Bayesian Empirical Likelihood Statistcal Inference Regression Model With Censored Data

The censored data in survival analysis has always been a hot topic in the field of statistics.Such data exists in many scientific research fields,such as demography,finance,biomedicine and social science.In survival analysis,the variables of interest are survival time,and there are many variables related to survival time,which affect the change of survival time in different forms.The ultimate goal of studying these factors is to find an optimal treatment plan to prolong the survival time of patients.Therefore,its research was very extensive and meaningful on such data.Meaningful.On the other hand,there are also high-dimensional data in survival analysis,that is,the number of unknown parameters to be estimated is close to or far beyond the size of the sample size,which may be tens or exponential times of the sample size.Because it is common in many fields,it is of great significance and practical value to carry out such data research.Regression model based on censored data is divided into parametric model,semi-parametric model and nonparametric model.Parametric model will produce deviation because of the incorrect probability distribution.Nonparametric model can overcome this shortcoming.However,the computational cost of nonparametric model is too high.Semiparametric model combines parametric model and nonparametric model to over-come the above shortcomings.Empirical likelihood function in nonparametric model can test hypothesis and construct confidence interval without estimating variance,be-cause the likelihood function does not involve the limit distribution of unknown variance and empirical likelihood.Likelihood function is also the core part of Bayesian theory.Bayesian statistics generates posterior inferences by updating prior mean,parameters and data.Bayesian empirical likelihood can combine nonparametric likelihood theory with prior information,taking into account the robustness and efficiency of statistical inference.Bayesian estimation method regards the parameters concerned as random variables and applies empirical information to them.In models,the parameters to be estimated can be better estimated.Therefore,the statistical model under the Bayesian empirical likelihood framework can be used to analyze the relationship between survival time and factors affecting survival time.Survival analysis based on Bayesian empirical likelihood can better model to estimate,and its application prospect is considerable.Therefore,parameter estimation and variable selection of regression model are t-wo important issues in survival analysis.This paper mainly studies Bayesian empirical likelihood and Bayesian empirical likelihood variable selection methods(parametric model,semiparametric model and transformation model)for several regression models with censored data.Through the design of specific priori and algorithm,Bayesian empirical likelihood method can effectively combine the information of prior and likeli-hood to obtain more efficient inference.The main content of this paper is divided into three parts,specifically.As follows:1 Bayesian Empirical Likelihood Inference of Linear Model with Right Censored Data.Firstly,Bayesian empirical likelihood confidence interval of model parameters is constructed,the corresponding algorithm is given,and then the asymptotic posterior distribution of Bayesian empirical likelihood is given.The convergence of the algorithm is demonstrated by simulation,and the coverage of the method is compared with the other two confidence intervals(empirical likelihood and Wald likelihood).Bayesian empirical likelihood confidence Interval has more accurate coverage.In the simulation study,we compare with the classical empirical likelihood method,the adjusted empiri-cal likelihood method and the normal approximation method,and obtain the coverage under different censored ratios and different distributions under some different sam-ple sizes,the trajectory graph of the sample chain and ergodic mean graph with QQ and M-H algorithms.The data of advanced acute myelogenous leukemia showed that there was a significant difference in the treatment failure rate between the two types of transplants.Under the same sampling chain length,the confidence interval of au-tologous transplantation is obviously smaller than that of allogeneic transplantation.Then based on spike-and-slab prior,the Bayesian variable selection of linear model parameters under censored data is given by using Bayesian hierarchical model.In the simulation,the effect of popular algorithm and Bayesian empirical likelihood variable selection algorithm in limited samples is demonstrated,which shows that Bayesian variable selection has higher accuracy and correct recognition rate.Using the Bayesian variable selection method and the existing variable selection methods(LASSO variable selection,ALASSO variable selection,SCAD variable selection).The average absolute deviation(MMAD)of the estimated values under different sample sizes and censored ratios was compared,and the mean values of true value(TP)and false value(FP)were selected.The right censored data of bone marrow transplantation in leukemia patients were validated in the empirical analysis,and the empirical results were very significant.2 Bayesian Empirical Likelihood Inference of Semiparametric Regres-sion Model with Right Censored Data.Firstly,the Bayesian empirical likelihood confidence interval and corresponding algorithm of model parameters are constructed,and the asymptotic posterior distribu-tion of Bayesian empirical likelihood is given.In semiparametric regression model with right censored data,the advantage of comparison between Bayesian empirical likeli-hood method and empirical likelihood method are obvious and the complex variance calculation are avoided.In the simulation,as the sample size increases exponentially,the deviation and mean square error decrease exponentially,indicating that the esti-mated value is a consistent estimate of the true value.We also compare the coverage rates of the four methods under different sample sizes,different censored ratios and different confidence levels.The results show that the coverage rates of the some meth-ods increase with the increase of sample sizes,and with the increase of censored ratios,the coverage rates of Bayesian empirical likelihood are obviously better than those of other methods.Given prior and proposed distributions,the extraction of Markov chains by M-H algorithm is also stable and convergent.Analysis of the example result shows that blood sugar is negatively correlated with the onset of diabetic complications stroke,and cholesterol is positively correlated with the onset of diabetic complications stroke.Under the same chain length,the confidence interval of blood sugar is obviously smaller than that of cholesterol.Cholesterol is more active in the onset of stroke.The output results are consistent with the actual analysis results,so the Bayesian empirical likelihood method proposed by us has practical value in the practical life.3 Bayesian Empirical Likelihood Inference of the Transformation Model with Interval Censored Data.Bayesian empirical likelihood based on interval censored data for transformation model is constructed.Compared with the classical empirical likelihood,the proposed method is more efficient.In the simulation,under the same conditions,the conver-gence accuracy of Bayesian empirical likelihood method is generally higher than that of ordinary empirical likelihood method.It shows that the Bayesian empirical likelihood method based on interval censored transformation model proposed by us has good properties and can estimate the model more accurately.Moreover,Bayesian empirical likelihood method can estimate the model more accurately.The estimated value under the method is closer to the true value,and the deviation and mean square error under the same basic condition are less than the results of empirical likelihood.According to the real medical data of HIV infection in hemophilia,the empirical analysis results show that the treatment of low dose factor VIII concentrate has a certain impact on HIV infection in hemophilia patients.With the increase of sampling chain length,the estimation of parameters becomes more and more accurate and the confidence interval length becomes shorter.The stationarity of sampling chain can also be seen from the path map.There is no doubt that the results obtained by applying these methods to the complete data model are significant,but in the case of high dimensional data and high censored,the results presented in this paper are significant.The results show that the Bayesian empirical likelihood method and Bayesian empirical likelihood variable selec-tion in linear regression model,semiparametric regression model and transformation model can be well applied,and more ideal results can be obtained.Therefore,Bayesian empirical likelihood method is used,it is of great practical significance to study the regression models of right censored data and interval censored data.