Abstract
This paper develops a methodology for extending multilevel modelling to incorporate spatial interaction effects. The motivation is that classic multilevel models are not specifically spatial. Lower level units may be nested into higher level ones based on a geographical hierarchy (or a membership structure—for example, census zones into regions) but the actual locations of the units and the distances between them are not directly considered: what matters is the groupings but not how close together any two units are within those groupings. As a consequence, spatial interaction effects are neither modelled nor measured, confounding group effects (understood as some sort of contextual effect that acts ‘top down’ upon members of a group) with proximity effects (some sort of joint dependency that emerges between neighbours). To deal with this, we incorporate spatial simultaneous autoregressive processes into both the outcome variable and the higher level residuals. To assess the performance of the proposed method and the classic multilevel model, a series of Monte Carlo simulations are conducted. The results show that the proposed method performs well in retrieving the true model parameters whereas the classic multilevel model provides biased and inefficient parameter estimation in the presence of spatial interactions. An important implication of the study is to be cautious of an apparent neighbourhood effect in terms of both its magnitude and statistical significance if spatial interaction effects at a lower level are suspected. Applying the new approach to a two-level land price data set for Beijing, China, we find significant spatial interactions at both the land parcel and district levels.
Highlights
Many geographical data sets have multilevel structures—for example, houses nested into districts into regions in an urban housing market, or cities nested into regions, that are further nested into countries
The intensity of the group dependence is quantified by cov(yij, yi’j) /var(yij) = σu2/(σu2+ σe2), which is known as the variance partitioning coefficient (VPC) [1,2]
Of particular note are the estimates for the intercept and the higher level covariate effect, which are highly biased when the spatial interaction effect at the lower level is large
Summary
Many geographical data sets have multilevel structures—for example, houses nested into districts into regions in an urban housing market, or cities nested into regions, that are further nested into countries. Using the language of the multilevel modelling literature, the finer spatial scale at PLOS ONE | DOI:10.1371/journal.pone.0130761. Multilevel Modelling with Spatial Interaction Effects which an outcome variable is measured is termed the lower level whereas the more aggregate spatial scale is called the higher level. The multilevel modelling anticipates both differences between the higher level units and correlations within those units. The correlations within units are expected because their members are assumed to be affected by the same aggregate effects. The within group correlation is usually termed group dependence. The existence of group dependence among lower level units violates the classic assumption of independence in a standard regression analysis, raising the risk of inefficient model estimation and incorrect inference [1]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
