Laplace approximation for conditional autoregressive models for spatial data of diseases

Conditional autoregressive (CAR) distributions are used to account for spatial autocorrelation in small areal or lattice data to assess the spatial risks of diseases. The intrinsic CAR (ICAR) distribution has been primarily used as the priori distribution of spatially autocorrelated random variables in the framework of Bayesian statistics. The posterior distributions of spatial variates and unknown parameters of Bayesian ICAR models are estimated with the Markov chain Monte Carlo (MCMC) methods or integrated nested Laplace approximation (INLA), which may suffer from failures in numeric convergence. This study used the Laplace approximation, a fast computational method available in software Template Model Builder (TMB), for the maximum likelihood estimation (MLEs) of the ICAR model parameters. This study used the TMB to integrate out the latent spatial variates for the fast computations of marginal likelihood functions. This study compared the runtime and performance among the TMB, MCMC, and INLA implementations with three case studies of human diseases in the United Kingdom and the United States. The MLEs of the ICAR model with TMB were similar to those by the MCMC and INLA methods. The TMB implementation was faster than the MCMC (up to 100–200 times) and INLA (nine times) models. • This study built conditional autoregressive models in template model builder • TMB implementation was 100-200 times faster than the MCMC method • TMB implementation was also faster than Bayesian approximation with R INLA