The Study On Bayesian Joint Model For Longitudinal And Survival Data And Its Application

Background and objective:In clinical medicine and epidemiology follow-up studies,longitudinal data and survival data are often collected simultaneously.There is often an association between these two types of data,making separate analyses produce biased estimates.The Bayesian joint model（JM）of longitudinal and survival data can take full advantage of all the information available for both types of data,resulting in more efficient parameter estimates.Based on case data and simulation studies,this study explores the applied strategies and statistical performance of Bayesian JMs when associated longitudinal and survival data,reveal the association between the two types of data,and provides methodological guidance for the analysis of similar data.Main research contents:Case analysis:Based on the Chinese Longitudinal Healthy Longevity Survey Database from 2002to 2014,four single-longitudinal Bayesian JMs（underlying process,underlying process and slope,cumulative effect,and random coefficient）were used to analyse four longitudinal data（systolic blood pressure,diastolic blood pressure,pulse pressure,and cognitive impairment）and all-cause mortality respectively.Three multi-longitudinal Bayesian JMs（multi-longitudinal underlying process,multi-longitudinal underlying process and slope,multi-longitudinal cumulative effect）were used for two relevant longitudinal measures（systolic blood pressure and diastolic blood pressure）and all-cause death data to explore the application of different types of Bayesian JMs in the analysis of case data.Simulation study:based on the simulated data generated for different scenarios（single-longitudinal data with different strength of association,censoring rate,sample size,distribution of random effects and error,times of measurements,and multiple longitudinal data with different correlations）,the statistical inference performance of four single-longitudinal Bayesian JMs and three multiple longitudinal Bayesian JMs under different sample characteristics were explored.Main results:（1）Results of the case studies.Of the four single-longitudinal JMs,the random coefficient JM had the best fit in the joint analysis of systolic,diastolic,pulse pressure,cognitive impairment and all-cause mortality processes.The results of the random coefficient model showed that the faster the systolic blood pressure increased[association coefficientα₂₁=0.23,95%CI=（0.03,0.41）;α₂₂=0.44,95%CI=（0.30,0.63）],the faster the diastolic blood pressure decreased[α₂₂=-0.36,95%CI=（-0.51,-0.14）],the faster the pulse pressure increased[α₂₁=0.21,95%CI=（0.07,0.39）],the faster the risk of cognitive impairment increased[α₂₂=1.06,95%CI=（0.37,1.94）]and the higher the risk of all-cause mortality in the elderly.The underlying process and slope JM showed that the faster the change in systolic blood pressure[α=0.025,95%CI=（0.002,0.049）]and the faster the change in diastolic blood pressure[α=0.111,95%CI=（0.071,0.154）],the higher the risk of all-cause mortality in the elderly.Multi-longitudinal JMs showed that the faster the changes in systolic and diastolic blood pressure,the higher the risk of all-cause mortality in the elderly[α_SBP=0.10,95%CI=（0.05,0.16）;α_DBP=0.11,95%CI=（0.04,0.19）].The survival sub-model in the multi-longitudinal JM has smaller covariate effects and standard deviations compared to the single-longitudinal JM.（2）Results of the simulation studyThe larger the association,the greater the deviation of the regression coefficients of the survival sub-models of four single-longitudinal joint model,the lower the 95%coverage probability（CP95）of each regression coefficient of the underlying process and slope JMs,and the lower the CP95 of the survival sub-models of the underlying process JM and cumulative effects JM.The larger the sample size,the smaller the bias and standard error（SE）of the regression coefficients,the larger the CP95,the smaller the difference in the SE estimates of the longitudinal sub-model regression coefficients among the four models,and the larger the difference in the deviation estimates of the survival sub-model regression coefficients among the four models.The estimated bias and SE of the regression coefficients of the survival sub-model increased with increasing censoring rates.In the vast majority of cases,of the four single-longitudinal JMs,the random coefficient JM has the smallest bias in estimating the survival process regression coefficient and the largest SE;the cumulative effects JM has the largest bias in estimating the survival process regression coefficient.Compared with the case where the random effects and errors obey the normal distribution,the estimated bias and SEs of the regression coefficients are significantly larger in the case of bimodal distribution;the estimated bias and SEs of the regression coefficients are more consistent with the normal distribution in the case of t-distribution;the estimated bias and SEs of most of the regression coefficients in the case of exponential distribution are close to the results in the case of normal distribution.The estimated bias of the regression coefficients of the longitudinal sub-model tends to increase with the number of measurements.For the survival sub-model,the random coefficient JM has the smallest bias and the highest CP95 for the same number of measurements scenario.The greater the correlation between multiple longitudinal outcomes,the smaller the estimation bias and SE of the longitudinal process and the greater the estimation bias and SE of the survival process.In the same multi-longitudinal correlation scenario,the estimated bias of the survival process in the multi-longitudinal cumulative effects JM is significantly higher than in the remaining two multi-longitudinal JMs.The multi-longitudinal JM has smaller biases and SEs in the parameter estimates of the survival sub-model than the single-longitudinal JM.Main conclusions:（1）Conclusions of the case studyBayesian JMs can well reveal associations between dynamic changes in blood pressure,risk of cognitive impairment,etc.,and risk of all-cause mortality in elderly populations,and obtain more valid parameter estimates.Single-longitudinal Bayesian random coefficient JM are better fitted to the data in a variety of practical situations.When correlations do exist between longitudinal endings,multi-longitudinal JMs can yield more accurate predictive models of the survival process than single-longitudinal JMs.（2）Conclusions of the simulation studyFor the four single-longitudinal Bayesian JMs:（a）From the overall performance of parameter estimation,in most cases,the random coefficient JM has the best parameter estimation performance,the underlying process and slope JM is the second,and the cumulative effect JM is the worst,and the Bayesian random coefficients JM has good generalizability and generalization value.（b）The distribution of random effects and errors has a strong influence on the parameter estimation results of the JM,especially when the random effects and errors are subject to a multi-modal distribution,the distribution of random effects and errors must be checked before the joint modelling analysis is used in practice to avoid serious bias from the true values.（c）For non-linear longitudinal and survival data,the times of longitudinal measurements is not as high as it should be,and the reasonable times of measurements needs to be set at the time of study design based on actual needs and design feasibility.For the three multi-longitudinal Bayesian JMs,the multi-longitudinal cumulative effects JM has the worst estimation accuracy and testing power for the survival process,and the multi-longitudinal underlying process JM and the multi-longitudinal underlying process and slope model have generally consistent estimation results.Correlations between multiple longitudinal outcomes affects the parameter estimation results of the survival process multi-longitudinal JMs.Therefore,in practice,when there is a clear correlation between multiple longitudinal outcomes,the researcher can rationally choose an appropriate joint analysis model by combining the real data,the purpose of the study,and the interpretability of the results.