Statistical Analysis Of Quantile Residual Life Models With Complex Data

Residual life refers to how long the individual can survive if the individual has lived for a period,ie how long the remaining life is.Residual life has a wide appli-cation in the biomedical and financial fields.For example,in medicine,the patients,especially those who suffer from a chronic disease like cancer,are interested in know-ing their remaining lifetime.Besides,doctors also need to know how these diseases develop,whether the treatments they are taking are effective,and whether applying new treatments can extend the life of patients.All this requires to know patient's residual life.Residual life is a conditional characteristic of the individual life.Re-cently,there are two main aspects of the research on residual life.One is the mean residual life and the other is the quantile residual life.Although there are many re-search results of the mean residual life,there are many disadvantages of the mean residual life.First,the mean residual life does not always exist when the distribution is heavy tailed.Secondly,when the distribution is skewed or non-symmetric,in this case,an individual with a longer life will have a greater impact on the mean residual life,and the mean residual life is sensitive to the outliers.But the quantile residual life is not very sensitive to this.And also residual life is easy to understand and has a wide range of applications.Therefor,the quantile residual life model has received a lot of attention.This thesis studies the quantile residual life regression models and semiparametric models under the length-biased right censored data and the biased-sampling right censored data.Firstly,we studies quantile residual life regression models under the length-biased and biased-sampling right censored data.We consider the two cases that the censored variables and covariates are independent or independent.And then we extend the models to a more general case,the residual life semiparametric models with length-biased right censored data and the biased-sampling right censored data.The semiparametric model is more flexible and can cover more models.In statistical analysis,we need to collect,organize,and analyze data firstly.Complex data structures will affect the establishment of statistical models,so it is necessary to establish appropriate statistical models based on different data charac-teristics.In chapter 1 we mainly introduces the characteristics of complex data,the background of residual life and the current research on it,as well as the motivation,main content,and some innovations of this thesis.When the probability that an individual is sampled depends on its own value,that is,the probability that each individual is sampled is different,the data is biased-sampling data.The original statistical inference procedure for simple data is no longer applicable with biased-sampling data.We must find new methods for biased-sampling data.Left truncation data means only the subject satisfies some condition and then it can be recruited into study.Moreover,length-biased data is a special kind of biased-sampling and left-truncated data,which satisfy the stationary condition and the probability of ob-serving a subject is proportional to his or her lifetime.Under this assumption,disease incidence is assumed to follow a stationary Poisson process and occurs at a constant rate during some period.The length-biased data with longer lengths or longer sur-vival times will over-represent the target population.In survival analysis,when the experiment or study ends,some individuals do not have events of interest,and there-fore cannot accurately observe the exact time at which the event occurs.Censored data arises when an individual's life length is known to occur only in a certain period of time.One of the possible censoring schemes is right censoring,where all that is known is that the individual is still alive at a given time.The censored data provides certain information for the study of the event.Simply removing the censored data or treating the censored data as complete data will cause bias.Besides,length-biased data can be further complicated by right-censoring.The biased-sampling data no longer obeys its overall model and changes the structure of the model.Residual life is easy to understand and has a wide range of applications.Previous studies on residual life frequently used the mean residual life and the quantile residual life.The quantile regression model directly models the conditional quantiles of the remaining life,and the results are easier to explain.The quantile regression is more flexible and robust than the mean and median of the remaining life.It can fully capture the characteristics of the survival distribution.Such as the effect of covariates are allowed to change,the survival distribution is allowed to different tails and it is easier to grasp data heteroscedasticity.In chapter 2,we consider a conditional log-linear regression model on the residual lifetimes at a fixed time point under right-censored and length-biased data for both covariate-independent censoring and covariate-dependent censoring.When censoring depends on the covariates,we assume that the censoring variable and the covariates satisfy the Cox model.Consistency and asymptotically normalities of the regression estimators are established.The estimation of the variance often is not straightforward since it involves the estimation of the density function under incomplete data.In our paper,the variance is obtained by a resampling approach proposed by Parzen,Wei&Ying(1994),which can be implemented easily and efficiently.Simmulation studies are performed to assess finite sample properties of the regression parameter estimator.Finally,we analyze the OSCAR real data by the proposed method.In chapter 3,we extend the method of chapter 2 to the case of biased-sampling and right censored data.First,we give the estimating equations for both covariate-independent censoring and covariate-dependent censoring.Because the estimating equations are not smooth functions of the parameters,it is difficult to solve the param-eters.In this chapter,we transform the problem of solving estimating equations into locally minimizing the L1 type convex function.Secondly,consistency and asymp-totically normalities of the regression estimators are given under certain conditions and the estimator of the asymptotic variance is given.To estimate the asymptotic variance of the parameter,we need to estimate the unbiased density function of the failure time.Generally we use non-parametric kernel estimator which makes the es-timation of the asymptotic variance too complete.So we use the bootstrap method to estimate the asymptotic variance of the parameters as in chapter 2.Further,the simulation results are given for different biased weight functions.Finally,we prove the related lemmas and theorems.In chapter 4,we first extend the residual life quantile regression model of chapter2 to a semiparametric model under the length-biased and right censored data.Sec-ondly,we similarly give the residual life semiparametric model under biased-sampling and right censored data.The semi-parametric model is between parametric regression model and non-parametric regression model and can accommodate flexibly when the data's distribution are unknown or not satisfying the given hypothesis.This chapter also gives the estimation equations for the cases where the censoring variable and the covariates are independent or dependent.The estimation equation is not a smooth function of the parameters,and contains nonparametric component8)₀().We solve the estimating estimation in two steps.In the first step,given the parameter,we derive an estimator of8)₀()point by point by constructing an estimation equation.Observed that the covariate effect ofis the same for different t.Making use of this information,another estimating equation is constructed to get a more reasonable estimator ofin the second step.It is well known that there are usually two difficul-ties in dealing with quantile problem.Firstly,because the estimating equation is a discontinuous function of,it causes great trouble to get the estimator.By apply-ing the majorize-minimize(MM)algorithm,introduced by Hunter&Lange(2000),to our discontinuous estimating equation,we obtain the estimator of.Secondly,another fact involving quantile residual is that the estimation of the variance often is not straightforward since it involves the estimation of the density function under incomplete data.In our paper,the variance is obtained by a resampling approach proposed as in chapter 2.In this chapter,we also give the large sample properties of the parameter estimates in both cases,namely the consistency and asymptotic normality.Finally,we give the simulation results and prove the related lemmas and theorems.In chapter 5,we summarized our work and made statements on further works.