Journal Article10.1007/s11222-023-10293-5
Spatial joint models through Bayesian structured piecewise additive joint modelling for longitudinal and time-to-event data
Anja Rappl,Thomas Kneib,Stefan Lang,Elisabeth Bergherr +3 more
TL;DR: Spatial joint models through Bayesian structured piecewise additive joint modelling for longitudinal and time-to-event data can effectively estimate spatial effects while being computationally efficient and stable in imbalanced data settings.
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Abstract: Abstract Joint models for longitudinal and time-to-event data simultaneously model longitudinal and time-to-event information to avoid bias by combining usually a linear mixed model with a proportional hazards model. This model class has seen many developments in recent years, yet joint models including a spatial predictor are still rare and the traditional proportional hazards formulation of the time-to-event part of the model is accompanied by computational challenges. We propose a joint model with a piecewise exponential formulation of the hazard using the counting process representation of a hazard and structured additive predictors able to estimate (non-)linear, spatial and random effects. Its capabilities are assessed in a simulation study comparing our approach to an established one and highlighted by an example on physical functioning after cardiovascular events from the German Ageing Survey. The Structured Piecewise Additive Joint Model yielded good estimation performance, also and especially in spatial effects, while being double as fast as the chosen benchmark approach and performing stable in an imbalanced data setting with few events.
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Figures
Table 3 Statistical overview of run times of 100 replications per setting and estimation method 
Fig. 4 Boxplots of mean squared error (MSE), bias and 95%-coverage per method by effect and simulation setting (Setting 1 - fgeo in ηls , Setting 2 - fgeo in ηs , Setting 3 - fgeo in ηl ). The orange horizontal line marks the reference value of each statistic 
Table 1 Illustration of data augmentation used for applying Poisson regression 
Table 2 BayesX estimates of linear effects of physical functioning after a caesura 
Fig. 3 BayesX estimates of the smooth effect of the age of onset aoo and the geographical location on physical functioning as well as the estimated baseline hazard of the model
Citations
Spatial Joint Modelling of Multivariate Longitudinal Outcomes and Cure Proportion using Latent Gaussian model with application to dataset on HIV/AIDS patients
Olayiwola Olaniyi Matthew,Dawodu Ganiyu Abayomi,Osinuga Idowu Ademola +2 more
- 30 Oct 2025
Abstract: Abstract Survival analysis has seen in recent times more of joint modelling of longitudinal and survival data using approximate hierarchical Bayesian method. This study modelled jointly multivariate and cure proportion with spatial variation using latent Gaussian models (LGMs). Spline specification was used to capture nonlinear trajectories of the multivariate longitudinal biomarkers in the LGM paradigm. Spatial dependence was assumed for neighbouring locations with an improper multivariate conditionally autoregressive prior on the spatial random effects, while inverse-Wishart prior was assumed for the covariance matrix of the random effects and Gaussian priors for the fixed effects. The penalised complexity prior was assumed for the Weibull shape parameters of the baseline hazard function for the event-time and binomial distribution for latent cure classification variable. Posterior distributions were evaluated using INLA. The study was applied to longitudinal and survival HIV/AIDS patients datasets. The random spatial variability was seen to be significant for both biomarkers, random intercept for BMI and cure probabilities. The full conditional distribution of latent variable gave a predicted cure proportion of 61%. The susceptible group on the hand, had a full conditional distribution of latent incidence variable predicting susceptible proportion of 39%. Flexibility of LGM for Bayesian hierarchical was demonstrated easily for multivariate joint model with spatial cure proportion.
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