A comparison of estimation methods for multilevel models of spatially structured data

Two recent contributions (Dong et al., 2015; Osland et al., 2016) point to the relevance of multilevel models for spatially structured data. In Osland et al. (2016) these models are used to examine the importance of district-level covariates for house prices in Stavanger, Norway, in Dong et al. (2015) similarly for land parcel prices in Beijing; we use these data sets in our comparison. In Osland et al. (2016), a district-level spatial random effect was fitted using an intrinsic CAR model estimated using WinBUGS. Dong et al. (2015) used R code provided in supplementary materials to their article, and subsequently improved in an R package (Dong et al., 2016a); computation there used custom MCMC C++ code to fit a SAR district-level spatial random effect. This article compares approaches to estimating models of this kind, using the R packages R2WinBUGS, HSAR, INLA, R2BayesX, hglm and the new package mclcar for Monte Carlo maximum likelihood estimation (Sha, 2016b). We show that multilevel models of spatially structured data may be estimated readily using a variety of approaches, not only the intrinsic CAR model more typically found in the existing literature. We also point to a range of issues for further research in situations in which data acquired at different levels of spatial resolution are combined.