The Study On Bayesian Proportional Hazard Model For Interval Censored Data And Its Application

In clinical follow-up study,interval censoring is a common censoring type besides right censoring.Cox proportional hazard model is an important basic method for survival analysis.The most easy-completing and understood methods,imputation can utility the traditional cox propotoinal harzard model by using left-point or mid-point to replace censored value.But the underestimation or overestimation problem can not be avoided.Therefore,traditional imputation method is not an ideal method to analyze interval-censoring data.From frequentist statistical method,baseline cumulative hazard function and the distribution of survival time should be defined for model fitting.For Bayesian statistical methods,the complicated likelihood fuction including both basiline cumulative hazard function and survival time distribution can bring heavy computing burden.This study explored the performance and practical application of a Bayesian proportional hazard model proposed in recent years by Lin,in which the monotone spline was applied to construct the baseline cumulative hazard function and poisson progress was taken for data augument for Gibbs posterior sampling.A large number of simulated samples with different characteristics were generated to compared Bayesian methods with multiple imputation and a classical MLE method.The Bayesian proportional hazards model method also was applied a clnical follow-up studyto explore the related risk factors of recurrence of cardiovascular disease after hospitalized treatment.in order to demonstrate the application of the method.Results(1)Simulated data analysisA.Influence of different sample characteristics on Bayesian proportional hazards modelMain results showed that the model had no obvious regularity and difference under the censored interval width of 10,50,100 and 200.In the interval-censoring rate groups of 0.2,0.5and 0.8,the bias of the model was also not different.There were other findings that with the increase of the interval width,the width of 95%confidence interval was also increasing.The Bias of regression coefficients of variable distributed as normal distribution was larger than those of the variable distributed as binomial distribution.However,the SE of regression coefficients of variable distributed as normal distribution was smaller than that distributed as binomial distribution.B.Comparisons among the bayesian method,the multiple imputation and the traditional methodThe absolute values of Bias increased with the increase of the right-censoring rate.In different groups of censor rate,the differences of the absolute values of Bias among three methods had no statistical significance.The SE decreased with the decreasing of the interval-censoring rate among three methods and the SE of regression coefficients estimated by the bayesian method was the the minimum value compared with other two methods.In terms of the operation speed,the computation time of the three methods all increased with the increase of interval-censoring rate.In addition,the bayesian method had the longest running time compared with other two methods.C.Influence of parameters settings on Bayesian proportional hazard modelThe absolute values of Bias,LPML and the SE had no obvious differences when the super parameters of gamma prior ranged from 0.001 to 1.In terms of the number of knots the absolute values of Bias usually got the minimum value when the number of knots is 10.(2)Real data analysisThrough the comparison among different parameter settings,we determined that the knots were 10 and the super parameters of gamma prior took a_?=b_?=1.Under this condition of the model,we indicated that premature coronary artery disease(HR=0.57,95%CI:(0.43,0.83))and diabetes(HR=1.75,95%CI:(1.42,2.29))were all statistically significant variables.The results of the multiple imputation method were that premature coronary artery disease(HR=0.60,95%CI:(0.50,0.71])and diabetes(HR=1.81,95%CI:(1.60,2.03))were all statistically significant variables.The results of the traditional method were that premature coronary artery disease(HR=0.59,95%CI:(0.43,0.83))and diabetes(HR=1.80,95%CI:(1.42,2.29))were also statistically significant variables.Conclusions(1)By simulation study,we suggested that the estimation for the covariate obeying binomial distribution by Bayesian proportional hazard model had smaller bias than that for the covariate obeying normal distribution,but the estimated results for the covariate obeying normal distribution was more stable than that for the covariate obeying binomial distribution.The different interval widths and interval-censoring rates had little influence on the estimation bias of regression coefficients,but with the increase of interval censoring rate and interval width,the accuracy of the estimation will decrease and the confidence interval will be wider.Compared with the multiple imputation and the conventional parameter method,the results of Bayesian method usually have smaller fluctuations,but the power of Bayesian method was lower than the other two methods.The model was robust when the super parameters of gamma prior ranged from 0.001 to 1 or when choosing different number of konts.However,the running time of Bayesian method was the longest among three methods.(2)By analyzing the follow-up data of patients with coronary heart disease,we believed that the prognosis of patients with premature coronary artery disease was better compared with common coronary artery disease,and diabetes mellitus was a risk factor for recurrent cardiovascular events.The simple imputation should be improved in the clinical research for interval-censored data and the Bayesian proportional hazard model could be recommended to researchers.