Abstract
Intrinsic conditional autoregressive modeling in a Bayeisan hierarchical framework has been increasingly applied in small-area ecological studies. This study explores the specifications of spatial structure in this Bayesian framework in two aspects: adjacency, i.e., the set of neighbor(s) for each area; and (spatial) weight for each pair of neighbors. Our analysis was based on a small-area study of falling injuries among people age 65 and older in Ontario, Canada, that was aimed to estimate risks and identify risk factors of such falls. In the case study, we observed incorrect adjacencies information caused by deficiencies in the digital map itself. Further, when equal weights was replaced by weights based on a variable of expected count, the range of estimated risks increased, the number of areas with probability of estimated risk greater than one at different probability thresholds increased, and model fit improved. More importantly, significance of a risk factor diminished. Further research to thoroughly investigate different methods of variable weights; quantify the influence of specifications of spatial weights; and develop strategies for better defining spatial structure of a map in small-area analysis in Bayesian hierarchical spatial modeling is recommended.
Highlights
In small-area ecological studies using a Bayesian approach, spatial dependence is accounted for at the prior level, often using the intrinsic conditional (Gaussian) autoregressive (ICAR) model as a means to capture the effects of unobserved spatially-structured latent covariates or measurement errors
Some common boundary vertices of adjacent polygons have different coordinates recorded in the digital map, which could have led to the incorrect adjacencies information
In Bayesian hierarchical spatial modeling, the ICAR model with prior information of adjacencies and spatial weights has been considered as one of the best choices available to smooth out random variation unrelated with underlying risk and stabilize estimated risks in small-area analysis [20]
Summary
In small-area ecological studies using a Bayesian approach, spatial dependence is accounted for at the prior level, often using the intrinsic conditional (Gaussian) autoregressive (ICAR) model as a means to capture the effects of unobserved spatially-structured latent covariates or measurement errors. Data for each area are assumed independent conditional on the spatial ICAR (or proper CAR) model. Within the framework for estimating risks and identifying risk factor(s), the ICAR model is used as part of a convolution prior for the (log) relative risk [1,2]. The convolution prior consists of a spatially-structured random effects term (S) and an unstructured random effects term (U) modeling the underlying (log) relative risks. It enables each area to borrow strength globally through the prior specification of a normal distribution for the unstructured random effects, and locally from neighboring observations through the prior specification of the ICAR model for spatial random effects. As a result, estimated risks are smoothed towards a combination of global (through U) and local (through S) risks. Unexpected changes in estimates of risks when compared to standardized ratios may reflect inappropriate smoothing
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