On Spatial and Platial Dependence: Examining Shrinkage in Spatially Dependent Multilevel Models

Levi John Wolf, Luc Anselin, Daniel Arribas-Bel and Lee Rivers Mobley

Annals of the American Association of Geographers, 2021, vol. 111, issue 6, 1679-1691

Abstract: Multilevel models have been applied to study many geographical processes in epidemiology, economics, political science, sociology, urban analytics, and transportation. They are most often used to express how the effect of a treatment or intervention might vary by geographical group, a form of spatial process heterogeneity. In addition, these models provide a notion of “platial” dependence: observations that are within the same geographical place are modeled as similar to one another. Recent work has shown that spatial dependence can be introduced into multilevel models and has examined the empirical properties of these models’ estimates. Systematic attention to the mathematical structure of these models has been lacking, however. This article examines a kind of multilevel model that includes both “platial” and “spatial” dependence. Using mathematical analysis, we obtain the relationship between classic multilevel, spatial multilevel, and single-level models. This mathematical structure exposes a tension between a main benefit of multilevel models, estimate shrinkage, and the effects of spatial dependence. We show, both mathematically and empirically, that classic multilevel models may overstate estimate precision and understate estimate shrinkage when spatial dependence is present. This result extends long-standing results in single-level modeling to multilevel models.

Date: 2021
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DOI: 10.1080/24694452.2020.1841602

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