Geographically Weighted Regression Modeling for Multiple Outcomes

Abstract

Geographically weighted regression (GWR) has been a popular tool applied in various disciplines to explore spatial nonstationarity for georeferenced data. Such a technique, however, typically restricts the analysis to a single outcome variable and a set of explanatory variables. When analyzing multiple interrelated response variables, GWR fails to provide sufficient information about the data because it only allows separate modeling for each response variable. This study attempts to address this gap by introducing a geographically weighted multivariate multiple regression (GWMMR) technique that not only explores spatial nonstationarity but also accounts for correlations across multivariate responses. We first present the model specification of the proposed method and then draw the associated statistical inferences. Several modeling issues are discussed. We also examine finite sample properties of GWMMR using simulation. For an empirical illustration, the new technique is applied to the stop-and-frisk data published by the New York Police Department. The results demonstrate the usefulness of the GWMMR.

地理加权回归（GWR）被广泛应用于各个学科, 用于探索地理数据的空间非静态性。然而, GWR常常局限于对单一结果变量和一组解释变量的分析。由于GWR只允许对每个响应变量进行单独建模, 因此在分析多个相关响应变量时, GWR无法提供数据的足够信息。本研究试图引入地理加权多元多重回归（GWMMR）技术来解决这个问题, 不仅能探索空间非静态性, 还能解释多变量响应之间的相关性。我们首先介绍了模型的特点, 然后得出有关的统计推断。本文讨论了几个建模问题。通过模拟检验了GWMMR的有限样本属性。做为实证, 该方法被应用于纽约市警察局的“叫停和搜查”（stop-and-frisk）数据。分析结果证明了GWMMR的有效性。

La regresión geográficamente ponderada (GWR) ha sido una herramienta estadística muy popular, aplicada en varias disciplinas para explorar la no estacionalidad espacial de los datos georreferenciados. Tal técnica, sin embargo, típicamente restringe el análisis a un resultado variable único y a un conjunto de variables explicativas. La GWR es incapaz de entregar suficiente información acerca de los datos porque solamente permite la modelización separada para cada variable de respuesta. Este estudio intenta salvar esa laguna introduciendo una técnica de regresión múltiple multivariada geográficamente ponderada (GWMMR) que no solo explora la no estacionalidad espacial, sino que también toma en cuenta las correlaciones a través de las respuestas multivariadas. Presentamos, en primer término, la especificación modelada del método propuesto y enseguida extraemos las inferencias estadísticas asociadas. Se discuten varias cuestiones de la modelización. También examinamos las propiedades de la muestra finita de la GWMMR, mediante simulación. A manera de ilustración empírica, se aplica la nueva técnica a los datos de parada y cacheo publicados por el Departamento de Policía de Nueva York. Los resultados demuestran la utilidad de la GWMMR.

Key Words:

• geographically weighted regression
• multiple outcomes
• multivariate multiple regression
• spatial nonstationarity

关键词:

• 地理加权回归
• 多重结果
• 多元多重回归
• 空间非静态性。

Palabras clave:

• no estacionalidad espacial
• regresión geográficamente ponderada
• regresión múltiple multivariada
• resultados múltiples.

Acknowledgment

The authors would like to thank the constructive comments and suggestions from the anonymous reviewers.

Notes

1 The calculations of Lee’s bivariate L statistics are implemented with functions lee and lee.mc in the R spdep package.

Additional information

Notes on contributors

Vivian Yi-Ju Chen

VIVIAN YI-JU CHEN is an Associate Professor in the Department of Statistics at Tamkang University, Tamsui District, New Taipei City 251301, Taiwan. E-mail: [email protected]. Her research interests include spatial data analysis, geographically weighted modeling, epidemiology, and public health statistics.

Tse-Chuan Yang

TSE-CHUAN YANG was an Associate Professor in the Department of Sociology, University at Albany, State University of New York, Albany at the time this article was drafted. Yang is now an Associate Professor in the Department of Preventive Medicine and Population Health, University of Texas Medical Branch, Galveston, TX 77551. E-mail: [email protected]. His research focuses on examining spatial health inequalities with advanced spatial methods, including both global and local models.

Hong-Lian Jian

HONG-LIAN JIAN is currently a Research Assistant in the Research Center for Humanities and Social Sciences at the Academic Sinica, Taipei 115, Taiwan. E-mail: [email protected]. His research interests are geography, urban research, and epidemiological studies.