Advertisement

Landmark estimation of transition probabilities in non-Markov multi-state models with covariates

  • Rune HoffEmail author
  • Hein Putter
  • Ingrid Sivesind Mehlum
  • Jon Michael Gran
Article
  • 9 Downloads

Abstract

In non-Markov multi-state models, the traditional Aalen–Johansen (AJ) estimator for state transition probabilities is generally not valid. An alternative, suggested by Putter and Spitioni, is to analyse a subsample of the full data, consisting of the individuals present in a specific state at a given landmark time-point. The AJ estimator of occupation probabilities is then applied to the landmark subsample. Exploiting the result by Datta and Satten, that the AJ estimator is consistent for state occupation probabilities even in non-Markov models given that censoring is independent of state occupancy and times of transition between states, the landmark Aalen–Johansen (LMAJ) estimator provides consistent estimates of transition probabilities. So far, this approach has only been studied for non-parametric estimation without covariates. In this paper, we show how semi-parametric regression models and inverse probability weights can be used in combination with the LMAJ estimator to perform covariate adjusted analyses. The methods are illustrated by a simulation study and an application to population-wide registry data on work, education and health-related absence in Norway. Results using the traditional AJ estimator and the LMAJ estimator are compared, and show large differences in estimated transition probabilities for highly non-Markov multi-state models.

Keywords

Multi-state models The Markov property Transition probabilities Landmarking Aalen–Johansen estimator 

Notes

Acknowledgements

This work was financially supported by the Research Council of Norway (Grant Nos. 237831, 218368 and 273674).

Supplementary material

10985_2019_9474_MOESM1_ESM.pdf (703 kb)
Supplementary material 1 (pdf 702 KB)

References

  1. Aalen OO, Borgan Ø, Fekjær H (2001) Covariate adjustment of event histories estimated from Markov chains: the additive approach. Biometrics 57(4):993–1001MathSciNetzbMATHGoogle Scholar
  2. Aalen OO, Borgan Ø, Gjessing H (2008) Survival and event history analysis: a process point of view. Springer, New YorkzbMATHGoogle Scholar
  3. Allignol A, Beyersmann J, Gerds T, Latouche A (2014) A competing risks approach for nonparametric estimation of transition probabilities in a non-Markov illness-death model. Lifetime Data Anal 20(4):495–513MathSciNetzbMATHGoogle Scholar
  4. Andersen PK, Keiding N (2002) Multi-state models for event history analysis. Stat Methods Med Res 11(2):91–115zbMATHGoogle Scholar
  5. Andersen PK, Pohar Perme M (2008) Inference for outcome probabilities in multi-state models. Lifetime Data Anal 14(4):405–431MathSciNetzbMATHGoogle Scholar
  6. Andersen PK, Pohar Perme M (2013) Multistate models. In: Klein JP, van Houwelingen HC, Ibrahim JG, Scheike TH (eds) Handbook of survival analysis. Chapman & Hall/CRC, Boca Raton, pp 417–439Google Scholar
  7. Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New YorkzbMATHGoogle Scholar
  8. Bender R, Augustin T, Blettner M (2005) Generating survival times to simulate Cox proportional hazards models. Stat Med 24(11):1713–1723MathSciNetGoogle Scholar
  9. Breslow NE (1972) Discussion of Professor Cox’s paper. J R Stat Soc Ser B 34:216–217MathSciNetGoogle Scholar
  10. Cole SR, Hernán MA, Anastos K, Jamieson BD, Robins JM (2007) Determining the effect of highly active antiretroviral therapy on changes in human immunodeficiency virus type 1 RNA viral load using a marginal structural left-censored mean model. Am J Epidemiol 166(2):219–227Google Scholar
  11. Datta S, Satten GA (2001) Validity of the Aalen-Johansen estimators of stage occupation probabilities and Nelson–Aalen estimators of integrated transition hazards for non-Markov models. Stat Probab Lett 55(4):403–411MathSciNetzbMATHGoogle Scholar
  12. Datta S, Satten GA (2002) Estimation of integrated transition hazards and stage occupation probabilities for non-Markov systems under dependent censoring. Biometrics 58(4):792–802MathSciNetzbMATHGoogle Scholar
  13. de Uña-Álvarez J, Meira-Machado L (2015) Nonparametric estimation of transition probabilities in the non-Markov illness-death model: a comparative study. Biometrics 71(2):364–375MathSciNetzbMATHGoogle Scholar
  14. de Wreede LC, Fiocco M, Putter H (2011) mstate: an R package for the analysis of competing risks and multi-state models. J Stat Softw 38(7):1–30Google Scholar
  15. Glidden DV (2002) Robust inference for event probabilities with non-Markov event data. Biometrics 58(2):361–368MathSciNetzbMATHGoogle Scholar
  16. Gran JM, Wasmuth L, Amundsen EJ, Lindqvist BH, Aalen OO (2008) Growth rates in epidemic models: application to a model for HIV/AIDS progression. Stat Med 27(23):4817–4834MathSciNetGoogle Scholar
  17. Gran JM, Lie SA, Øyeflaten I, Borgan Ø, Aalen OO (2015) Causal inference in multi-state models-sickness absence and work for 1145 participants after work rehabilitation. BMC Public Health 15(1):1082Google Scholar
  18. Hernan M, Robins JM (2018) Causal inference. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  19. Hoff R, Corbett K, Mehlum IS, Mohn FA, Kristensen P, Hanvold TN, Gran JM (2018) The impact of completing upper secondary education - a multi-state model for work, education and health in young men. BMC Public Health 18(1):556Google Scholar
  20. Hougaard P (1999) Multi-state models: a review. Lifetime Data Anal 5(3):239–264MathSciNetzbMATHGoogle Scholar
  21. Keiding N, Klein JP, Horowitz MM (2001) Multi-state models and outcome prediction in bone marrow transplantation. Stat Med 20(12):1871–1885Google Scholar
  22. Lin DY, Wei LJ (1989) The robust inference for the Cox proportional hazards model. J Am Stat Assoc 84(408):1074–1078MathSciNetzbMATHGoogle Scholar
  23. Meira-Machado LF, de Uña-Álvarez J, Cadarso-Suárez C, Andersen PK (2008) Multi-state models for the analysis of time-to-event data. Stat Methods Med Res 18(2):195–222MathSciNetGoogle Scholar
  24. Putter H, Spitoni C (2018) Non-parametric estimation of transition probabilities in non-Markov multi-state models: the landmark Aalen–Johansen estimator. Stat Methods Med Res 27(7):2081–2092MathSciNetGoogle Scholar
  25. Putter H, Fiocco M, Geskus RB (2007) Tutorial in biostatistics: competing risks and multi-state models. Stat Med 26(11):2389–2430MathSciNetGoogle Scholar
  26. Robins JM, Hernan MA, Brumback B (2000) Marginal structural models and causal inference in epidemiology. Epidemiology 11(5):550–560Google Scholar
  27. Rosenbaum PR (1987) Model-based direct adjustment. J Am Stat Assoc 82(398):387–394zbMATHGoogle Scholar
  28. Røysland K (2011) A martingale approach to continuous-time marginal structural models. Bernoulli 17(3):895–915MathSciNetzbMATHGoogle Scholar
  29. Scheike TH (2002) The additive nonparametric and semiparametric Aalen model as the rate function for a counting process. Lifetime Data Anal 8(3):247–262MathSciNetzbMATHGoogle Scholar
  30. Sundet JM, Barlaug DG, Torjussen TM (2004) The end of the Flynn effect? A study of secular trends in mean intelligence test scores of Norwegian conscripts during half a century. Intelligence 32(4):349–362Google Scholar
  31. Titman AC (2015) Transition probability estimates for non-Markov multi-state models. Biometrics 71(4):1034–1041MathSciNetzbMATHGoogle Scholar
  32. Wechsler D (1955) Manual for the Wechsler adult intelligence scale. Psychological Corp., Oxford, EnglandGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Oslo Centre for Biostatistics and EpidemiologyUniversity of Oslo and Oslo University HospitalOsloNorway
  2. 2.National Institute of Occupational HealthOsloNorway
  3. 3.Department of Medical Statistics and BioinformaticsLeiden University Medical CenterLeidenThe Netherlands

Personalised recommendations