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Semi-Markov and non-homogeneous Markov models in medicine

  • Daniel Commenges

Abstract

Longitudinal studies form the most interesting part of medical studies. All therapeutical trials and the majority of epidemiological studies can be considered as longitudinal studies but most often this aspect is ignored or grossly simplified when the data are analysed. For instance a very simple therapeutical trial can be described as follows. Patients are included in the trial, each at a certain moment of their history. When included they are assigned randomly to one of two treatment groups. A particular variable R, called here the response variable, can be observed till the end of the study. A simple method of analysis consists in comparing the values of R in the two groups observed after a time Te of treatment. A Pearson’s X2 or a Student’s t test for instance can be used according to the statistical nature of the variable.

Keywords

Sojourn Time Markov Chain Model Hyperthyroidic Patient Homogeneous Markov Chain Marginal Total 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Aalen, O.O. and S. Johansen (1978). An empirical transition matrix for non-homogeneous Markov chains based on censored observations. Scand.J.Statist. 5, 141–150.MathSciNetzbMATHGoogle Scholar
  2. Aaling, D.W. (1958). The after-history of pulmonary tuberculosis: a stochastic model. Biometrics 14, 527–547.CrossRefGoogle Scholar
  3. Andersen, P.K., O. Borgan, R. Gill and N. Keiding (1982). Linear non-parametric tests for comparison of counting processes, with applications to censored data. Int.Statist.Rev. 50, 219–258.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Anderson, T.W. and L.A. Goodman (1957). Statistical inference about Markov chains. Ann.Math.Statist. 28, 89–110.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Commenges, D., P. Barberger-Gateau, J.F. Dartigues, P. Loiseau and R. Salamon (1984). A non-homogeneous Markov chain model for follow-up studies with application to epilepsy. Meth.Inform. Med. 23, 109–114.Google Scholar
  6. Cox, D.R. (1972). Regression models and life tables. J.Roy.Stat. Soc. B 34, 187–220.zbMATHGoogle Scholar
  7. Cox, D.R. (1975). Partial likelihood. Biometrika 62, 269–276.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Flemming, T.R. (1978). Nonparametric estimation for non-homogeneous Markov processes in the problem of competing risks. Ann. Statist. 6, 1057–1070.MathSciNetCrossRefGoogle Scholar
  9. Gail, M.H., T.J. Santner and C.C. Brown (1980). An analysis of comparative carcinogenesis experiments based on multiple times to tumor. Biometrics 36, 255–266.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Gill, R.D. (1980). Nonparametric estimation based on censored observations of Markov renewal process. Z.Wahr.Verw.Geb. 53, 97–116.zbMATHCrossRefGoogle Scholar
  11. Iosifescu, M. and P.Tautu (1973). Stochastic processes and applications in biology and medicine. Springer-Verlag.Google Scholar
  12. Hsieh, F.Y., J. Crowley and D.C. Tormey (1983). Some test statistics for use in multistate survival analysis. Biometrika 70, 111–119.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Kaplan, E.L. and P. Meier (1958). Nonparametric estimation from incomplete observations, J.Amer.Stat.Assoc. 53, 457–481.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Kay, R. (1984). Multistate survival analysis: an application to breast cancer. Meth.Infor.Med. 23, 157–162.Google Scholar
  15. Lagakos, S.W., C.J. Sommer and M. Zelen (1978). Semi-Markov models for partially censored data. Biometrika 65, 311–317.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Mantel, N. (1966). Evaluation of survival data and two rank-order statistics arising in its consideration. Cancer Chemotherapy Report 50, 163–170.Google Scholar
  17. Mantel, N. and W. Haenszel (1959). Statistical aspects of the analysis of data from retrospective studies of disease. J.Nat.Cancer Inst. 22, 719–740.Google Scholar
  18. Peto, R. and J. Peto (1972). Asymptotically efficient rank invariant test procedures. J.Roy.Statist.Soc. A 135, 185–207.CrossRefGoogle Scholar
  19. Temkin, N. (1978). An analysis for transient states with application to tumor skrinkage. Biometrics 34, 571–580.zbMATHCrossRefGoogle Scholar
  20. Voelkel, J. and J. Crowley (1984). Nonparametric inference for a class of semi-Markov processes with censored observations. Ann.Stat. 12, 142–160.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Weiss, G.H. and M. Zelen (1965). A semi-Markov model for clinical trials. J.Appl.Prob. 2, 269–285.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Daniel Commenges
    • 1
  1. 1.Université de Bordeaux IIBordeaux CedexFrance

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