Evaluating the impact of misspecified spatial neighboring structures in Bayesian CAR models

Abstract

Spatial neighboring graphs play a crucial role in accounting for global spatial dependency, particularly in spatial models that utilize the Conditional Autoregressive (CAR) covariance structure. The Bayesian modified Besag–York–Molliè (BYM2) model, which falls under the category of CAR models, introduces a precision parameter to quantify the variability not captured by the fixed risk components and a mixing parameter to decipher the proportion of random effects attributed to the spatial component and the aspatial random noise. Despite the advantages these extra features bring, misspecification of BYM2 model components is common, and its effects are not well understood. Previous studies often avoid simulations due to computational demands, relying instead on performance metrics for inferences and model comparisons using empirical data.

This study uses comprehensive simulations to examine the impact of erroneously specified spatial neighborhood structures on the BYM2 model. We considered three different neighborhood structures: a first-order adjacency-based structure and two minimum distance-based structures with threshold distances of 70 km and 140 km at various sparsity levels. For each structure, we simulate data under that structure and then analyze it using the remaining two structures as misspecified cases to evaluate their impact on model fit. Fixed PC prior settings were applied to control for prior specification effects in examining bias and MSE. The study was further validated through practical analyses of road crash incidents in Ghana and a lip cancer cases data in Scotland, UK.

Our findings reveal that incorrect specification of the neighboring structure does not significantly impact the fixed effects. However, it affects the estimates of the mixing parameter and precision term, thus impacting the spatial component. In cases of high spatial dependency and misspecified neighborhood structures, the BYM2 model tends to underestimate the mixing parameter. Under-specifying the neighborhood structure results in underestimated hyper-parameter values while over-specifying it leads to an overfitted spatial smooth. The empirical application results which were consistent with the simulation also emphasized the critical importance of accurately specifying spatial structures in BYM2 models. Relying solely on metrics like the Watanabe-Akaike Information Criterion (WAIC), Deviance Information Criterion (DIC), and Conditional Predictive Ordinate (CPO) estimates to determine an optimal spatial structure can be misleading. Instead, the Moran’s Index (MI) statistic is more reliable for identifying the most suitable neighborhood structure.