https://orcid.org/0000-0003-0274-599X
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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.
许多学科在研究地理过程中都用到了多层次模型, 包括流行病学、经济学、政治学、社会学、城市分析和交通。多层次模型能够描述治理或介入的效果在地理群组上的差异, 这种差异是空间过程异构性的一种形式。此外, 这些模型提供了“位置”依赖关系：同一个地理位置的观测采用类似模型。现有的研究, 已经将空间依赖关系引入到多层次模型, 也探讨了模型估算的经验特征。然而, 文献中缺乏对模型数学结构的系统性研究。本文研究了一种多层次模型, 包括“位置”和“空间”依赖关系。利用数学分析, 本文获得了传统多层次、空间多层次和单层次模型的相关性。该数学结构表明, 多层次模型主要优势（压缩估计）和空间依赖关系之间存在着矛盾。本文从数学上和经验上显示, 当存在空间依赖关系时, 传统多层次模型高估了估计精度、低估了压缩估计。该结果把存在已久的单层次建模扩展到多层次模型。
Los modelos de nivel múltiple se ha aplicado al estudio de muchos procesos geográficos en epidemiología, economía, politología, sociología, analítica urbana y transporte. Más a menudo se usan para expresar el modo como el efecto de un tratamiento o intervención podría variar por grupo geográfico, una forma de heterogeneidad del proceso espacial. Además, estos modelos proveen una noción de dependencia “placial”: las observaciones que se hallan dentro del mismo lugar geográfico son modeladas como similares las unas con las otras. En trabajo reciente se ha mostrado que la dependencia espacial puede incorporarse en modelos de nivel múltiple, y se han examinado las propiedades empíricas de los cálculos de estos modelos. Sin embargo, se nota la falta de atención sistemática a la estructura matemática de los modelos. Este artículo examina un tipo de modelo de nivel múltiple que incluye dependencia tanto “placial” como “espacial”. Utilizando análisis matemático, obtenemos la relación entre los modelos de nivel múltiple clásico, nivel múltiple espacial y de nivel sencillo. Esta estructura matemática expone una tensión entre un beneficio principal de los modelos de nivel múltiple, el encogimiento estimado y los efectos de la dependencia espacial. Mostramos, tanto matemática como empíricamente, que los modelos de nivel múltiple clásicos pueden sobrestimar la precisión de la estimación, y subestimar el encogimiento estimado en presencia de la dependencia espacial. Este hallazgo extiende los resultados de larga duración en el modelado de nivel singular a los modelos de nivel múltiple.
Supplemental Material
Supplemental data for this article can be accessed on the publisher's website.
The supplemental material contains more mathematical detail on the results in the article. Specifically, Bayesian estimators for the models are defined, an elaboration of the distinctions between different kinds of multilevel models is presented, and alternative forms of the shrinkage matrix are specified.
Notes
1 Consult Diez-Roux (Citation2000) or Pickett and Pearl (Citation2001) for a review of early work in this field.
2 Generally, we refer to regions when discussing geographical groups, and these constitute the upper level part of a multilevel model. In contrast, we refer to observations when discussing the members of regions, and these constitute the lower level part of a multilevel model.
3 This model is sometimes stated with region-level data, Z, and region-level marginal effects, but this can be restated into a form equivalent to Equation 1. We focus only on X to simplify notation and emphasize the focus of the variance components model: distinct sources of variability for groups and observations.
4 Detail on how this is done is provided in the Supplemental Material. A fully Bayesian treatment makes the shrinkage matrix very straightforward to specify but is not the only way to obtain this result nor estimate these models. Alternative methods to examine the mathematical structure of the model include expectation maximization (Dempster, Laird, and Rubin Citation1977) or restricted maximum likelihood estimation (Bates et al. Citation2015), and integrated nested Laplace approximation (Bivand, Gómez-Rodriguez, and Rue Citation2014; Lindgren and Rue Citation2015) can be very efficient for the purposes of estimation.
5 Spatial Markov random field structures for spatial dependence (Cressie and Wikle Citation2011, 4.2.1) with separable covariance (discussed in Supplemental Material) can use the same derivation.
6 A full derivation for this is shown in Supplemental Material.
7 For the multilevel models, four Gibbs samplers are simulated from random starts for 22,000 iterations. The spatial parameters were sampled using Metropolis-within-Gibbs sampling. After 2,000 iterations, the chains all converged to the same posterior distribution (Gelman and Rubin Citation1992), with less than a 1 percent improvement on estimate variance likely from more iterations. Effective size was well over 1,000 in all parameters in each chain. The last 10,000 iterations were pooled for analysis. Single-level models are fit via maximum likelihood. All estimators are distributed in PySAL (Rey and Anselin Citation2007; Wolf Citation2018).
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Notes on contributors
Levi John Wolf
LEVI JOHN WOLF is a Senior Lecturer in Quantitative Human Geography at the University of Bristol and Fellow at the Alan Turing Institute, University of Bristol, School of Geographical Sciences, Clifton BS8 1SS, UK. E-mail: [email protected]. His research interests include Bayesian spatial statistics and unsupervised learning in the analysis of elections, redistricting, segregation, and inequality.
Luc Anselin
LUC ANSELIN is the Stein-Freiler Distinguished Service Professor of Sociology and the College Director of the Center for Spatial Data Science at the University of Chicago, Chicago, IL 60637. E-mail: [email protected]. He researches topics in the analysis of spatial data, ranging from exploration to visualization and modeling. He has published on topics from environmental and natural resource economics to criminology, public health, and international relations.
Daniel Arribas-Bel
DANIEL ARRIBAS-BEL is a Senior Lecturer in Geographic Data Science at the University of Liverpool and Fellow at the Alan Turing Institute, University of Liverpool, Department of Geography & Planning, Liverpool L69 7ZT, UK. E-mail: [email protected]. He researches the spatial configuration of cities, in economic, political, and morphological processes, which instigated an interest in spatial modeling, machine learning, and scientific computing.
Lee Rivers Mobley
LEE RIVERS MOBLEY is an Associate Professor of Spatial Science and Health Economics at Georgia State University, Atlanta, GA 30303. E-mail: [email protected]. Her research includes analyses of disparities among populations and across geography, examining socioecological problems where place and space are important; studies in spatial demography; and analysis of health care markets and behaviors of both consumers and providers.