CHECKING FOR THE GAMMA FRAILTY DISTRIBUTION UNDER THE MARGINAL PROPORTIONAL HAZARDS FRAILTY MODEL

The marginal proportional hazards frailty model for multivariate failure time data characterizes the intracluster dependency with the frailty model while formulating the marginal distributions with the proportional hazards model. The gamma frailty distribution has been widely used to model intracluster dependency because of its simple interpretation and mathematical tractability. Glidden (2000) proposed a two-stage method for estimating the dependence parameter under the marginal proportional hazards frailty model when the frailty follows a gamma dis- tribution. The goodness of fit test for the marginal proportional hazards model has been proposed by Spiekerman and Lin (1996). In this paper, we provide a graphical as well as a numerical method for checking the adequacy of the gamma frailty distribution. The test process is derived from the posterior expectation of the frailty given the observable data. The critical value can be obtained by a Monte Carlo simulation. Two examples from genetics studies are provided to illustrate the proposed testing procedure.

[1]  O. Aalen,et al.  Heterogeneity in survival analysis. , 1988, Statistics in medicine.

[2]  David V. Glidden.  Checking the adequacy of the gamma frailty model for multivariate failure times , 1999 .

[3]  P. Hougaard.  Survival models for heterogeneous populations derived from stable distributions , 1986 .

[4]  D. Oakes,et al.  Semiparametric inference in a model for association in bivanate survival data , 1986 .

[5]  D G Clayton,et al.  A Monte Carlo method for Bayesian inference in frailty models. , 1991, Biometrics.

[6]  Susan A. Murphy,et al.  Asymptotic Theory for the Frailty Model , 1995 .

[7]  D. Oakes,et al.  Bivariate survival models induced by frailties , 1989 .

[8]  P. Hougaard.  A class of multivanate failure time distributions , 1986 .

[9]  G. Vogler,et al.  Estimating effects of proband characteristics on familial risk: II. The association between age at onset and familial risk in the Maryland schizophrenia sample , 1991, Genetic epidemiology.

[10]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[11]  D. Lin,et al.  Cox regression analysis of multivariate failure time data: the marginal approach. , 1994, Statistics in medicine.

[12]  Checking the marginal Cox model for correlated failure time data , 1996 .

[13]  S. Self,et al.  Semiparametric Likelihood Estimation in the Clayton–Oakes Failure Time Model , 1999 .

[14]  Z. Ying,et al.  Checking the Cox model with cumulative sums of martingale-based residuals , 1993 .

[15]  T. Louis,et al.  Inferences on the association parameter in copula models for bivariate survival data. , 1995, Biometrics.

[16]  Modeling Multivariate Survival Data by a Semiparametric Random Effects Proportional Odds Model , 2002, Biometrics.

[17]  Richard D. Gill,et al.  A counting process approach to maximum likelihood estimation in frailty models , 1992 .

[18]  David V. Glidden,et al.  A Two-Stage Estimator of the Dependence Parameter for the Clayton-Oakes Model , 2000, Lifetime data analysis.

[19]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[20]  L. J. Wei,et al.  Regression analysis of multivariate incomplete failure time data by modeling marginal distributions , 1989 .

[21]  D. Clayton,et al.  Multivariate generalizations of the proportional hazards model , 1985 .

[22]  Susan A. Murphy,et al.  Consistency in a Proportional Hazards Model Incorporating a Random Effect , 1994 .

[23]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[24]  Odd O. Aalen,et al.  Modelling Heterogeneity in Survival Analysis by the Compound Poisson Distribution , 1992 .

[25]  N Mantel,et al.  Mantel-Haenszel analyses of litter-matched time-to-response data, with modifications for recovery of interlitter information. , 1977, Cancer research.

[26]  P. Billingsley,et al.  Probability and Measure , 1980 .

[27]  I. McKeague,et al.  Identification of Nonlinear Time Series from First Order Cumulative Characteristics , 1994 .

[28]  D. Clayton.  A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence , 1978 .

[29]  Thomas A. Louis,et al.  Assessing gamma frailty models for clustered failure time data , 1995, Lifetime data analysis.

[30]  D. Pollard.  Convergence of stochastic processes , 1984 .

[31]  N. Mantel,et al.  Use of logrank scores in the analysis of litter-matched data on time to tumor appearance. , 1979, Cancer Research.

[32]  D. Oakes.  A Model for Association in Bivariate Survival Data , 1982 .

[33]  Danyu Lin,et al.  Marginal Regression Models for Multivariate Failure Time Data , 1998 .