Advertisement

Modelling Times Between Events with a Cured Fraction Using a First Hitting Time Regression Model with Individual Random Effects

  • S. MalefakiEmail author
  • P. Economou
  • C. Caroni
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 136)

Abstract

The empirical survival function of time-to-event data very often appears not to tend to zero. Thus there are long-term survivors, or a “cured fraction” of units which will apparently never experience the event of interest. This feature of the data can be incorporated into lifetime models in various ways, for example, by using mixture distributions to construct a more complex model. Alternatively, first hitting time (FHT) models can be used. One of the most attractive properties of a FHT model for lifetimes based on a latent Wiener process is that long-term survivors appear naturally—corresponding to failure of the process to reach the absorbing boundary—without the need to introduce special components to describe the phenomenon. FHT models have been extended recently in order to incorporate individual random effects into their drift and starting level parameters and also to be applicable in situations with recurrent events on the same unit with possible right censoring of the last stage. These models are extended here to allow censoring to occur at every intermediate stage. Issues of model selection are also considered. Finally, the proposed FHT regression model is fitted to a dataset consisting of the times of repeated applications for treatment made by drug users.

Keywords

Recurrent events Wiener process Long-term survivors 

References

  1. 1.
    Aalen, O.O., Gjessing, H.K.: Understanding the shape of the hazard rate: a process point of view. Stat. Sci. 16, 1–22 (2001)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Ando, T.: Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models. Biometrika 94, 443–458 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ando, T.: Predictive Bayesian model selection. Am. J. Math. Manag. Sci. 31, 13–38 (2011)Google Scholar
  4. 4.
    Balka, J., Desmond, A.F., McNicholas, P.D.: Review and implementation of cure models based on first hitting times for Wiener processes. Lifetime Data Anal. 15, 147–176 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Barr, D.R., Sherrill, E.T.: Mean and variance of truncated normal distributions. Am. Stat. 53, 357–361 (1999)Google Scholar
  6. 6.
    Berkson, J., Gage, R.P.: Survival curves for cancer patients following treatment. J. Am. Stat. Assoc. 47, 501–515 (1952)CrossRefGoogle Scholar
  7. 7.
    Boag, J.W.: Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J. R. Stat. Soc., Ser. B 11, 15–34 (1949)zbMATHGoogle Scholar
  8. 8.
    Caroni, C., Economou, P.: A hidden competing risk model for censored observations. Braz. J. Probab. Stat. 28, 333–352 (2014)Google Scholar
  9. 9.
    Cox, D.R., Miller, H.D.: The Theory of Stochastic Processes. Chapman and Hall, New York (1965)zbMATHGoogle Scholar
  10. 10.
    Economou, P., Malefaki, S., Caroni, C.: A threshold regression model with random effects for recurrent events. Meth. Comput. Appl. Prob. (2015)Google Scholar
  11. 11.
    Griffiths, W.: A Gibbs’ sampler for the parameters of a truncated multivariate normal distribution. Working Paper Series, 856. Department of Economics, The University of Melbourne, Melbourne (2002)Google Scholar
  12. 12.
    Kaplan, E.L., Meier, P.: Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc. 53, 457–481 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lawless, J.F.: Statistical Models and Methods for Lifetime Data, 2nd edn. Wiley, New York (2003)zbMATHGoogle Scholar
  14. 14.
    Lee, M.-L.T., Garshick, E., Whitmore, G.A., Laden, F., Hart, J.: Assessing lung cancer risk to rail workers using a first hitting time regression model. Environmetrics 15, 501–512 (2004)CrossRefGoogle Scholar
  15. 15.
    Lee, M.-L.T., Whitmore, G.A.: First Hitting Time Models for Lifetime Data. In: Balakrishnan, N., Rao, C.R. (eds.) Handbook of Statistics, vol. 23, pp. 537–543. Elsevier, Amsterdam (2004)Google Scholar
  16. 16.
    Lee, M.-L.T., Whitmore, G.A.: Threshold regression for survival analysis: modeling event times by a stochastic process reaching a boundary. Stat. Sci. 21, 501–513 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Linhart, H., Zucchini, W.: Model Selection. Wiley, New York (1986)zbMATHGoogle Scholar
  18. 18.
    Maller, R.A., Zhou, X.: Survival Analysis with Long-term Survivors. Wiley, Chichester (1992)Google Scholar
  19. 19.
    Pennell, M.L., Whitmore, G.A., Lee, M.-L.T.: Bayesian random-effects threshold regression with application to survival data with nonproportional hazards. Biostatistics 11, 111–126 (2010)CrossRefGoogle Scholar
  20. 20.
    Raftery, A.E.: Bayesian model selection in social research (with discussion by Andrew Gelman, Donald B. Rubin and Robert M. Hauser). In: Marsden, P.V. (ed.) Sociological Methodology, pp. 111–196. Blackwell, Oxford (1995)Google Scholar
  21. 21.
    Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A.: Bayesian measures of model complexity and fit (with discussion). J. R Stat. Soc., Ser. B 64, 583–639 (2002)CrossRefzbMATHGoogle Scholar
  22. 22.
    Stogiannis, D., Caroni, C.: Issues in fitting inverse Gaussian first hitting time regression models for lifetime data. Commun. Stat. - Simul. Comput. 42, 1948–1960 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Stogiannis, D., Caroni, C., Anagnostopoulos, C.E., Toumpoulis, I.K.: Comparing first hitting time and proportional hazards regression models. J. Appl. Stat. 38, 1483–1492 (2011)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Yakovlev, A.Y., Tsodikov, A.D., Bass, L.: A stochastic model of hormesis. Math. Biosci. 116, 197–219 (1993)CrossRefzbMATHGoogle Scholar
  25. 25.
    Whitmore, G.A.: An inverse Gaussian model for labour turnover. J. R Stat. Soc., Ser. A 142, 468–478 (1979)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mechanical Engineering and AeronauticsUniversity of PatrasRion-PatrasGreece
  2. 2.Department of Civil EngineeringUniversity of PatrasRion-PatrasGreece
  3. 3.Department of Mathematics, School of Applied Mathematical and Physical SciencesNational Technical University of AthensAthensGreece

Personalised recommendations