Developing a Locally Adaptive Spatial Multilevel Logistic Model to Analyze Ecological Effects on Health Using Individual Census Records

Abstract

Geographical variable distributions often exhibit both macroscale geographic smoothness and microscale discontinuities or local step changes. Nonetheless, accounting for both effects in a unified statistical model is challenging, especially when the data under study involve a multiscale structure and non-Gaussian response variables. This study develops a locally adaptive spatial multilevel logistic model to examine binomial response variables that integrates an innovative locally adaptive spatial econometric model with a multilevel model. It takes into account global spatial autocorrelation, local step changes, and vertical dependence effects arising from the multiscale data structure. Another appealing feature is that the spatial correlation structure, implied by a spatial weights matrix, is learned along with other model parameters via an iterative estimation algorithm, rather than being presumed to be invariant. Bayesian Markov chain Monte Carlo (MCMC) samplers are derived to implement this new spatial multilevel logistic model. A data augmentation approach, drawing on recently devised Pólya-gamma distributions, is adopted to reduce computational burdens of calculating binomial likelihoods with a logit link function. The validity of the developed model is evaluated by a set of simulation experiments, before being applied to analyze self-rated health for the elderly in Shijiazhuang, the capital city of Hebei Province, China. Model estimation results highlight a nuanced geography of self-rated health and identify a range of individual- and area-level correlates of health for the elderly. Key Words: geography of health, local spatial modeling, multilevel models, spatial autocorrelation, spatial econometrics.

地理变因分佈经常同时展现在巨观尺度的地理平稳性，以及在微观尺度的不持续性或地方的阶跃变化。然而在一个统一的统计模型中同时考量两种效应具有挑战，特别是当研究的数据涉入多重尺度结构与非高斯反应变项时。本研究发展一个在地调适的空间多层级罗吉特模型，以检视二项式反应变项，整合创新的在地调适空间计量经济模型与多层级模型。本研究考量全球空间自相关、地方阶跃变化，以及从多重尺度数据结构而生的垂直依赖效应。另一项令人感兴趣的特徵是，由空间加权矩阵所包含的空间相关结构，通过互动式评估演算与其他模型的参数一同习得，而非预设不变。本研究导出马可夫链蒙地卡罗（MCMC）抽样器，以执行此一崭新的空间多层级罗吉特模型。本研究採用晚近设计的Polya-Gamma分佈之数据增大法，用来降低以罗吉特链结函数计算二项式可能性的演算负担。在运用至分析中国河北省会石家庄的老人健康自我评分之前，此一发展模型的效力由一组模拟实验进行评估。模型评估结果强调具细微差别的自我健康评分地理，并指认一系列老人在个人和地区层级的健康相关事物。关键词：健康地理学，地方空间模式化，多层级模型，空间自相关，空间计量经济学。

Las distribuciones de variables geográficas a menudo exhiben uniformidad geográfica a la macroescala y discontinuidades en microescala, o cambios de paso locales. Sin embargo, tomar en cuenta ambos efectos en un modelo estadístico unificado es todo un reto, especialmente cuando los datos bajo estudio implican una estructura de escala múltiple y variables de respuesta no gaussiana. Este estudio desarrolla un modelo logístico espacial de nivel múltiple adaptable localmente para examinar variables de respuesta binomial, que integra un modelo econométrico espacial innovador localmente adaptable con un modelo de nivel múltiple. Se toman en cuenta la autocorrelación espacial global, los cambios de paso locales y los efectos de dependencia vertical que surgen de la estructura de datos a multiescala. Otro rasgo atrayente es que la estructura de la correlación espacial implícita en una matriz de pesos espaciales, es aprendida junto con otros parámetros del modelo por medio de un algoritmo de estimación iterativa, en vez de que se le presuma de invariable. Se derivaron muestrarios Monte Carlo de las cadenas bayesianas de Markov (MCMC) para implementar este nuevo modelo logístico espacial de nivel múltiple. Se adoptó un enfoque de acrecentamiento de datos, basado en las distribuciones Pólya-gamma, recientemente diseñadas, para reducir las cargas computacionales de calcular las probabilidades binomiales con una función de enlace logit. La validez del modelo desarrollado se evalúa con un conjunto de experimentos de simulación, antes de aplicarse para analizar la salud auto calificada de ancianos en Shijiazhuang, ciudad capital de la Provincia Hebei, China. Los resultados estimados por el modelo destacan una geografía matizada de la salud autoevaluada, e identifican un rango de correlatos a nivel individual y de área de la salud de los adultos mayores. Palabras clave: autocorrelación espacial, econometría espacial, geografía de la salud, modelado espacial local, modelos de nivel múltiple.

Acknowledgments

The authors are grateful for the helpful comments of the reviewers and the editor, which have greatly improved the content of the article. They also appreciate the insightful comments on environmental pollution in Hebei Province by Professor Bifeng Wang and Professor Hui Song from Hebei Geo Univeristy.

Notes

Notes

1 Although SAR and CAR models have been used almost in parallel in different fields (e.g., spatial econometrics and geographical analysis in general vs. spatial statistics), they are closely linked, and detailed descriptions of the similarities between the two models are provided in Assunção and Krainski (2009).

2 ${\stackrel{\sim }{x}}_k$ is an area-wise mean-centered variable so the expectation of the product between ${\stackrel{\sim }{x}}_k$ and $\mathit{\mu }$ E[${\stackrel{\sim }{x}}_k\mathit{\mu }$] = 0. Originally, the Mundlak correction is proposed in the panel data modeling context and deals with potential dependence between time-variant predictors and individual random effects. This approach, however, is readily applicable to general multilevel models (Raudenbush and Bryk 2002; Bell and Jones 2015) and spatial panel econometrics models (Debarsy 2012).

3 The urban–rural divide here is more of institutional than physical landscape separation of the population, although the vast majority of people with agricultural hukou live in rural areas. This hukou system, implemented in 1958, has supported and strengthened a rural–urban dual structure in China that results in unequal distribution of resources (e.g., health services and facilities) and thus large gaps in terms of health outcomes in rural and urban areas (e.g., Chan 2009).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 41822104 and 41601148), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA23100301), and the UK Economic and Social Research Council (Grant No. ES/P003567/1).

GUANPENG DONG is a Lecturer in Geographic Data Science at the Department of Geography & Planning, University of Liverpool, Liverpool L69 7ZT, UK. E-mail: [email protected] or [email protected]. His core research interests include the development of spatial/spatiotemporal statistical and multilevel modeling approaches and the applications in urban studies.

JING MA is an Associate Professor of Human Geography in the Faculty of Geographical Science, Beijing Normal University, Beijing 100875, China. E-mail: [email protected]. Her main research interests include activity and travel behavior, subjective well-being, environmental justice, and health.

DUNCAN LEE is a Professor of Statistics at the School of Mathematics & Statistics, University of Glasgow, Glasgow G12 8SQ, UK. E-mail: [email protected]. His core research interests are in developing statistical methods for use in epidemiology, focusing on disease mapping and studies that investigate the effects of air pollution on human health.

MINGXING CHEN is a Professor of Economic Geography at the Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China. E-mail: [email protected]. His core research interest is urban and regional development.

GWILYM PRYCE is a Professor of Social Statistics at Sheffield Methods Institute, University of Sheffield, Western Bank, Sheffield S10 2TN, UK. E-mail: [email protected]. His research interests include spatial inequality, urban economics, and environmental sorting.

YU CHEN is a Lecturer in Chinese Studies at the School of East Asian Studies, University of Sheffield, Sheffield S10 2TN, UK. E-mail: [email protected]. Her core research interests include China’s urbanization and urban development policies.