EXPLORING THE MODIFIED GAMMA FRAILTY DISTRIBUTION: AN OPTIMAL DESIGN APPROACH USING PYTHON
Abstract
A suction/injection controlled mixed convection flow of an incompressible and viscous fluid in a vertical SurvivalAnalysis is pivotal in understanding the effects of covariates on potentially censored failure times and in the joint modelling of clustered data. It is used in the context of incomplete repeated measures and failure times in longitudinal studies. Survival data are often subject to right censoring and to a subsequent loss of information about the effect of explanatory variables. Frailty models are one common approach to handle such data.Three frailty models are used to analyze bivariate time-to-event data. All approaches accommodate right censored lifetime data and account for heterogeneity in the study population. A Modified Gamma Frailty Model is compared with two existing Frailty Models. The survival-analysis was performed using the Python.The newly derived MGF was analyzed using Python which is more robust when sample size is more than forty.The MGF model performs better than the existing models in the presence of clustering. However the CGF is preferable in the absence of clusters in a given data set.
References
Abdulazeez, S.A (2020). An Optimal Design for Inference via Modified Gamma FrailtyDistribution.KASU Journal of Mathematical Sciences (KJMS) VOL.1, ISSUE 1.
Anthony I. W.,, Isaac D. E., Victor A. K., (2019): Survival Analysis of Under –five Mortality and Its Associated Determinants in Nigeria: Evidence from a Survey Data International Journal of Statistics and Applications 9(2): 59-66
Cox, D.R. (1972). Regression models and life tables, Journal of the Royal StatisticalSociety, B 34, 187-400. DOI: https://doi.org/10.1111/j.2517-6161.1972.tb00899.x
Getachew Y. & Bekele S., (2016). Survival Analysis of Under-Five Mortality of Children and its Associated Risk Factors in Ethiopia. J Biosens Bioelectron7: 213. DOI: https://doi.org/10.4172/2155-6210.1000213
Iachine, I., Holm, N., Harris, J., Begun, A., Iachina, M., Laitinen, M., Kaprio, J., Yashin, A. (1998). How heritable is individual susceptibility to death? The results of an analysis of survival data on Danish, Swedish and Finnish twins. Twin Research1, 196 – 205 DOI: https://doi.org/10.1375/twin.1.4.196
Iachine, I. (2002). The Use of Twin and Family Survival Data in the Population Studies of Aging: Statistical Methods Based on Multivariate Survival Models. Ph.D. Thesis. Monograph 8, Department of Statistics and Demography, University of Southern Denmark.
Kalbfleisch, J. D. and R.L. Prentice, (1980). The statistical analysis of failure timedata(John Wiley & Sons Inc., New York).
Kiefer, N.M., (1988). Economic duration data and hazard functions, Journal of Economic Literature 26, 646-679.
Lancaster, T., (1990). The econometric analysis of transit data (Cambridge UniversityPress, Cambridge, U.K).
Lee Y. & Song J.K. (2001). Hierarchical likelihood approach for frailty models. Biometrika 88(1):233–33 DOI: https://doi.org/10.1093/biomet/88.1.233
Longini, I. M., and Halloran, M. E. (1996). A frailty mixture model for estimating vaccineefficacy. Applied Statistics, 45, 165-173. DOI: https://doi.org/10.2307/2986152
Moger, T. A., and Aalen, O. O. (2005). A distribution for multivariate frailty based on the compound poisson distribution with random scale. Lifetime Data Analysis, 11,41-59. DOI: https://doi.org/10.1007/s10985-004-5639-z
Petersen J. H (1998). An Additive Frailty Model for Correlated Life Times," Biometrics 54 : 646-661. DOI: https://doi.org/10.2307/3109771
Pickles, A., Crouchley, R.,Simono, E., Eaves, L., Meyer, J., Rutter, M., Hewitt, J., Silberg,J. (1994). Survival models for developmental genetic data: age of onset of puberty and antisocial behavior in twins. Genetic Epidemiology11, 155 - 170 DOI: https://doi.org/10.1002/gepi.1370110206
Price, D. L., and Manatunga, A. K. (2001). Modelling survival data with a cured fraction using frailty models.Statistics in Medicine, 20, 1515-1527. DOI: https://doi.org/10.1002/sim.687
Rakhmawati T.W, Ha I.D, Lee H, Lee Y.(2021). Penalized variable selection for cause-specific hazard frailty models with clustered competing-risks data. Stat. Med. 40(29):6541–57 DOI: https://doi.org/10.1002/sim.9197
Rueten-Budde A.J, Putter H., Fiocco M. (2019). Investigating hospital heterogeneity with a competing risks frailty model. Stat. Med. 38(2):269–88 DOI: https://doi.org/10.1002/sim.8002
Vermunt, J.K., (1996). Log-linear event history analysis (Tilburg University Press,Netherlands).
Vilcassim, N. J. and D.C. Jain, (1991). Modeling purchase timing and brand switchingbehavior incorporating explanatory variables and unobserved heterogeneity,Journal of Marketing Research, 28, 29-41. DOI: https://doi.org/10.1177/002224379102800103
Wedel, M., W.A. Kamakura, W.S. DeSarbo and F. ter Hofstede, (1995). Implicationsfor asymmetry, non proportionality and heterogeneity in brand switching modelsfrom piece-wise exponential mixture hazard models, Journal of MarketingResearch, 32, 457-463. DOI: https://doi.org/10.2307/3152180
Wienke, A., Arbeev, K., Locatelli, I., Yashin, A.I. (2005). A comparison of different correlated frailty models and estimation strategies. Mathematical Biosciences198, 1 – 13 DOI: https://doi.org/10.1016/j.mbs.2004.11.010
Wienke, A., Christensen, K., Holm, N., Yashin, A. (2000). Heritability of death from respiratory diseases:an analysis of Danish twin survival data using a correlated frailty model. IOS Press, Amsterdam.
Wienke, A., Holm, N., Skytthe, A., Yashin, A.I. (2001). The heritability of mortality due toheart diseases: a correlated frailty model applied to Danish twins. Twin Research4, 266 - 274 DOI: https://doi.org/10.1375/twin.4.4.266
Wienke, A., Christensen, K., Skytthe, A., Yashin, A.I. (2002). Genetic analysis of cause of death in a mixture model with bivariate lifetime data. Statistical Modelling 2, 89 - 102 DOI: https://doi.org/10.1191/1471082x02st030oa
Wienke, A., Lichtenstein, P., Yashin, A.I. (2003a). A bivariate frailty model with a cure fraction for modelling familial correlations in diseases. Biometrics59, 1178 – 1183 DOI: https://doi.org/10.1111/j.0006-341X.2003.00135.x
Wienke, A., Holm, N., Christensen, K., Skytthe, A., Vaupel, J., Yashin, A.I. (2003b). Theheritability of cause-specific mortality: a correlated gamma-frailty model applied to mortality due to respiratory diseases in Danish twins born 1870 - 1930. Statistics in Medicine 22, 3873 - 3887 DOI: https://doi.org/10.1002/sim.1669
Yashin, A.I., Iachine, I.A. (1995a). Genetic analysis of durations: Correlated frailty model applied to survival of Danish twins. Genetic Epidemiology12, 529 – 538 DOI: https://doi.org/10.1002/gepi.1370120510
Yashin, A.I., Iachine, I.A. (1995b). Survival of related individuals: an extension of some fundamental results of heterogeneity analysis. Mathematical Population Studies5, 321-39 DOI: https://doi.org/10.1080/08898489509525410
Yashin, A.I., Manton, K.G., Iachine, I.A. (1996). Genetic and environmental factors in durationstudies: multivariate frailty models and estimation strategies. Journal of Epidemiology and Biostatistics1, 115 – 120
Yashin, A.I., Iachine, I.A. (1997). How frailty models can be used for evaluating longevity limits:Taking advantage of an interdisciplinary approach. Demography34, 31 - 48 DOI: https://doi.org/10.2307/2061658
Yashin, A.I.,Iachine, I. (1999a). Dependent hazards in multivariate survival problems. Journal of Multivariate Analysis71, 241 – 261 DOI: https://doi.org/10.1006/jmva.1999.1848
Yashin, A.I., Iachine, I. (1999b). What difference does the dependence between durations make? Insights for population studies of aging. Lifetime Data Analysis5, 5 – 22 DOI: https://doi.org/10.1023/A:1009622214567
Zdravkovic, S., Wienke, A., Pedersen, N.L., Marenberg, M.E., Yashin, A.I., de Faire, U. (2002). Heritability of death from coronary heart disease: a 36 years follow-up of 20,966 Swedish twins.Journal of Internal Medicine252, 247 – 54 DOI: https://doi.org/10.1046/j.1365-2796.2002.01029.x
Zdravkovic, S., Wienke, A., Pedersen, N.L., Marenberg, M.E., Yashin, A.I., de Faire, U. (2004). Genetic influences on CHD-death and the impact of known risk factors: Comparison of two frailty models.Behavior Genetics34, 585 – 591 DOI: https://doi.org/10.1007/s10519-004-5586-1
Zhu R., Kosorok M.R. (2012). Recursively imputed survival trees. J. Am. Stat. Assoc. 107(497):331–40 DOI: https://doi.org/10.1080/01621459.2011.637468
Copyright (c) 2023 FUDMA JOURNAL OF SCIENCES
This work is licensed under a Creative Commons Attribution 4.0 International License.
FUDMA Journal of Sciences