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.

多层次模型已被应用于研究流行病学、经济学、政治学、社会学、城市分析和交通等领域的许多地理过程。它们最常用于表达治疗或干预的效果如何因地理群体而异，这是一种空间过程异质性的形式。此外，这些模型提供了一种“柏拉图”依赖性的概念：同一地理位置内的观测被建模为彼此相似。最近的工作表明，空间相关性可以引入多级模型，并检查了这些模型估计的经验属性。然而，一直缺乏对这些模型的数学结构的系统关注。本文研究了一种包括“平面”和“空间”依赖性的多级模型。通过数学分析，我们得到了经典的多层次模型、空间多层次模型和单层次模型之间的关系。这种数学结构暴露了多级模型的主要好处、估计收缩率和空间依赖性的影响之间的紧张关系。我们在数学和经验上都表明，当存在空间依赖性时，经典的多级模型可能会高估估计精度并低估估计收缩。这一结果将单级建模中的长期结果扩展到了多级模型。当存在空间依赖性时，经典的多级模型可能会夸大估计精度并低估估计收缩。这一结果将单级建模中的长期结果扩展到了多级模型。当存在空间依赖性时，经典的多级模型可能会夸大估计精度并低估估计收缩。这一结果将单级建模中的长期结果扩展到了多级模型。