On estimation and influence diagnostics for a Bivariate Promotion Lifetime Model Based on the FGM Copula: A Fully Bayesian Computation

Adriano Kamimura Suzuki, Francisco Louzada, Vicente Garibay Cancho

Abstract


In this paper we propose a bivariate long-term model based on the Farlie-Gumbel-Morgenstern copula to model, where the marginals are assumed to be long-term promotion time structured. The proposed model allows for the presence of censored data and covariates. For inferential purpose a Bayesian approach via Markov Chain Monte Carlo is considered. Further, some discussions on the model selection criteria are given. In order to examine outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback-Leibler divergence. The newly developed procedures are illustrated on artificial and real data.

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H. Aslanidou, D.K. Dey, and D. Sinha, Bayesian analysis of multivariate survival data using Monte Carlo methods, Canadian Journal of Statistics 1 (2008), pp. 33-48.

J. Berkson and R.P. Gage, Survival cure for cance patients following treatment, Journal of the American Statistical Association 47(259) (1952), pp. 501-515.

J.W. Boag, Maximum likelihood estimates of the proportion of patients cured by cancer therapy, Journal of the Royal Statistical Society B 11(1) (1949), pp.15-53.

S.P. Brooks, Discussion on the paper by Spiegelhalter, Best, Carlin, and van der Linde, Journal Royal Statistical Society B 64(3) (2002), pp. 616-618.

B.P. Carlin and T.A. Louis, Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed., Chapman & Hall/CRC, Boca Raton, 2001.

D.G. Clayton, A model for association in bivariate life-tables and its application in epidemiological studies of familial tendency in chronic disease incidence, Biometrika 65 (1978), pp. 141-151.

M.H. Chen, J.G. Ibrahim and D. Sinha, A new Bayesian model for survival data with a surviving fraction, Journal of the American Statistical Association 94(447) (1999), pp. 909-919.

M.H. Chen, Q.M. Shao, and J. Ibrahim, Monte Carlo Methods in Bayesian Computation, Springer-Verlag, New York, 2000.

S. Chib and E. Greenberg, Understanding the metropolis-hastings algorithm, The American Statistician 49 (1995), pp. 327-335.

S.C. Chiou and R.S. Tsay, A Copula-based Approach to Option Pricing and Risk Assessment, Journal of Data Science 6 (2008), pp. 273-301.

G. Colby and P. Rilstone, Simplied estimation of multivariate duration models with unobserved heterogeneity, Computational Statistics 22 (2007), pp. 17-29.

D.A. Conway, Farlie-Gumbel-Morgenstern distributions, in Encyclopedia of Statistical Sciences 3, S. Kotz and N.L. Johnson, eds., John Wiley & Sons, New York, 1983, pp. 2831.

R.D. Cook and S. Weisberg, Residuals and Inuence in Regression, Chapman & Hall/CRC, Boca Raton, 1982.

M.K. Cowless and B.P. Carlim, Markov chain Monte Carlo convergence diagnostics: a comparative review, Journal of the American Statistical Association 91 (1996), pp. 883-904.

G. De Luca and G. Rivieccio, Archimedean copulae for risk measurement, Journal of Applied Statistics 36(8) (2009), pp. 907-924.

P. Embrechts, F. Linskog, and A. McNiel, Modelling dependence with copulas and applications to risks management. available at

http://www.math.ethz.ch/baltes/ftp/papers.html (2003).

D. Gamerman and H.F. Lopes, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, 2nd ed., Chapman & Hall, Boca Raton, 2006.

A.E. Gelfand and A.F.M. Smith, Sampling-Based Approaches to Calculating Marginal Densities, Journal of the American Statistical Association 85(410) (1990), pp. 398-409.

A.E. Gelfand, D.K. Dey, and H. Chang, Model determination using predictive distributions with mplementation via sampling-based methods, Bayesian Statistics 4, Peñíscola, 1991, Oxford Univ. Press, New York, pp. 147-167.

A. Gelman, J. Carlin, and D. Rubin, Bayesian Data Analysis, Chapman & Hall/CRC, New York, 2006.

W.R. Gilks, S. Richardson, and D.J. Spiegelhater, Markov Chain Monte Carlo in Practice, Chapman & Hall, London, 1996.

K. Goethals, P. Janssen, and L. Duchateau, Frailty models and copulas: similarities and dierences, Journal of Applied Statistics 35(9) (2008), pp. 1071-1079.

P. Gustafson, D. Aeschliman, and A.R. Levy, A simple approach to tting Bayesian survival models, Lifetime Data Analysis 9 (2003), pp. 519.

D. Hanagal, Modeling Survival Data Using Frailty Models, Chapman & Hall/CRC, Boca Raton, 2011.

P. Hougaard, A class of multivariate failure time distributions, Biometrika 73(1986), pp. 671-678.

P. Hougaard, Fitting a multivariate failure time distribution, IEEE Transactions on Reliability 38 (1989), pp. 444-448.

P. Hougaard, Analysis of Multivariate Survival Data, Springer, Heidelberg, 2000.

J.G. Ibrahim, M-H. Chen, and D. Sinha, Bayesian Survival Analysis, Springer Verlag, New York, 2001.

P. Jaworski, Copula Theory and Its Applications: Proceedings of the Workshop Held in Warsaw, 25-26 September 2009 Volume 198 de Lecture Notes in Statistics, eds. P. Jaworski, F. Durante, W. Härdle and T. Rychlik, Springer, 2010.

R.A. Maller and X. Zhou, Survival Analysis with Long-Term Survivors, Wiley, New York, 1996.

A.K. Manatunga and D. Oakes, Parametric Analysis of Matched Pair Survival Data, Lifetime Data Analysis 5 (1999), pp. 371-387.

R. Nelsen, An Introduction to Copulas, 2nd ed., Springer, New York, 2006.

D. Oakes, Bivariate survival models induced by frailties, Journal of the American Statistical Association 84 (1989), pp. 487-493.

D. Oakes, On Frailty Models and Copulas Recent Advances in Statistical Methods, Proceedings of Statistics 2001 Canada: The 4th Conference in Applied Statistics, Montreal, 2001, pp. 218-224.

F. Peng and D. Dey, Bayesian analysis of outlier problems using divergence measures, The Canadian Journal of Statistics. La Revue Canadienne de Statistique 23(2) (1995), pp. 199-213.

Y. Peng, K.B.G. Dear, and J.W. Denham, A generalized F mixture model for cure rate estimation, Statistics in Medicine 17 (1998), pp. 813-830.

J. Rodrigues, M. de Castro, V.G. Cancho, and F. Louzada Neto, On the unification of long-survival models, Statistics and Probabilities Letters 79(1) (2009), pp. 753-759.

J.S. Romeo, N.I. Tanaka, and A.C. Pedroso de Lima, Bivariate survival modeling: a Bayesian approach based on Copulas, Lifetime Data Analysis 12 (2006), pp. 205-222.

S.K. Sahu and D.K. Dey, A comparison of frailty and other models for bivariate survival data, Lifetime Data Analysis 6 (2000), pp. 207-228.

J.H. Shih and T.A. Louis, Inferences on the association parameter in copula models for bivariate survival data, Biometrics 51 (1995), pp. 1384-1399.

D.J. Spiegelhalter, N.G. Best, B.P. Carlin, and A. van der Linde, Bayesian measures of model complexity and t Journal of the Royal Statistical Society B 64 (2002), pp. 583-639.

D. Spiegelhalter, A. Thomas, N. Best, and D. Lunn, OpenBUGS User Manual, version 3.0.2, MRC Biostatistics Unit, Cambridge; software available at http://mathstat.helsinki./openbugs.

The Diabetic Retinopathy Study Research Group, Preliminary report on the efeect of photo-coagulation therapy, American Journal of Ophthalmology 81 (1976), pp. 383-396.

P.K. Trivedi and D.M. Zimmer, Copula modelling: an introduction for practitioners, Foundations and Trends in Econometrics 1 (2005), pp. 1-111.

J.W. Vaupel, K.G. Manton, and E. Stallard, The impact of heterogeneity in individual frailty on the dynamics of mortality, Demography 16 (1979), pp. 439-454.

A. Wienke, Frailty Models in Survival Analysis, Chapman & Hall/CRC, Boca Raton, 2011.

A.Y. Yakovlev and A.D. Tsodikov, Stochastic Models of Tumor Latency and Their Biostatistical Applications, World Scientic, Singapore, 1996.

S. Zhang, Y. Zhang, K. Chaloner, and J.T. Stapleton, A copula model for bivariate hybrid censored survival data with application to the MACS study, Lifetime Data Analysis 16 (2010), pp. 231-249.




DOI: https://doi.org/10.5540/tema.2013.014.03.0441

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TEMA - Trends in Applied and Computational Mathematics

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