Spatial Random Slope Multilevel Modeling Using Multivariate Conditional Autoregressive Models: A Case Study of Subjective Travel Satisfaction in Beijing

Abstract

This article explores how to incorporate a spatial dependence effect into the standard multilevel modeling (MLM). The proposed method is particularly well suited to the analysis of geographically clustered survey data where individuals are nested in geographical areas. Drawing on multivariate conditional autoregressive models, we develop a spatial random slope MLM approach to account for the within-group dependence among individuals in the same area and the spatial dependence between areas simultaneously. Our approach improves on recent methodological advances in the integrated spatial and MLM literature, offering greater flexibility in terms of model specification by allowing regression coefficients to be spatially varied. Bayesian Markov chain Monte Carlo (MCMC) algorithms are derived to implement the proposed model. Using two-level travel satisfaction data in Beijing, we apply the proposed approach as well as the standard nonspatial random slope MLM to investigate subjective travel satisfaction of residents and its determinants. Model comparison results show strong evidence that the proposed method produces a significant improvement against a nonspatial random slope MLM. A fairly large spatial correlation parameter suggests strong spatial dependence in district-level random effects. Moreover, spatial patterns of district-level random effects of locational variables have been identified, with high and low values clustering together.

本文探讨如何将空间依赖效应纳入标准多层级模式化（MLM）。本文所提出的方法, 特别适合在地理上聚集的调研数据分析, 其中个人在地理区域中套叠。我们运用多变量条件式自迴归模型, 发展空间随机斜率 MLM 方法, 以解释在同一区域内的个人对群体内部的依赖, 以及同时在区域之间的空间依赖。我们的方法, 改善晚近整合式的空间与MLM文献中的方法学进展, 并透过让迴归係数在空间上具有差异, 提供模型特殊化方面更大的弹性。该方法衍生出贝叶斯的马可夫链蒙地卡罗（MCMC）演算法, 以执行提出的模型。我们运用在北京的二层旅行满意度数据, 并应用提出的方法以及标准非空间随机斜率 MLM, 探讨居民的主观旅行满意度及其决定因素。模式比较的后果, 显示出强健的证据, 支持提出的方法对非空间随机斜率 MLM 而言产生显着的改进。相当大的空间相关参数, 显示出在行政区层级随机影响的大幅空间依赖。此外, 本研究指认区位变数中的行政区层级随机影响的空间模型, 其中高数值与低数值聚共同聚集。

En este artículo se explora el modo de incorporar un efecto de dependencia espacial en un procedimiento de modelado estándar a nivel múltiple (MLM). El método propuesto es particularmente adecuado para el análisis de datos de estudios aglomerados geográficamente, donde los elementos individuales están anidados en áreas geográficas. Basándonos en modelos de auto-regresión condicionales multivariados, desarrollamos un enfoque espacial MLM de inclinación aleatoria con el cual explicar simultáneamente la dependencia al interior del grupo entre individuos de la misma área y la dependencia entre áreas. Se destacan las mejoras de nuestro enfoque en el contexto de avances metodológicos recientes en la literatura espacial integrada y del MLM, ofreciendo una mayor flexibilidad en términos de la especificación del modelo al permitir que los coeficientes de regresión varíen espacialmente. Se derivan algoritmos bayesianos de la cadena de Markov Monte Carlo (MCMC) para implementar el modelo propuesto. Usando datos de satisfacción de viaje a dos niveles para Beijing, aplicamos el enfoque propuesto lo mismo que el MLM de inclinación aleatoria estándar no-espacial para investigar la satisfacción subjetiva de viaje de los residentes y sus determinadores. Los resultados de la comparación de los modelos muestran una fuerte evidencia de que el método propuesto produce una mejora significativa frente al enfoque MLM de inclinación aleatoria no-espacial. Un parámetro de correlación espacial bastante grande sugiere una fuerte dependencia espacial de los efectos aleatorios a nivel de distrito. Aún más, los patrones espaciales de los efectos aleatorios a nivel de distrito de las variables locacionales han sido identificados, con los valores altos y bajos agrupándose entre sí.

Key Words:

• Bayesian estimation
• Beijing
• conditional autoregressive model
• multilevel modeling
• subjective travel satisfaction

关键词：:

• 贝叶斯评估法
• 北京
• 条件式自迴归模型
• 多层级模式化
• 主观旅行满意度。

Palabras clave:

• estimación bayesiana
• Beijing
• modelo de auto-regresión condicional
• modelado a nivel múltiple
• satisfacción subjetiva de viaje

Acknowledgments

The authors are grateful for the comments of three reviewers and the editor, which have greatly improved the content of this article. They also would like to thank Professor Bill Browne and Dr. George Leckie from the Centre for Multilevel Modeling (CMM), University of Bristol for providing valuable comments on the previous draft of the article. Thanks are also due to Professor Wenzhong Zhang and Dr. Jianhui Yu from the Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences for providing the data used in the study.

Funding

This work was funded by the Economic and Social Research Council (ESRC) through the Applied Quantitative Methods Network: Phase II, grant number ES/K006460/1. The first author also gratefully acknowledges the ESRC for funding his doctoral research during 2011 and 2014 and support from the National Natural Science Foundation of China (Project No. 41201169).

Notes

1. For each of the spatial multilevel models implemented, the convergence of the MCMC sampler is diagnosed using the CODA package (Plummer et al. 2006) in R. In terms of efficient computation, a crucial part is updating the spatially random effect, a JP by 1 (384 by 1 in this case) vector based on its full posterior conditional distribution. We take advantage of a desirable characteristic of the GMRFs, the sparsity of their precision matrix, and therefore some fast sparse matrix Cholesky factorization can be carried out. More specifically, a canonical parameterization of the posterior distribution of the spatial random effect is used to draw samples of them via a very useful and computationally efficient function in an R package, SPAM, created by Furrer and Sain (2010). Details on fast sampling algorithms for GMRFs are provided in Rue and Held (2005) and Furrer and Sain (2010). It takes about four minutes for the MCMC sampler of the MLM-MLCAR model to produce 10,000 samples on a laptop with an Intel Core 2.5 GHz processor. Before applying the code to the travel satisfaction data, we conducted a series of simulation studies with known data generating processes and using the geography (data structure) of the travel data to test the code. The results show that spatial multilevel models can accurately retrieve the true model parameters. The R code for implementing the spatial multilevel models and the simulation study results are available on request.

2. Based on estimates from MLM-MLCAR in Table 4, the marginal effects of commuting by car and public transports (with other transport modes as baseline category) are 0.492 + (–0.158) × Commuting time and 0.791 + (–0.218) × Commuting time, respectively. Equating the two marginal effects to zero (and exponentiating the solutions) gives the commuting time thresholds in the main text.

Additional information

Notes on contributors

Guanpeng Dong

GUANPENG DONG is a Postdoctoral Research Associate of Quantitative Methods in Sheffield Methods Institute, the University of Sheffield, Sheffield S1 4DP, UK. E-mail: [email protected]. His core research interests include developing integrated spatial statistical and multilevel modeling methodologies for properly analyzing geographically hierarchical data and the application of a wide range of spatial statistical and econometric approaches in housing markets, environmental evaluation, housing behavior analysis, and urban economics.

Jing Ma

JING MA is an Assistant Professor of Human Geography in the School of Geography, Beijing Normal University, Beijing, 100875, P.R. China. E-mail: [email protected]. Her main research interests include spatial microsimulation, activity-travel behavior, urban geography, environmental pollution, and sustainable development.

Richard Harris

RICHARD HARRIS is a Professor of Quantitative Social Geography in the School of Geographical Sciences, University of Bristol, Bristol BS8 1SS, UK. E-mail: [email protected]. His core research interests include spatial statistics and geodemographics in marketing, education, public policy, and urban geography.

Gwilym Pryce

GWILYM PRYCE is a Professor of Urban Economics and Social Statistics and the Director of Sheffield Methods Institute, University of Sheffield, Sheffield S1 4DP, UK. E-mail: [email protected]. His core research interests are largely in the broad field of urban economics, including housing and mortgage markets analysis, urban segregation, economic valuation of environmental amenities, and neighborhood effect inference.