A Diagnostic for Association in Bivariate Survival Models Article Received: 17 June 2003 Revised: 12 November 2004 Accepted: 29 November 2004 DOI:
10.1007/s10985-004-0386-8 Cite this article as: chen, M. & Bandeen-Roche, K. Lifetime Data Anal (2005) 11: 245. doi:10.1007/s10985-004-0386-8 Abstract
We propose exploratory, easily implemented methods for diagnosing the appropriateness of an underlying copula model for bivariate failure time data, allowing censoring in either or both failure times. It is found that the proposed approach effectively distinguishes gamma from positive stable copula models when the sample is moderately large or the association is strong. Data from the Women’s Health and Aging Study (WHAS, Guralnik et al.,
The Womens’s Health and Aging Study: Health and Social Characterisitics of Older Women with Disability. National Institute on Aging: Bethesda, Mayland, 1995) are analyzed to demonstrate the proposed diagnostic methodology. The positive stable model gives a better overall fit to these data than the gamma frailty model, but it tends to underestimate association at the later time points. The finding is consistent with recent theory differentiating ‘catastrophic’ from ‘progressive’ disability onset in older adults. The proposed methods supply an interpretable quantity for copula diagnosis. We hope that they will usefully inform practitioners as to the reasonableness of their modeling choices. Keywords Archimedean copula bivariate failure time copula models frailty References Bandeen-Roche, K.J., Liang, K.Y. 2002 “Modelling multivariate failure time associations in the presence of a competing risk,” Biometrika. 89 299 314 Google Scholar Brown, B.W., Hollander, M., Korwar, R.M. 1974 “Nonparametric tests of independence for censored data, with application to heart transplant studies” Proschan, F. Serfling, R. J. eds. in Reliability and Biometry: Statistical Analysis of Lifelength SIAM Philadelphia 327 353 Google Scholar Clayton, D.G. 1978 “A model for association in bivariate life-tables and its application in epidemiological studies of familiar tendency in chronic disease incidence,” Biometrika vol. 65 141 151 Google Scholar Dabrowska, D.M. 1988 “Kaplan–Meier Estimate on the Plane,” The Annals of Statistics. 16 1475 1489 Google Scholar Ferrucci, L., Guralnik, J.M., Simonsick, E., Salive, M.E., Corti, C., Langlois, J.R. 1996 “Progressive versus catastrophic disability: a longitudinal view of the disablement process,” Journal of Gerontology. Series A, Biological Sciences & Medical Sciences. 51 123 130 Google Scholar Fried, L.P., Ettinger, W.H., Hermanson, B., et al. 1994 “Physical disability in older adults: a physiologic approach,” Journal of Clinical Epidemiology. 47 747 760 Google Scholar Geman, S., Hwang, C.-R. 1982 “ Nonparametric maximum likelihood estimation by the method of sieves,” Annals of Statistics. 10 401 411 Google Scholar Genest, C., MacKay, R.J. 1986 “The joy of copulas: bivariate distributions with given marginals,” The American Statistician. 40 280 283 Google Scholar Genest, C., Rivest, L.P. 1993 “Statistical inference procedures for bivariatearchimedan copulas,” Journal of the American statistical Association. 88 1034 1043 Google Scholar Glidden, D.V. 1999 “Checking the adequacy of the gamma frailty model for multivariate failure times,” Biometrika. 86 381 393 Google Scholar Gong, G., Samaniego, F.J. 1981 “Pseudo maximum likelihood estimation: theory and applications,” The Annals of Statistics. 9 861 869 Google Scholar Guralnik, J.M. Fried, L.P. Simonsick, E.M. Kasper, J. Lafferty, M.E. eds. 1995The Women’s Health and Aging Study: Health and Social Characteristics of Older Women with Disability National Institute on Aging Bethesda, Maryland Google Scholar Hall, P., Park, B.U., Turlach, B.A. 1998 “A note on design transformation and binning in nonparametric curve estimation,” Biometrika. 85 469 476 Google Scholar Hougaard, P. 1986a “Survival models for heterogeneous population derived from stable distributions,” Biometrika. 73 387 396 Google Scholar Hougaard, P. 1986b “A class of multivariate failure time distribution,” Biometrika 73 671 673 Google Scholar Huster, W.J., Brookmeyer, R, Self, S.G. 1989 “Modelling paired survival data with covariates,” Biometrics. 45 145 156 Google Scholar Kaplan, E.L., Meier, P. 1958 “Nonparametric estimation from incomplete observations,” Journal of the American Statistical Association. 53 457 481 Google Scholar Kendall, M.G. 1938 “A new measure of rank correlation,” Biometrika. 30 81 93 Google Scholar Liang, K.-Y. 1991 “Estimating effects of probands” characteristics on familial risk: I Adjustment for censoring and correlated ages at onset Genetic Epidemiology. 8 329 338 Google Scholar Liang, K.-Y., Self, S.G., Bandeen-Roche, K.J., Zeger, S.L. 1995 “Some recent developments for regression analysis of multivariate failure time data,” Lifetime Data Analysis. 1 403 415 Google Scholar Lonergan, E.T., Krevans, J.R. 1991 “A national agenda for research on aging,” New England Journal of Medicine. 1 1825 1828 Google Scholar Manatunga, A.K., Oakes, D. 1996 “A measure of association for bivariate frailty distributions,” Journal of Multivariate Analysis. 56 60 74 Google Scholar Marshall, A.W., Olkin, I. 1988 “Families of multivariate distributions,” Journal of the American Statistical Association. 83 834 841 Google Scholar Oakes, D. 1989 “Bivariate survival models induced by frailties,” Journal of the American Statistical Association. 84 487 493 Google Scholar Pizer, S.M. 1975“Numerical Computing and Mathematical Analysis,” Science Research Associates, Inc. Chicago Google Scholar Prentice, R.L., Kalbfleisch, J.D., Peterson, A.V., Jr., Flournoy, N., Farewell, V.T., Breslow, N.E. 1978 “The analysis of failure times in the presence of competing risk,” Biometrics. 34 541 554 Google Scholar Reinsch, C.H. 1967 “Smoothing by spline functions,” Numerische Mathematik. 10 177 183 Google Scholar
Schweizer B., Sklar A. (1983). Probabilistic Metric Spaces, New York: North-Holland, 1983.
Shih, J.H., Louis, T.A. 1995 “ Inference on the association parameter in copula models for bivariate survival data,” Biometrics. 51 1384 1399 Google Scholar Shih, J.H., Louis, T.A. 1995 “Assessing gamma frailty models for clustered failure time data,” Lifetime Data Analysis. 1 205 220 Google Scholar Shih, J.H. 1998 “A goodness-of-fit test for association in a bivariate survival model,” Biometrika. 85 189 200 Google Scholar Viswanathan, B., Manatunga, A.K. 2001 “Diagnostic plots for assessing the grailty distribution in multivariate survival data,” Lifetime Data Analysis. 7 143 155 Google Scholar Wand, M.P. 1997 “Data-based choice of histogram bin width,” The American Statistician. 51 59 64 Google Scholar Wang, W., Wells, M.T. 2000a “Model selection and semi-parametric inference for bivariate censored data,” Journal of the American Statistical Association. 95 62 72 Google Scholar Copyright information
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