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

Segregation in the Twenty First Century: Processes, Complexities and Future Directions

“…Advances in other literatures might help us here. For example, spatial proximity and spatial hierarchy have recently been integrated into a unified modelling framework and applied to individual‐level data by Dong et al () using Bayesian multivariate conditional autoregressive models. It may be possible to use this approach to extend the multi‐level modelling method presented by Manley et al to decompose different spatial facets of segregation in the context of individual‐level longitudinal data.…”

Comparing definitions of spatial relations for the analysis of geographic disparities in mortality within a Bayesian mixed-effects framework

“…However, those models tend to be applied to aggregate data and have disease mapping purposes rather than analytical purposes with individual data. An exception can be found in recent work by Dong et al where the model is extended to individual nested data 5 . In this paper, a hybrid approach to geographic disparity analyses with elements from social epidemiology and spatial econometrics were tested in an attempt to face spatial relation misspecification and ecological bias, the main limitations of each individual methodology.…”

“…As such, the impact of weighting schemes in the SMM model fit must be explored in depth. In this regard, the CAR model has the advantage of an unambiguous definition of weights, which is equal to one for adjacent municipalities, and to zero otherwise 5 . Additionally, the SMM chain convergence for 5,000 iterations was not optimal; 3.…”

“…It models area-level spatial autocorrelation of residuals using a conditional autoregressive prior 5 with functions for mean and variance, which involve the neighborhoods residuals ( Figure 1, Equation 6). Similar to spatial autocorrelation that is modeled in the distribution function, the CAR model does not apply weights to area-level residuals like the SMM model does, but it uses the same equation for the CM model (Figure 1, Equation 4).…”

“…In other words, these models do not take into consideration the spatial configuration of geographic units. The validity of interpretations about geographical phenomena that arise from non-spatial mixed-effects analyses have been questioned 5 . Even when the standard hierarchical MM only takes into account individual membership to a single geographic unit, interpretations tend to be done in terms of geographical effects.…”