| Objective:The Bayesian inference is adopted to comprehensively evaluate the impact of the different specifications of the baseline hazard function on the parameter estimation and dynamic prediction of the joint model.The purpose of this study is to provide a methodological support for improving the predictive accuracy of the dynamic prediction of the model,which is conducive to optimizing individualized treatment strategies and medical decisions.Methods:(1)This research constructed a Bayesian joint model that consisted of a linear mixed-effects sub-model for the longitudinal outcome as well as a cox proportional hazards sub-model for the time-to-event data.The two submodels were then assumed to be linked via the current value association structure.The joint model was defined via baseline hazard specified by a Weibull function and a B-spline function.(2)The simulation study jointly generated time-varying covariate data and survival data from either a standard Weibull distribution with an increasing hazard function or a two-component mixture Weibull distribution.To investigate the impact of baseline hazard functions(Weibull and B-spline)in the joint model on the performance of parameters estimation,the performance of the parameter estimation was assessed by bias,standard error,standard deviation,and coverage probability.The rationality of the baseline hazard function specification was measured through the root mean square deviation of the difference between the posterior estimation of baseline hazard and true hazard functions.(3)Both No-U-Turn sampler and Gibbs sampler were used to approximate the posterior distributions of the parameters in the Bayesian joint model.Then the estimated parameters were used to evaluate the impact of different sampler algorithms on the predictive accuracy of dynamic prediction of the joint model for longitudinal and time-to-event data.(4)The instance data came from a multicenter,randomized,open-labelled and parallel-group clinical study.The aim of this study was to evaluate the prevention of an experimental drug in the recurrence and metastasis of hepatocellular carcinoma patients after radical resection.The joint model was constructed by fitting the data of 315 primary hepatocellular carcinoma patients in the randomized controlled trial,comparing Weibull and B-spline functions,to illustrate the insensitivity of parameter estimation to the baseline hazard function specification.The convergence of the model was assessed by trace plot,Gelman Rubin statistics and autocorrelation plot.The expected log predictive density(ELPD),which was calculated by the Pareto-smoothed importance sampling leave-one-out cross-validation(PSIS-LOO-CV)method,was used to evaluate the overall prediction accuracy of the Bayesian model as well as the impact of the baseline hazard function specification on the predictive accuracy of dynamic prediction of the joint model.Results:(1)The posterior distributions of model parameters were obtained according to the joint likelihood and the joint prior distribution.p(Ti,di,yi|bi;θ)=p(Ti,di|bi;θ)p(yi|bi;θ)denoted the joint likelihood function for the joint model.Joint prior distribution was p(θ).where θ=(θtT,θyT,θbT)T denoted the full parameter vector,withθt denoting the parameters for the event time outcome,θy being the parameters for the longitudinal outcomes,and θb representing the unique parameters of the random-effects covariance matrix.The log posterior for the i th individual was logp(θ,bi|Ti,di,yi)∝∑i=1n∑j=1 ni logp(Ti,di|bi,θ)+log p(yij|bi,θ)+logp(bi |θ)+logp(θ).(2)The results from the simulation study suggested that the Bayesian joint model was able to accurately estimate the true parameters because their coverage probabilities were generally very close to 95%in most of the simulation scenarios.Under simple or complex survive data scenario,there was a certain bias when the sample size(N)was only 100.However,when the sample size(N)increased to 300,the joint model with different specifications of the baseline hazard function could estimate all parameters as effectively,especially the association parameters.The bias value fluctuated between 0.001 and 0.004,and the coverage probabilities were very close to the desired 95%.Finally,the simulation study showed the estimates of parameters were generally insensitive to the specification of the baseline hazard function.At the same time,under each scenario,the Gaussian Kronrod quadrature was used to evaluate the likelihood of model defined by 7,11,and 15 quadrature nodes.The results showed that when the sample size was small,the choice of the number of quadrature nodes had a remarkable impact on the parameter estimation,which was shown as the smaller the number of nodes,the greater the bias.However,when the sample size became large,the estimates of the parameters were nearly unbiased,regardless the numbers of the quadrature node.(3)When the sample size(N)was 300,under a simple survival data,the Weibull function provided the best fit to the true function with the lowest root mean square deviation values(RMSD=0.024).Compared with the Weibull function,there was little difference in the root mean square deviation of the B-spline function with two internal nodes(RMSD=0.035),which showed that the B-spline function could also fit the data well in the simple survive data scenario.Under a complex survival data,the B-spline function with two internal nodes presented a lower root mean square deviation estimates(RMSD=0.076)compared to that from the Weibull function(RMSD=0.287).B-splines function seemed flexible in capturing complex baseline shapes in more complicated survival data.At the same time,for the dynamic prediction of the survival probability of the joint model,the B-spline function had a smaller prediction error than that of the Weibull function at different time points.(4)Comparing with the Gibbs sampler,the No-U-Turn sampler has a faster convergence rate.In the joint model,the No-U-Turn sampler only needed about 2000 iterations to converge,while the Gibbs sampler had 50000 iterations.Corresponding to the parameters,the autocorrelation plot of the Markov chain showed that compared with the Gibbs sampler,the autocorrelation coefficient of the No-U-Turn sampler decayed more rapidly to 0.In the analysis of the joint model,the dynamic prediction performance of the No-U-Turn sampler at different time points was better than the Gibbs sampler.This is because the former had a smaller prediction error and higher discrimination ability.(5)The results of real data analysis showed that the results of parameter estimation was not very sensitive to the baseline hazard function specification.In model comparison,the values of ELPD were obtained by the PSIS-LOO-CV method.Based on the estimated values of ELPD,the overall prediction accuracy of the B-spline function with two internal nodes was higher than that of the Weibull function.This study obtained an association ofα=-0.097(95%CI:-0.147～-0.048),with a unit decrease in the albumin marker corresponding to a exp(-α)=1.102-fold increase in the risk for recurrence or death(95%CI:1.049～1.158).At the same time,the B-spline function model showed a better performance than the Weibull function model in the dynamic prediction of survival probability,which was consistent with the simulation study.Conclusions:(1)The study shows in general how the estimates of parameters are insensitive to the specification of the baseline hazard in the Bayesian joint model for longitudinal and time-to-event data.However,these predictions will heavily rely on the accuracy of the model in estimating the baseline hazard function.In the case of small sample size,the small number of orthogonal nodes has a certain deviation to the parameter estimation.In the joint model,there are some differences in the dynamic prediction obtained between the No-U-Turn sampler and the Gibbs sampler.The convergence speed of the No-U-Turn sampler is generally faster and its dynamic prediction performance is also higher than the Gibbs sampler.(2)Serum albumin can be used as a biomarker to dynamically predict the primary hepatocellular carcinoma patients after radical resection.Increasing the serum albumin level of patients after operation will help to reduce the risk of recurrence or death. |
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