Abstract
Social and physical processes often exhibit both macro-level geographic smoothness – implying positive spatial dependence – and micro-level discontinuities – suggesting implicit step changes or boundaries in the data. However, a simultaneous treatment of the two features in a unified statistical model poses great challenges. This study extends an innovative locally adaptive spatial auto-regressive modelling approach to a multi-level modelling framework in order to explore multiple-scale geographical data. It develops a Bayesian locally adaptive spatial multi-level model that takes into account horizontal global spatial dependence and local step changes, as well as a vertical group dependency effect imposed by the multiple-scale data structure. At its heart, the correlation structures of spatial units implied by a spatial weights matrix are learned along with other model parameters using an iterative estimation algorithm, rather than being assumed to be invariant and exogenous. A Bayesian Markov chain Monte Carlo (MCMC) sampler for implementing this new spatial multi-level model is derived. The developed methodology is applied to infer neighbourhood quality using property transaction data, and to examine potential correlates of neighbourhood quality in Liverpool. The results reveal a complex and fragmented geography of neighbourhood quality; besides an overall smoothness trend, boundaries delimiting neighbourhood quality are scattered across Liverpool. Socio-economics, built environment, and locational characteristics are statistically significantly associated with neighbourhood quality.
Highlights
Multi-level modelling has been demonstrated as a useful tool to derive summary statistics for higher-level units from outcomes measured for low-level units
Before interpreting the estimated geography of neighbourhood quality and covariate effects, we first discuss the comparison of the two model specifications and its implications
The estimated global spatial autoregressive parameter ρ is slightly larger in the non-adaptive spatial multi-level model than in the adaptive model. This is driven by the step changes identified in the neighbourhood quality surface, since quite a few neighbourhoods are
Summary
Multi-level modelling has been demonstrated as a useful tool to derive summary statistics for higher-level (or more aggregated) units from outcomes measured for low-level units (e.g. individual). Leckie & Goldstein, 2009) Such a model for two levels (or scales) is shown in Eq 1. Two important advantages pertain to the multi-level model-based estimates. There is great flexibility in terms of controlling for pupil-level characteristics (e.g. prior education achievement) and understanding the links of school-level characteristics to effectiveness. The estimates on school effectiveness are reliable because of the borrowing strength from other data points (Goldstein, 2011; Raudenbush & Bryk, 2002). The estimator of θj, θ j, is shrunk to the global mean of y after controlling for pupil-level covariate effects, depending on the magnitudes of σζ, σε2 and nj, less subject to sampling uncertainties in particular schools Model (1) and its variants have been applied to derive health statistics for aggregated spatial units by using data available at a fine-resolution spatial scale (e.g. Arcaya, Brewster, Zigler, & Subramanian, 2012; Ma, Mitchell, Dong, & Zhang, 2017), and serve as an important approach in the model-based small area estimation literature (e.g. Rao, 2003)
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