemiology applied to Reproduction and Respiratory HealthInserm-Univ. Grenoble AlpesGrenobleFrance
  4. 4.Department of Public HealthUniversity of CopenhagenCopenhagenDenmark

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