Mathematics and Computer Science

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Frailty Models Under Xgamma Distribution with Application to Survival Data

Received: 26 July 2023    Accepted: 14 August 2023    Published: 31 August 2023
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Abstract

Frailty models provide an alternative to proportional hazards models, which are designed to discover the properties of the unobserved heterogeneity in individual risks of disease and death. In spite of this distribution of the frailty is normally assumed to be continuous. In some circumstances, it is appropriate to recollect discrete frailty distributions. Generally, Gamma, Weibull, Exponential, Lognormal, and Log-logistic baseline distributions have fitted with frailty distribution. The Xgamma distribution among a unique finite aggregate of exponential and gamma distribution and allowance for the different shapes of the hazard function. The study aims to fit the above four distributions with the Xgamma baseline distribution and apply them to test popular actual-lifestyles statistics set. The study result revealed that Xgamma with Positive Stable (PS) frailty model is a good choice for the Veterans' Administration Lung Cancer study data set and Xgamma with Log-Normal (LN) frailty model is the best fit for the Culling dairy heifer cow’s data set. Additionally, Xgamma identifies the baseline distribution with the lowest Akaike's Information Criteria (AIC) and Bayesian Information Criteria (BIC) values. The study result proved Xgamma distribution and its extended model for frailty distribution is the possible approach in a real-life time or survival analysis.

DOI 10.11648/j.mcs.20230804.11
Published in Mathematics and Computer Science (Volume 8, Issue 4, July 2023)
Page(s) 87-93
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Xgamma Distribution, Hazard Function, Survival Analysis, Parametric Frailty Models, Marginal LOG-Likelihood, Clustered Data Analysis

References
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[2] David G. Klenbaum and Mitchel Klein, Survival Analysis. A Self-Learning Text, Third Edition. Series of Statistics for Biology and health. Springer-Verlag New York. DOI: 10.1007-1-4419-6646-9, 2012.
[3] Duchateau L, Janssen P, Legrand C, Nguti R, Sylvester R, “The shared Frailty Model and the Power for Heterogeneity Test in Multicenter Trials”. Computational Statistics & Data Analysis, 40 (3), 603-620, 2002.
[4] Vaupel JW, Manton KG, Stallard E, “The Impact of Heterogeneity in Individual Frailty on the Dynamics of Mortality”. Demography, 16 (3), 439-454, 1979.
[5] Hougaard P, Analysis of Multivariate Survival Data. Lifetime Data analyses, 1 (3), 255-283, 2000.
[6] Dunhateau L, Janssen P, The Frailty Model. Series of Statistics for Biology and Health. Springer-Verlag. DOI: 10.1007/978-0-387-72835-3, 2008.
[7] Wienke A, Frailty Models in Survival Analysis. Chapman & Hall/CRC, Boca Raton, 2010.
[8] Clayton, D., Cuzick, J, Multivariate generalizations of the proportional hazard model. Journal of the Royal Statistical Society (A) 148, 82-117, 1983a.
[9] Ibrahim J. G., Chen MH., Sinha D, Frailty Models. In Bayesian Survival Analysis. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3447-84, 2001.
[10] Lindley DV. Fiducially distribution and Bayes theorem. Journal of Royal Statistical Society A. 1958; 20 (1): 102-107.
[11] J. Nagaraj, S. Parthasarathy, C. Ponnuraja, “Lindley Distribution as Frailty Models with Application to Life Time Data”. Advances and Applications in Statistics. 75, 119-134. http://dx.doi.org/10.17654/0972361722031, 2022
[12] Subhradev Sen, Sudhansu S. Maiti, N. Chandra “The Xgamma Distribution: Statistical Properties and Application”. Journal of Modern Applied Statistical Methods. Vol. 15, No. 1, 774-788, 2016.
[13] Subhradev Sen, Sudhansu S. Maiti, N. Chandra, “Survival estimation in Xgamma distribution under the progressively type-II right censored scheme”. Model Assisted Statistics and Applications 13 (2018) 107-121. DOI 10.3233/MAS-180423 2018.
[14] Cox DR, “Regression Model and Life-Tables”. Journal of Royal Society B, 34 (2), 187-220, 1972.
[15] Van den Berg Gj, Drepper PM, “Inference for shared-Frailty Survival Models with Left-Truncated Data”. Working Papers 12-5, University of Mannheim, Department of Economics. URL http://ideas.repec.org/p/mnh/wpaper/30729.html. 2012.
[16] Balan, TA, Putter H “Frailty EM: An R Package for Estimating Semi Parametric Shared Frailty Model”. Journal of Statistical Software. 90 (7): 2019. DOI: 10.18637/jss.v090.i07. 2019.
[17] David D. Hanagal, “Modeling Survival Data Using Frailty Models. Industrial and Applied Mathematics”. Springer Nature Singapore Pte Ltd (295 Pages). DOI: 10.1007/978-981-15-1181-3, 2011.
[18] Munda, Marco & Rotolo, Federico & Legrand, Catherine, “Parfm: Parametric Frailty Models in R”, “Journal of Statistical Software, Foundation for Open Access Statistics, Vol. 51 (i11), 2012. http://hdl.handle.net.10.18637/jss.v051.i11.
[19] Hougaard P, “Frailty Models for Survival Data”. Lifetime Data Analysis, 1 (3), 255-273, 1995.
[20] Duchateau L, Janssen P, The frailty model. Springer. New York: Springer-Verlag, 2008.
[21] De Vliegher. S, Barkema. H. W, Opsomer. G, et al., Association between somatic cell count in early lactation and culling of dairy heifers using Cox frailty models. J. Dairy Sci. 88, 560-568, 2005.
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[23] Byar, D. P., The Veterans Administration study of chemoprophylaxis of recurrent stage I bladder tumors: comparisons of a placebo, pyridoxine, and topical thiotepa. In Bladder Tumors and Other Topics in Urological Oncology (M. Pavone-Macaluso, P. H. Smith, and F. Edsmyn, eds.). New York: Plenum, pp. 363-370, 1980.
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  • APA Style

    Ashok Kumar Palanisamy, Muthukumar Madaswamy. (2023). Frailty Models Under Xgamma Distribution with Application to Survival Data. Mathematics and Computer Science, 8(4), 87-93. https://doi.org/10.11648/j.mcs.20230804.11

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    ACS Style

    Ashok Kumar Palanisamy; Muthukumar Madaswamy. Frailty Models Under Xgamma Distribution with Application to Survival Data. Math. Comput. Sci. 2023, 8(4), 87-93. doi: 10.11648/j.mcs.20230804.11

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    AMA Style

    Ashok Kumar Palanisamy, Muthukumar Madaswamy. Frailty Models Under Xgamma Distribution with Application to Survival Data. Math Comput Sci. 2023;8(4):87-93. doi: 10.11648/j.mcs.20230804.11

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  • @article{10.11648/j.mcs.20230804.11,
      author = {Ashok Kumar Palanisamy and Muthukumar Madaswamy},
      title = {Frailty Models Under Xgamma Distribution with Application to Survival Data},
      journal = {Mathematics and Computer Science},
      volume = {8},
      number = {4},
      pages = {87-93},
      doi = {10.11648/j.mcs.20230804.11},
      url = {https://doi.org/10.11648/j.mcs.20230804.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20230804.11},
      abstract = {Frailty models provide an alternative to proportional hazards models, which are designed to discover the properties of the unobserved heterogeneity in individual risks of disease and death. In spite of this distribution of the frailty is normally assumed to be continuous. In some circumstances, it is appropriate to recollect discrete frailty distributions. Generally, Gamma, Weibull, Exponential, Lognormal, and Log-logistic baseline distributions have fitted with frailty distribution. The Xgamma distribution among a unique finite aggregate of exponential and gamma distribution and allowance for the different shapes of the hazard function. The study aims to fit the above four distributions with the Xgamma baseline distribution and apply them to test popular actual-lifestyles statistics set. The study result revealed that Xgamma with Positive Stable (PS) frailty model is a good choice for the Veterans' Administration Lung Cancer study data set and Xgamma with Log-Normal (LN) frailty model is the best fit for the Culling dairy heifer cow’s data set. Additionally, Xgamma identifies the baseline distribution with the lowest Akaike's Information Criteria (AIC) and Bayesian Information Criteria (BIC) values. The study result proved Xgamma distribution and its extended model for frailty distribution is the possible approach in a real-life time or survival analysis.},
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - Frailty Models Under Xgamma Distribution with Application to Survival Data
    AU  - Ashok Kumar Palanisamy
    AU  - Muthukumar Madaswamy
    Y1  - 2023/08/31
    PY  - 2023
    N1  - https://doi.org/10.11648/j.mcs.20230804.11
    DO  - 10.11648/j.mcs.20230804.11
    T2  - Mathematics and Computer Science
    JF  - Mathematics and Computer Science
    JO  - Mathematics and Computer Science
    SP  - 87
    EP  - 93
    PB  - Science Publishing Group
    SN  - 2575-6028
    UR  - https://doi.org/10.11648/j.mcs.20230804.11
    AB  - Frailty models provide an alternative to proportional hazards models, which are designed to discover the properties of the unobserved heterogeneity in individual risks of disease and death. In spite of this distribution of the frailty is normally assumed to be continuous. In some circumstances, it is appropriate to recollect discrete frailty distributions. Generally, Gamma, Weibull, Exponential, Lognormal, and Log-logistic baseline distributions have fitted with frailty distribution. The Xgamma distribution among a unique finite aggregate of exponential and gamma distribution and allowance for the different shapes of the hazard function. The study aims to fit the above four distributions with the Xgamma baseline distribution and apply them to test popular actual-lifestyles statistics set. The study result revealed that Xgamma with Positive Stable (PS) frailty model is a good choice for the Veterans' Administration Lung Cancer study data set and Xgamma with Log-Normal (LN) frailty model is the best fit for the Culling dairy heifer cow’s data set. Additionally, Xgamma identifies the baseline distribution with the lowest Akaike's Information Criteria (AIC) and Bayesian Information Criteria (BIC) values. The study result proved Xgamma distribution and its extended model for frailty distribution is the possible approach in a real-life time or survival analysis.
    VL  - 8
    IS  - 4
    ER  - 

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Author Information
  • Department of Statistics, PSG College of Arts and Science, Coimbatore, India

  • Department of Statistics, PSG College of Arts and Science, Coimbatore, India

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