Statistical Inference Of Semi-varying Coefficients Frailty Models For Clustered Failure Time Data

The frailty models and the marginal models are the most popular models in analyzing clustered failure time data,where frailty models describe the dependence among individuals in the same cluster by introducing the random effect term into the traditional survival models.Frailty models can not only improve the accuracy of the estimation for covariate effects,but also provide the estimation for the variability of random effect.Traditional survival models often assume the covariate effects are constants in practical applications.However,the covariate effects may not be fixed but vary with the value of the covariates in many practical problems,and even rely on other covariates or the survival time.The non-parametric survival models can be used to handle this issue with enough flexibility,but we will encounter the curse of dimensionality when the variable dimensions increase.The varying coefficients models assume that the effects of some covariates are univariate functions of other covariate,thus effectively avoiding this problem effectively while maintaining the flexibility of the models.In this paper,a set of semi-varying coefficients frailty models are proposed and investigated,including the proportional hazard frailty models,the additive hazard frailty model and the accelerated failure time frailty model.Firstly,we study the semi-varying coefficients gamma frailty models,and achieve the estimators for model parameters based on B-spline,penalized partial likelihood and the profile likelihood.The performance of the proposed method in finite sample is evaluated by the simulation studies.Then,we apply the method to analyze the advanced lung cancer data,which provides a more accurate interpretation of covariate effects than the traditional models.Secondly,we study multivariate normal frailty models with semi-varying coefficients.To solve the problem that there is no explicit expression of the integrated likelihood function,the Laplace approximation is used to obtain the approximate expression of the likelihood function.The varying coefficients and constant coefficients are fitted using B-splines and the penalized partial likelihood.Moreover,the frailty parameter is estimated by maximizing an approximate profile likelihood function.The performance of the proposed methods is assessed with simulation studies and real data.Thirdly,we develop an additive hazards frailty model with time semi-varying coefficients.We show that the constant coefficients and time-varying coefficients could be obtained from the estimating equations,whereas the frailty parameter being estimated by cross moment methods.The consistency and asymptotic normality of the estimators are proved by the empirical process theory.The finite sample properties of the estimators are studied in a simulation study.The proposed model is illustrated with rehospitalization colorectal cancer study data,and a new interpretation of the effect of covariates on the risk of readmission in patients is presented.Finally,a kind of accelerated failure time frailty models with semi-varying coefficients are proposed,we utilize the pseudo-response variable to replace the log survival time and adopt the B-spline method to approximate the varying coefficients.Then,the hierarchical likelihood function and the iterative weighted least squares estimating equations are secured.The frailty parameter and error parameter can be estimated by restrictive hierarchical likelihood function.Simulation studies reveal that the estimators of the method are stable,with very small deviation,and suitable for high censoring rate data.Then,the proposed model is used to analyze the data of elderly patients with bladder cancer in SEER database,describing the dynamics of the influence of sex on survival time with the ages of patients getting the disease.