Bayesian Factor Analysis for Spatially Correlated Data, With Application to Summarizing Area-Level Material Deprivation From Census Data

This article describes a Bayesian hierarchical model for factor analysis of spatially correlated multivariate data. The Ž rst level speciŽ es, for each area on a map, the distribution of a vector of manifest variables conditional on an underlying latent factor; at the second level, the areaspeciŽ c latent factors have a joint distribution that incorporates spatial correlation. The framework allows for both marginal and conditional (e.g., conditional autoregressive) speciŽ cations of spatial correlation. The model is used to quantify material deprivation at the census tract level using data from the 1990 U.S. Census in Rhode Island. An existing and widely used measure of material deprivation is the Townsend index, an unweighted sum of four standardized census variables (i.e., Z scores) corresponding to area-level proportions of unemployment, car ownership, crowding, and home ownership. The Townsend and many related indices are computed as linear combinations of measured census variables, which motivates the factor-analytic structure adopted here. The model-based index is the posterior expectation of the latent factor, given the census variables and model parameters. Index construction based on a model allows several improvements over Townsend’s and similarly constructed indices: (1) The index can be represented as a weighted sum of (standardized) census variables, with data-driven weights; (2) by using posterior summaries, the indices can be reported with corresponding measures of uncertainty; and (3) incorporating information from neighboring areas improves precision of the posterior parameter distributions. Using data from Rhode Island census tracts, we apply our model to summarize variations in material deprivation across the state. Our analysis entertains various spatial covariance structures. We summarize the relative contributions of each census variable to the latent index, suggest ways to report material deprivation at the area level, and compare our model-based summaries with those found by applying the standard Townsend index.