A Multivariate Spatial Mixture Model for Areal Data: Examining Regional Differences in Standardized Test Scores

Motivated by a study exploring geographic disparities in test scores among fourth graders in North Carolina, we develop a multivariate mixture model for the spatial analysis of correlated continuous outcomes. The responses are modelled as a finite mixture of multivariate normal distributions, which accommodates a wide range of marginal response distributions and allows investigators to examine covariate effects within subpopulations of interest. The model has a hierarchical structure incorporating both individual and areal level predictors as well as spatial random effects for each mixture component. Conditional auto-regressive priors on the random effects provide spatial smoothing and allow the shape of the multivariate distribution to vary flexibly across geographic regions. By integrating over this distribution, we obtain region-specific joint, marginal and conditional inferences of interest. We adopt a Bayesian modelling approach and develop an efficient posterior sampling algorithm that relies primarily on closed form full conditionals. Our results show that students in the central and coastal counties of North Carolina demonstrate higher achievement on average than students in the other parts of the state. These findings can be used to guide county level initiatives, such as school-based literacy programmes, to improve elementary education.