Bayesian Empirical Likelihood Inference For Censored Data With Missing Information

In the survival analysis,the failure time is usually censored,and missing data generally exist.Analyzing such complex data has vital practical significance.Several semi-parametric models were included in the survival analysis to establish the relationship between survival time and covariates.For the censored data with missing information,it is difficult to perform Bayesian estimation on semi-parametric models.This is mainly due to the method to handling missing variables under the Bayesian framework is single,and it is easy to get biased estimates.Bayesian empirical likelihood has the potential to solve this problem.Bayesian empirical likelihood combines the empirical likelihood function with the prior information and does not need to make assumptions about the semi-parametric model and missing covariates.Therefore,this paper uses the Bayesian empirical likelihood and Bayesian jackknife empirical likelihood to infer the semi-parametric model under the right-censored data with missing covariates and right-censored data with missing censoring indicator.This paper provides a practical way to perform Bayesian estimation of the semi-parametric model under censored data with missing information and carries out three parts of research,as follows:1.Bayesian empirical likelihood of accelerated failure models for right-censored data.Firstly,a semi-parametric accelerated failure time model was established between the failure time and covariates.Based on the Buckley-James estimation equation,the constraint condition in empirical likelihood function was obtained by changing the order of summation.An empirical likelihood similar to the likelihood function of right-censored data was established.After introducing the prior distribution,Bayesian empirical likelihood was constructed.Furthermore,an efficient Metropolis-Hasting sampling algorithm was given.In the simulation studies,we compare the performance of Buckley-James estimation,Bayesian,and Bayesian empirical likelihood on point estimation.Then we compared empirical likelihood,Bayesian,and Bayesian empirical likelihood on interval estimation in finite samples.To examine the dependence of Bayesian empirical likelihood on prior,we consider informative and uninformative priors respectively.Bayesian empirical likelihood estimation has shown more accuracy and stability in point estimation,better than empirical likelihood in interval estimation,and less dependence on the prior.Finally,the Bayesian empirical likelihood method was applied to analyze the gastric cancer data.2.Bayesian empirical likelihood of accelerated failure time models with missing covariates.In the case of right-censored data with missing covariates at random,the unbiased transformation of right-censored data was first considered.The inverse probability weighted least squares estimation equation was obtained by adjusting the missing covariates with inverse probability weighting.Then,a inverse probability weighted Buckley-James estimation equation was given.and changed the order of summation was to obtain a form consistent with the empirical likelihood assumptions.The empirical likelihood was established based on the two equations.The Bayesian empirical likelihood was constructed by introducing the prior,and the proof of the large sample properties for posterior distribution was given.In the simulation study,the results of Bayesian estimation and these two kinds of Bayesian empirical likelihood on semi-parametric accelerated failure time model under different types of censored ratio,missing ratio,and prior distribution were compared,and the implementation of these methods under a setting like the actual data from mouse leukemia study was given.The Bayesian estimation and Bayesian empirical likelihood based on inverse probability weighted least squares estimation can only obtain accurate estimation results in some cases,while Bayesian empirical likelihood estimation based on inverse probability weighted Buckley-James estimation shows strong stability and accuracy.Bayesian empirical likelihood estimation based on inverse probability weighted Buckley-James estimation can effectively perform Bayesian inference on the semi-parametric accelerated failure model with missing covariates.In the real data analysis,two studies were performed on the mouse leukemia dataset by assuming different endpoints.3.Bayesian jackknife empirical likelihood of linear transformation model with missing censoring indicators.Firstly,a semi-parametric linear transformation model was established for right-censored data with the censoring indicators missing at random.The model included several standard semi-parametric models.Then regression imputation and kernel imputation were used to complete the missing censoring indicator.Then the empirical likelihood of the regression coefficient was established based on a U-type estimation equation,and the Bayesian empirical likelihood and its sampling algorithm were given.In addition,based on the same estimation equation,the jackknife empirical likelihood was constructed to adjust the deviation caused by nonlinear constraints in the empirical likelihood optimization.The Bayesian jackknife empirical likelihood and its sampling algorithm were established.In the finite sample,under different distributions of error,types of covariable,censoring ratios,and missing ratios,the Bayesian jackknife empirical likelihood has shown excellent estimation performance,especially in interval estimation.Moreover,the selection strategies of regression imputation and kernel imputation in different scenarios were obtained by simulation results.Finally,the methods were applied to analyze a stage II breast cancer data.The results are consistent with medical knowledge and provide some practical value.This paper verifies the feasibility of Bayesian empirical likelihood in the semi-parametric model estimation of censored data and censored data with missing information through three parts of research.Bayesian empirical likelihood does not need to make assumptions about missing variables and unknown items in the model.It can effectively combine weighted adjustment and imputaion to construct the likelihood function more flexibly.By comparing with Bayesian estimation methods,it is found that Bayesian empirical likelihood has significant advantages,especially in interval estimation.