Joint Modelling of Multivariate Longitudinal and Time-to-Event under Bayesian Inference: With Application to Type 2 Diabetes and Hypertension Disease.

Many medical and epidemiological studies record longitudinal measurements until a time to-event outcome occurs. When there is an association between the time-to-event and the longitudinal outcomes, separately modelling them may lead to biased estimates. Joint modelling of longitudinal and survival data is an effective method for analysing their relationship. In most cases, univariate joint modelling of longitudinal and time-to-event outcomes is an effective method to evaluate their association. However, this model-based analysis can yield biased estimates when multiple longitudinal outcomes are highly correlated and do not follow a multivariate normal distribution. To overcome the problem, we develop a bivariate joint model with a skewed multivariate normal distribution, providing a flexible approach for non-symmetric and correlated longitudinal outcomes. The proposed model specification consists of two sub-models linked by shared random effects. This involved the Cox proportional hazards model for time-to-event data and the multivariate mixed-effects model for correlated longitudinal data following bivariate skew-normal distributions. We estimate the parameters using a Bayesian framework and implement Markov chain Monte Carlo methods in R with JAGS. We assess the performance of the proposed models via simulations and apply the methodology to a data set to assess the association between longitudinal blood sugar and blood pressure measures and time to chronic complications. Our studies suggest a strong, significant positive relationship between patterns of blood glucose levels and systolic blood pressure. Over time, increases in systolic blood pressure raised the risk of microvascular and macrovascular complications. In bivariate joint modelling, employing a skewed multivariate normal distribution instead of a standard multivariate normal distribution improves model fit and yields more accurate parameter estimates for the longitudinal biomarker.

In medical research, it is common to collect multivariate data by measuring subjects multiple times across different outcomes. Researchers often use univariate mixed-effects models to analyse this data by assuming that both the random effects and errors follow a normal distribution. Additionally, the response variables are assumed to be linear functions of the unknown regression parameters. However, the assumption of normal distribution may not always provide reliable results if the data exhibit skewness; outcome variables may have a nonlinear relationship with covariates, such as time. Furthermore, a univariate mixed-effects model applied to correlated multivariate longitudinal outcomes without considering their correlation may lead to biased parameter estimates. To simultaneously overcome these issues, we presented a flexible semiparametric multivariate mixed-effects model that incorporates multiple correlated longitudinal outcomes, exhibits skewness, and uses a nonparametric function to capture nonlinear time effects. The proposed models are illustrated through an application to correlated glucose concentration and blood pressure data, aiming to study the association between glucose concentration and blood pressure in individuals with type 2 diabetes and hypertension. A simulation study is conducted to evaluate the performance of the proposed models. The results from both the application and simulation studies suggest that the semiparametric mixed effect model, which utilizes a multivariate normal distribution for the random errors, has better performance than other proposed models since it accommodates the nonlinear effects of covariates and asymmetrical characteristics of longitudinal measurements. In our application, we found a strong association between the changes in glucose concentration and blood pressure, with the rate of change increasing over time.