Univariate frailty distributions: A Review

S.G.Parekh ¹ , S.R. Patel²

Section:Review Paper, Product Type: Isroset-Journal
Vol.5 , Issue.6 , pp.352-356, Dec-2018

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.352356

Online published on Dec 31, 2018

Copyright © S.G.Parekh, S.R. Patel . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
The main idea of this review article is to review different univariate frailty models discussed by various authors. Particularly Discrete, Gamma, Inverse Gaussian, Generalized exponential, Log normal, Compound Poisson and Compound Negative Binomial frailty distributions are discussed. So far we have furnished the development of different frailty distributions with some important features and we have also provided the recent developments of many frailty distributions. By generating the observations of some distributions, we have obtained maximum likelihood estimators of the parameters with their standard errors involved in these frailty distributions.

Key-Words / Index Term :
Frailty models, Gamma, Inverse Gaussian, Generalized exponential, Log normal, Compound Poisson, Compound Negative Binomial, distributions

References :
Aalen, O.O., “Two examples of modelling heterogeneity in survival aanalysis”, Scandinavian Journal of Statistics, Vol.14, pp.19 – 25, 1987.
Aalen, O.O.,“Modelling heterogeneity in survival analysis by the compound Poisson distri¬bution”, Annals of Applied Probability, Vol.4, pp.951 – 972, 1992
Aalen, O.O., Tretli, S., “Analyzing incidence of testis cancer by means of a frailty model”, Cancer Causes and Control, Vol.10, pp.285 – 292, 1999.
Aalen O.O. Borgan 0, Gjessing HK, “Survival and event history analysis. A process point of view”, Springer, New York, 2008.
Abbring, J., van den Berg, G.J., “The unobserved heterogeneity distribution in duration analysis”, Working Paper, Free University Amsterdam, 2005
Benerjee, A. K. and Bhattacharyya, G. K.“A purchase incidence model with inverse Gaussian inter purchase times”, J. Amer. Statist. Asso.,Vol.71, pp.823-829, 1976.
Berretoni, J.N.,“Practical applications of the Weibull distribution”, Ind. Qual. Control, Vol. 21, pp.71-79, 1976
Carlo C. H. and Vonta I., “Frailty models for arbitrarily censored and truncated data”, Lifetime data Analysis, Vol.10, pp.369-388, 2004.
Chamberlain, G., “Heterogeneity, omitted variable bias, and duration dependence”, In: J.Heckman, B. Singer (eds.). Longitudinal analysis of labour market data, pp. 3 - 38, Cambridge University Press, 1992.
Chhikara, R.S., Folks, J.L., “The Inverse Gaussian distribution”. Marcel Dekker: New York, 1989.
Clayton, D.G., “A model for association in bivariate life tables and its application in epi- demiological studies of familial tendency in chronic disease Incidence”, Biometrika, Vol.65, pp.141 – 151, 1978.
Clegg A, Young J, Iliffe S et al., “Frailty in elderly people”, Lancet, Vol.381, pp.752-762, 2013.
Congdon, P.,“Modelling frailty in area mortality”, Statistics in Medicine, Vol.14, pp.1859-74, 1995.
Epstein, B., and M. Sobel (1955). Sequential life tests in the exponential case. Ann.Math. Stat., 26, 82-93.
Epstein, B.,“Statistical life test acceptance procedures”, Technometrics, Vol.2, pp.435-446, 1960.
Fried LP, Tangen CM, Walston J et al., “Frailty in older adults: evidence for a phenotype”, J Gerontol A Biol Sci Med Sci, Vol. 56, pp.M146- 56, 2001.
Govindarajulu, Z., “A supplement to Mendenhall’s bibliography on life testing and related topics”, J, Am. Stat. Assoc., Vol.59, pp.1231-1291, 1964.
Gustafson, P., “Large hierarchical Bayesian analysis of multivariate survival data”, Biometrics Vol.53, pp.230 – 242, 1997.
Gupta R. D. & Kundu, D., “Exponentiated exponential family: an alternative to gamma and weibull distribution”, Technical report. Dept. of Math., Stat. & Comp. Sci., University of New Brusnwick, Saint John, NB, Canada, 1997.
Gupta, R. D., and Kundu D.,“Generalized exponential distributions, Austral. & New Zealand J. Statist”, Vol. 41, Issue.2, pp.173-188, 1999.
Hanagal D.D., “Modeling heterogeneity for bivariate survival data by the Weibull distribution”, Stat Papers, Vol.51, pp.947-958, 2010.
Hanagal D.D. and Alok D.D., “Modeling of Inverse Gaussian Frailty Model for Bivariate Survival Data”, Communication in Statistics- Theory and methods, Vol. 42, pp.3744-3769, 2013.
Hougaard, P., “Life table methods for heterogeneous populations Distributions describing the het¬erogeneity”, Biometrika, Vol.71, Issue.1, pp.75-83, 1984.
Hougaard, P., “A class of multivariate failure time distributions”, Biometrika, Vol.73, pp. 671 – 678, 1986b.
Hougaard, P., Myglegaard, P., Borch-Johnsen, K.,“Heterogeneity models of disease suscep¬ tibility, with application to diabetic nephropathy”, Biometrics, Vol.50, pp.1178 – 1188, 1994.
Hougaard, P., “Analysis of multivariate survival data”, Springer, New York, 2000.
Jeong, J.-H., Jung, S.-H., Wieand, S., “A parametric model for long-term follow-up data from phase III breast cancer clinical trials”, Statistics in Medicine, Vol. 22, pp. 339 – 352, 2003.
Jeremy Balka, Anthony F. Desmond, Paul D. McNicholas, “Review and implementation of cure models based on first hitting times for Wiener process”, Lifetime Data Anal, Vol.15, pp.147-176, 2009.
Krisztina Mekli, James Y. Nazroo, et.al., “Proinflammatory genotype is associated with frailty phenotype in the English Longitudinal Study of Aging”, Aging Clin Exp.Res, DOI 10.1007/s40520-015-0419-z, 2015.
Krisztina Mekli, Alan Marshall, James Nazroo, Bram Vanhoutte, Neil Pendleton, “Genetic variant of interleukin-18 gene is associated with the Frailty Index in the English Longitudinal Study of Aging”, Age and Aging, Vol.0,pp.1-5, doi: 10.1093/ageing/afv122, 2015.
Lancaster, T., “Econometric methods for the duration of unemployment”, Econometrica, Vol. 47, pp.939 – 956, 1989.
Lillard, L.A., Brian, M.J., Waite, M.J., “Premarital cohabitation and subsequent marital dissolution: a matter of self-selection”, Demography, Vol. 32, pp. 437 – 457, 1995.
Macdonald, A.S., “Modeling the impact of genetics on insurance”, North American Actuarial Journal, Vol.3, pp.83 – 105, 1999.
Manton, K.G., Stallard, E., Vaupel, J.W., “Methods for comparing the mortality experience of heterogeneous populations”, Demography, Vol. 18, pp.389 – 410, 1989.
Manton, K., Stallard, E., Vaupel, J., “Alternative models for heterogeneity of mortality risks among the aged”, Journal of the American Statistical Association, Vol.81, pp.635 – 644, 186.
McGilchrist, C.A., Aisbett, C.W.,“Regression with frailty in survival analysis”, Biometrics, Vol.47, pp.461 – 466, 1991.
Mendenhall, W., “A bibliography on life testing and related topics”, Biometrika, Vol.45, pp.521- 543, 1958.
Moger, T.A., Aalen, O.O.,“A distribution for multivariate frailty based on the compound Poisson distribution with random scale”, Lifetime Data Analysis Vol.11, pp.41 – 59, 2005.
Mogar T.A, Marion Haugen, Benjamin H. K. Y.P., Gjessing K. H., Borgan O., “A hierarchical frilty model applied to two-generation melanoma data”, Lifetime Data Anal, Vol.19, pp.445-460, 2011.
Morley, E., Perry, H.M., Miller, D.K., “Something about frailty”, Journal of Gerontology: MEDICAL SCIENCES, Vol. 57A, pp.M698 - M704, 2002.
Murphy, S.A., Rossini, A., van der Vaart, A.W. , “Maximum likelihood estimation in the proportional odds model”, Journal of the American Statistical Association, Vol.92, pp.968 – 976, 1997.
Nelson, W. B., “Graphical analysis of accelerated life test data with the inverse power law model”, IEEE Trans. Reliab., R21, pp.2-11, 1972.
Nickell, S.J. (1979). “Estimating the probability of leaving unemployment”, Econometrica, Vol.47, pp.1249-1266, 1979.
Parekh S.G., Patel S.R., Ghosh D.K., and Raykundaliya D.P., “Discrete Frailty Models”, Proceedings of the 10thINDIACom; INDIACom-2016; IEEE conference ID: 37465. 2016 3rd International Conference on “Computing for Sustainable Global Development”, pp.180-182, 2016.
Parekh S.G. and Patel S.R.,“A study of compound Poisson and compound negative Binomial frailty models”, International Journal of Statistics and Applied Mathematics, Vol.3, Issue.2, pp.41- 44, 2018a.
Parekh S.G. and Patel S.R., “On estimation of Univariate Bayesian frailty models”, International Journal of Statistics and Applied Mathematics, Vol.3, Issue.2, pp.199- 205, 2018b.
Parekh S.G. and Patel S.R., “Some Univarite Continuous Frailty Models”, International Journal of Computational and Theoretical Statistics, Vol.5, Issue.1, pp. 29-37, 2018c
Price, D.L., Manatunga, A.K., “Modelling survival data with a cured fraction using frailty models”, Statistics in Medicine Vol.20, pp.1515 – 1527, 2001.
P.Ecomomou, C.Caroni, “Graphical Tests for the Assumption of Gamma and Inverse Gaussian Frailty Distributions”, Lifetime Data Analysis, Vol.11, pp.565-582, 2005.
Ripatti, S., Palmgren, J.,“Estimation of multivariate frailty models using penalised partial likelihood”, Biometrics, Vol.56, pp.1016 – 1022, 2000.
Ripatti, S., Larsen, K., Palmgren, J., “Maximum likelihood inference for multivariate frailty models using an automated MCEM algorithm”, Lifetime Data Analysis, Vol.8, pp.349 – 360, 2002.
Santos dos, C. A. and Achcar, J. A., “A Bayesian analysis for multivariate survival data in the presence of covariates”, Journal of Statistical Theory and Applications, Vol.9, pp.233- 253, 2010.
Schmoor, C., Schumacher, M., “Effect of covariate omission and categorization when ana¬ lyzing randomized trials with the Cox model”, Statistics in Medicine, Vol.16, pp.225 – 237, 1997.
Schumacher, M., Olschewski, M., Schmoor, C., “The impact of heterogeneity on the com¬parison of survival times”, Statistics in Medicine, Vol.6, pp.773 – 784, 1987.
Schumacher, M., “Frailty-Modelle und ihre Anwendungen in der Medizin”, unpublished draft, 1989.
Seshadri V., “The Inverse Gaussian Distribution”, Statistical Theory and Applica¬ tions, Springer, New York, 1999.
Shin SY, Fauman EB, Petersen AK et al., “An atlas of genetic influences on human blood metabolites” Nat Genet, Vol.46, pp.543-550, 2014.
Sukhatme, P.V., “Tests of significance for samples of the χ^2 population with two degree of freedom”, Ann. Eugen., Vol.8, pp.52-56, 1937.
Vaupel, J.W., Manton, K.G., Stallard, E., “The impact of heterogeneity in individual frailty on the dynamics of mortality”, Demography, Vol.16, Issue.3, pp.439 – 454, 1979.
Vaupel, J.W., Yashin, A.I., “Heterogeneity`s ruses: Some surprising effects of selection on population dynamics”, The American Statistician, Vol. 39, pp.176 –185, 1985.
Weibull, W., “A statistical distribution function of wide applicability”, J. Appl. Mech., Vol.18, pp.293-297, 1951.
Whitemore, A., and B. Altschuler, “Lung cancer incidence in cigarette smokers: further analysis of Doll and Hill’s data for British physicians”, Biometrics, Vol.32, pp.805-816, 1976.
Xue, X., Brookmeyer, R., “Bivariate frailty model for the analysis of multivariate survival time”, Lifetime Data Analysis, Vol.2, pp.277 – 290, 1996.
Yu, B., “Estimation of shared gamma frailty models by a modified EM algorithm”, Computational Statistics and Data Analysis, Vol. 50, pp.463-474, 2006.