Advertisement

Journal of Statistical Theory and Practice

, Volume 6, Issue 4, pp 698–724 | Cite as

EM Algorithm-Based Likelihood Estimation for Some Cure Rate Models

Article

Abstract

In the recent work of Rodrigues et al. (2009), a flexible cure rate survival model was developed by assuming the number of competing causes of the event of interest to follow the Conway-Maxwell Poisson distribution. This model includes as special cases some of the well-known cure rate models discussed in the literature. As the data obtained from cancer clinical trials are often subject to right censoring, the expectation maximization (EM) algorithm can be used as a powerful and efficient tool for the estimation of the model parameters based on right censored data. In this paper, the cure rate model developed by Rodrigues et al. (2009) is considered and assuming the time-to-event to follow the exponential distribution, exact likelihood inference is developed based on the EM algorithm. The inverse of the observed information matrix is used to compute the standard errors of the maximum likelihood estimates (MLEs). An extensive Monte Carlo simulation study is performed to illustrate the method of inference developed here. Finally, the proposed methodology is illustrated with real data on cutaneous melanoma.

Keywords

Cure rate models Conway-Maxwell Poisson (COM-Poisson) distribution Maximum likelihood estimators EM algorithm Profile likelihood Lifetime data Exponential distribution Asymptotic variances 

AMS Subject Classification

62N02 62P10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Balakrishnan, N., and Y. Peng. 2006. Generalized gamma frailty model. Stat. Med., 25, 2797–2816.MathSciNetCrossRefGoogle Scholar
  2. Berkson, J., and R. P. Gage. 1952. Survival cure for cancer patients following treatment. J. Am. Stat. Assoc., 47, 501–515.CrossRefGoogle Scholar
  3. Claeskens, G., R. Nguti, and P. Janssen. 2008. One-sided tests in shared frailty models. Test, 17, 69–82.MathSciNetCrossRefGoogle Scholar
  4. Conway, R. W., and W. L. Maxwell. 1961. A queuing model with state dependent services rates. J. Ind. Eng., 12, 132–136.Google Scholar
  5. Cox, D., and D. Oakes. 1984. Analysis of survival data. London, Chapman & Hall.Google Scholar
  6. Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B, 39, 1–38.MathSciNetMATHGoogle Scholar
  7. Farewell, V. T., 1982. The use of mixture models for the analysis of survival data with long-term survivors. Biometrics, 38, 1041–1046.CrossRefGoogle Scholar
  8. Hoggart, C. J., and J. E. Griffin. 2001. A Bayesian partition model for customer attrition. In Bayesian methods with applications to science, policy, and official statistics (Selected Papers from ISBA 2000), Proceedings of the Sixth World Meeting of the International Society for Bayesian Analysis, ed. E. I. George, 61–70. Creta, Greece, International Society for Bayesian Analysis.Google Scholar
  9. Ibrahim, J. G., M.-H. Chen, and D. Sinha. 2001. Bayesian survival analysis, New York, Springer-Verlag.CrossRefGoogle Scholar
  10. Kadane, J. B., G. Shmueli, T. P. Minka, S. Borle, and P. Boatwright. 2006. Conjugate analysis of the Conway-Maxwell-Poisson distribution. Bayesian Anal., 1, 363–374.MathSciNetCrossRefGoogle Scholar
  11. Kokonendji, C. C., D. Mizère, and N. Balakrishnan. 2008. Connections of the Poisson weight function to overdispersion and underdispersion. J. Stat. Plan. Inference, 138, 1287–1296.MathSciNetCrossRefGoogle Scholar
  12. Lange, K. 1995. A gradient algorithm locally equivalent to the EM algorithm. J. R. Stat. Soc. Ser. B, 57, 425–437.MathSciNetMATHGoogle Scholar
  13. Louis, T. A. 1982. Finding the observed information matrix when using the EM algorithm. J. R. Stat. Soc. Ser. B, 44, 226–233.MathSciNetMATHGoogle Scholar
  14. Maller, R. A., and X. Zhou. 1996. Survival analysis with long-term survivors. New York, John Wiley & Sons.MATHGoogle Scholar
  15. McLachlan, G. J., and T. Krishnan. 2008. The EM algorithm and extensions, 2nd ed., Hoboken, NJ, John Wiley & Sons.CrossRefGoogle Scholar
  16. Meeker, W. Q., and L. A. Escobar. 1998. Statistical methods for reliability data. New York, John Wiley & Sons.MATHGoogle Scholar
  17. Rodrigues, J., M. de Castro, V. G. Cancho, and N. Balakrishnan. 2009. COM-Poisson cure rate survival models and an application to a cutaneous melanoma data. J. Stat. Plan. Inference, 139, 3605–3611.MathSciNetCrossRefGoogle Scholar
  18. Self, S. G., and K.-Y. Liang. 1987. Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J. Am. Stat. Assoc., 82, 605–610.MathSciNetCrossRefGoogle Scholar
  19. Shmueli, G., T. P. Minka, J. B. Kadane, S. Borle, and P. Boatwright. 2005. A useful distribution for fitting discrete data: revival of the Conway-Maxwell-Poisson distribution. J. R. Stat. Soc. Ser. C, 54, 127–142.MathSciNetCrossRefGoogle Scholar
  20. Sy, J. P., and J. M. G. Taylor. 2000. Estimation in a Cox proportional hazards cure model. Biometrics, 56, 227–236.MathSciNetCrossRefGoogle Scholar
  21. Yakovlev, A. Y. 1994. Parametric versus nonparametric methods for estimating cure rates based on censored survival-data. Stat. Med., 13, 983–985.CrossRefGoogle Scholar
  22. Yakovlev, A. Y., and A. D. Tsodikov. 1996. Stochastic models of tumor latency and their biostatistical applications, Singapore, World Scientific.CrossRefGoogle Scholar
  23. Yakovlev, A. Y., A. D. Tsodikov, and L. Bass. 1993. A stochastic-model of hormesis. Math. Biosci., 116, 197–219.CrossRefGoogle Scholar
  24. Yin, G., and J. G. Ibrahim. 2005. Cure rate models: a unified approach. Can. J. Stat., 33, 559–570.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations