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Exponential Mixture Models with Long-Term Survivors and Covariates

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  • Ghitany, M. E.
  • Maller, R. A.
  • Zhou, S.

Abstract

Suppose a population contains individuals who may be subject to failure with exponentially distributed failure times, or else are "immune" to failure. We do not know which individuals are immune but we can infer their presence in a data set if many of the largest failure times are censored. We also have explanatory vectors containing covariate information on each individual. Models for data with such immune or "cured" individuals are of great interest in medical and criminological statistics, for example. In this paper we provide sufficient conditions for the existence, consistency, and asymptotic normality of maximum likelihood estimators for the parameters in a useful parameterization of these models. The theory is then applied to derive the asymptotic properties of the likelihood ratio test for a difference between immune proportions in a "one-way" classification. A procedure for testing the "boundary" hypothesis, that there are in fact no immunes present in data with a one-way classification, is also discussed.

Suggested Citation

  • Ghitany, M. E. & Maller, R. A. & Zhou, S., 1994. "Exponential Mixture Models with Long-Term Survivors and Covariates," Journal of Multivariate Analysis, Elsevier, vol. 49(2), pages 218-241, May.
    • Handle: RePEc:eee:jmvana:v:49:y:1994:i:2:p:218-241
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    Cited by:

    1. Shen, Pao-sheng, 2000. "Testing for sufficient follow-up in survival data," Statistics & Probability Letters, Elsevier, vol. 49(4), pages 313-322, October.
    2. Sean Yiu & Vernon T. Farewell & Brian D. M. Tom, 2017. "Exploring the existence of a stayer population with mover–stayer counting process models: application to joint damage in psoriatic arthritis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(4), pages 669-690, August.
    3. Hirose, Hideo, 2012. "Estimation of the number of failures in the Weibull model using the ordinary differential equation," European Journal of Operational Research, Elsevier, vol. 223(3), pages 722-731.
    4. Pons, O. & Lemdani, M., 2003. "Estimation and test in long-term survival mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 41(3-4), pages 465-479, January.
    5. López-Cheda, Ana & Cao, Ricardo & Jácome, M. Amalia & Van Keilegom, Ingrid, 2017. "Nonparametric incidence estimation and bootstrap bandwidth selection in mixture cure models," Computational Statistics & Data Analysis, Elsevier, vol. 105(C), pages 144-165.
    6. Choi, K. C. & Zhou, X., 2002. "Large Sample Properties of Mixture Models with Covariates for Competing Risks," Journal of Multivariate Analysis, Elsevier, vol. 82(2), pages 331-366, August.
    7. Morbiducci, Marta & Nardi, Alessandra & Rossi, Carla, 2003. "Classification of "cured" individuals in survival analysis: the mixture approach to the diagnostic-prognostic problem," Computational Statistics & Data Analysis, Elsevier, vol. 41(3-4), pages 515-529, January.
    8. Barreto-Souza, Wagner, 2015. "Long-term survival models with overdispersed number of competing causes," Computational Statistics & Data Analysis, Elsevier, vol. 91(C), pages 51-63.
    9. Lopez-Cheda , Ana & Cao, Ricardo & Jacome, Maria Amalia & Van Keilegom, Ingrid, 2015. "Nonparametric incidence and latency estimation in mixture cure models," LIDAM Discussion Papers ISBA 2015014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. H. Vu & R. Maller & X. Zhou, 1998. "Asymptotic Properties of a Class of Mixture Models for Failure Data: The Interior and Boundary Cases," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(4), pages 627-653, December.
    11. Varadan Sevilimedu & Shuangge Ma & Pamela Hartigan & Tassos C. Kyriakides, 2021. "An Application of the Cure Model to a Cardiovascular Clinical Trial," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 13(3), pages 402-430, December.
    12. Yingwei Peng & Keith B. G. Dear, 2000. "A Nonparametric Mixture Model for Cure Rate Estimation," Biometrics, The International Biometric Society, vol. 56(1), pages 237-243, March.
    13. Francisco Louzada & Juliana Cobre, 2012. "A multiple time scale survival model with a cure fraction," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 355-368, June.
    14. Amico, Mailis & Van Keilegom, Ingrid, 2017. "Cure models in survival analysis," LIDAM Discussion Papers ISBA 2017007, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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