Regression Analysis And Goodness-of-fit Test For Bivariate Interval-censored Data Under Copula Models

In recent years,the model development and regression analysis of interval-censored failure time data have received extensive attention.Among those,the semiparametric model is one of the hot research topics.Interval-censored data usually refers to the situation where the failure time of interest cannot be exactly observed and instead is known to occur in an interval,which often occurs in many areas including biomedical or epidemiological research.If there is only one event of interest,it is often referred to as univariate interval-censored data.However,in real life,there are often two correlated failure events of interest,and the observations on both failure events suffer interval censoring.We often refer to such data as bivariate interval-censored data.For example,in clinical trials of ophthalmic diseases,the lesion times of the left and right eyes of the same patient are likely to be correlated,which can easily generate bivariate interval-censored data.Many scholars have proposed some models that can be used for regression analysis of univariate interval-censored data,such as the additive hazards model,the proportional hazards(PH)model,the proportional odds(PO)model,the linear transformation model,and so on.However,for regression analysis of bivariate interval-censored data,an important goal is to deal with the correlation between the data,and the model building thus becomes complicated,so scholars often suppose the covariates only have linear effects to simplify the problem and ignore nonlinear covariates effects.However,in fact,nonlinear covariate effects are common in practice,and ignoring nonlinear covariates can easily lead to inaccurate analysis results.This thesis discusses two copula-based semiparametric partly linear models.They are partly linear additive hazards models and partly linear transformation models.The proposed method not only can flexibly handle dependence structures but also allows covariates that may have nonlinear effects.In addition,a goodness-of-fit test for testing the copula model related to bivariate interval-censored failure time data is considered.The main contents are divided into three chapters as follows.In the second chapter of this thesis,a kind of copula-based partly linear additive hazards model is proposed,which considers both time-dependent covariates and covariates that may have nonlinear effects.For the estimation of the model parameters,a sieve maximum likelihood estimation approach based on Bernstein polynomials is proposed to approximate the baseline cumulative hazard functions and nonlinear covariate effects.The resulting estimators of regression parameters are shown to be consistent,asymptotically efficient and normal.Subsequently,simulation study is conducted to assess the performance of this method and the results show that it is effective in practice.Finally,we apply the proposed method to Age-related Eye Disease Study.In the third chapter of this thesis,two-parameter copula-based partly linear transformation models are proposed.This model contains not only the commonly used PH model and PO model,but also a broader class of models,and is more applicable to situations where there is a multiplicative effect of covariates on risk.Similarly,the model also considers the potential nonlinear effects and makes use of Bernstein polynomials,also a sieve maximum likelihood estimation approach is provided.The resulting estimators of regression parameters are shown to be consistent and asymptotically efficient and normal.Simulation results show that the proposed method is effective in finite samples.Finally,this paper also applies it to a set of real data arising from dental research.In the fourth chapter of this thesis,we discuss the goodness-of-fit test based on semi-parametric copula models for bivariate interval-censored data.The copula-based approach is one of the common methods used to deal with correlation between bivariate data and is also the method used in the first two chapters.However,there are many kinds of copula functions,and the use of inappropriate copula functions can lead to wrong analysis results.For the problem,we propose three test statistics or procedures,two based on the pseudo in-and-out-sample(PIOS)likelihood approach and the other based on the information ratio(IR)method.The asymptotic properties of the proposed test procedures are established and in particular,the three methods are shown to be asymptotically equivalent.Numerous numerical simulations show that the proposed tests can better control the size.In addition,the power of the tests performs well under the given alternative hypotheses.Finally,the three tests are validated with examples.