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

Abstract

This article describes a Bayesian hierarchical model for factor analysis of spatially correlated multivariate data. The rst level speci 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 areaspeci c latent factors have a joint distribution that incorporates spatial correlation. The framework allows for both marginal and conditional (e.g., conditional autoregressive) speci 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.KEY WORDS: Conditional autoregressive model; Health inequalities; Latent variable; Posterior rank; Small-area estimation; Socioeconomic status; Townsend index. FACTOR ANALYSIS AND SPATIAL DATAFactor-analytic models are useful for summarizing variance and covariance patterns in multivariate data. A common formulation of factor analysis assumes that measurable variables, such as scores on a test, are manifestations of an underlying latent construct, such as ability or intelligence. The latent variable formulation can be useful for data reduction, that is, summarizing multivariate observations using a lowerdimensional variable. A thorough review has been given by Bartholomew and Knott (1999, chaps. 1-3). Recent work from a Bayesian perspective has been done by Geweke and Zhou (1996), Press and Shigamesu (1997), Aguilar and West (2000), and Rowe (2002).Multivariate spatial data can arise in a number of applied contexts. Wang andWall (2001, 2003) studied multivariate indicators of cancer ris...

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