Abstract
In biomedical studies, frailty models arecommonly used in analyzing multivariate survival data, wherethe objective of the study is to estimate both the covariateeffect and the dependence between the multivariate survival times.However, inference based on these models are dependent on thedistributional assumption of frailty. We propose a diagnosticplot for assessing the frailty assumption. The proposed methodis based on the cross-ratio function and the diagnostic plotsuggested by Oakes (1989). We use kernel regression smoothingwith bandwidth choice by cross-validation, to obtain the proposedplot. The resulting plot is capable of differentiating betweenthe gamma and positive stable frailty models when strong associationis present. We illustrate the feasibility of our method usingsimulation studies under known frailty distributions. The approachis applied to data on blindness for each eye of diabetic patientswith adult onset diabetes and a reasonable fit to the gamma frailtymodel is found.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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 vol. 65 pp. 141–151, 1978.
D. M. Dabrowska, “Kaplan-Meier estimate on the plane,” Ann. Statist. vol. 16 pp. 1475–1489, 1988.
C. Genest and L.-P. Rivest, “Statistical inference procedures for bivariate Archimedean copulas,” J. Amer. Statist. Assoc. vol. 88 pp. 1034–1043, 1993.
C. Genest and R. J. MacKay, “The joy of copulas: Bivariate distributions with given marginals,” The American Statistician vol. 40 pp. 280–283, 1986.
D. V. Glidden, “Checking the adequacy of the gamma frailty model for multivariate failure times,” Biometrika vol. 86 pp. 381–393, 1999.
E. J. Gumbel and C. K. Mustafi, “Some analytic properties of bivariate external distributions,” J. Amer. Statist. Assoc. vol. 62 pp. 569–588, 1967.
W. Härdle, Applied Nonparametric Regression, Cambridge University Press, 1990.
P. Hougaard, “Survival models for heterogeneous populations derived from stable distributions,” Biometrika vol. 73 pp. 387–396, 1986a.
P. Hougaard, “A class of multivariate failure time distributions,” Biometrika vol. 73 pp. 671–678, 1986b.
W. J. Huster, R. Brookmeyer and S. G. Self, “Modeling paired survival data with covariates,” Biometrics vol. 45 pp. 145–156, 1989.
M. G. Kendall, “A new measure of rank correlation,” Biometrika vol. 30 pp. 81–93, 1938.
J. F. Lawless, Statistical Models and Methods for Lifetime Data, Wiley: New York, 1982.
L. Lee, “Multivariate distributions having Weibull properties,” J. Mult. Anal. vol. 9 pp. 267–277, 1979.
K.-Y. Liang, “Estimating effects of probands' characteristics on familial risk: I. Adjustment for censoring and correlated ages at onset,” Genetic Epidemiology vol. 8 pp. 329–338, 1991.
A. K. Manatunga and D. Oakes, “A measure of association for bivariate frailty distributions,” J. of Multivariate Analysis vol. 56 pp. 60–74, 1996.
D. Oakes, “A model for association in bivariate survival data,” J. R. Statist. Soc. B vol. 44 pp. 414–422, 1982.
D. Oakes, “Semiparametric inference in a model for association in bivariate survival data,” Biometrika vol. 73 pp. 353–361, 1986.
D. Oakes, “Bivariate survival models induced by frailties,” J. Amer. Statist. Assoc. vol. 84 pp. 487–493, 1989.
D. Oakes and A. K. Manatunga, “A Fisher information for a bivariate extreme value distribution,” Biometrika vol. 79 pp. 827–832, 1992.
A. E. Pulver and K.-Y. Liang, “Estimating effects of proband characteristics on familial risk: II. The association between age at onset and familial risk in the Maryland schizophrenia sample,” Genetic Epidemiology vol. 8 pp. 339–350, 1991.
P. J. Rousseeuw, “Least median of squares regression,” J. Amer. Statist. Assoc. vol. 79 pp. 871–880, 1984.
J. H. Shih and T. A. Louis, “Inferences on the association parameter in copula models for bivariate survival data,” Biometrics vol. 51 pp. 1384–1399, 1995a.
J. H. Shih and T. A. Louis, “Assessing gamma frailty models for clustered failure time data,” Lifetime Data Analysis vol. 1 pp. 205–220, 1995b.
G. C. Stone, Statistical analysis of accelerated aging tests on solid electrical insulation, Unpublished M.A. Sc. thesis, University of Waterloo, Waterloo, Ontario, Canada, 1978.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Viswanathan, B., Manatunga, A.K. Diagnostic Plots for Assessing the Frailty Distribution in Multivariate Survival Data. Lifetime Data Anal 7, 143–155 (2001). https://doi.org/10.1023/A:1011348823081
Issue Date:
DOI: https://doi.org/10.1023/A:1011348823081