Using Bayesian Methods to Control for Spatial Autocorrelation in Environmental Justice Research: An Illustration Using Toxics Release Inventory Data for a Sunbelt County

ABSTRACT:

Many previous environmental justice (EJ) studies have argued that there is disproportionate collocation of environmental disamenities with racial and ethnic minorities, even holding constant other factors such as income and political action. However, most of the EJ studies do not account for the presence of spatial autocorrelation, especially those that also include nonnormal distributions. Using the location of new Toxics Release Inventory facilities (TRIFs) in Maricopa County, Arizona in the 1990s, we illustrate a finding of spatial autocorrelation and the use of Bayesian spatial models to accommodate the issue. The results show that the relationship between Asian minority status in a census tract and new TRIF establishments found with regression models does not remain statistically significant once spatial autocorrelation is accounted for. Instead, three variables, the percentage of American Indians in the tract, population density, and the percentage of residents aged 55–74, statistically significantly explained new TRIF establishments. This illustrates that failure to control for spatial autocorrelation can lead to incorrect policy understanding.

Notes

1 Ordinary least squares (OLS) regression estimation is technically suitable for any interval-level data. In spite of only having 0 or positive values, count data with many possible outcomes (such as income data) can be estimated well using OLS. However, in cases where the distribution of the dependent variable is constrained in some way, such as when estimating relatively rare counted events, other estimators can be more efficient because they take into account the limits of the dependent variable. The establishments of new TRIFs in Maricopa County in the 1990s are very limited. They are truncated at 0, and the maximum number in a single census tract is no more than 3.

2 Theoretically, the variance of a Poisson distribution is the same as its mean. However, empirical data sets often show more variability than the theoretical variance. This phenomenon, which is called overdispersion, occurs for a distribution whose variance is a function of its mean such as Poisson and binomial distributions (Breslow, 1984).

3 Although modified likelihood-based methods such as the maximizing pseudo-likelihood method have been proposed, these methods basically sacrifice efficiency to achieve a closed form (e.g., pseudo-likelihood). In addition, this method can accommodate only negative spatial autocorrelation but not positive spatial autocorrelation (Besag, 1974). It is a critical issue in that most empirical data sets show positive spatial autocorrelation (e.g., Griffith, 2006).

4 Though the data used here are not the newest—and are not all of the data in Campbell et al. (2010)—they are suitable for analyzing effects of controlling for spatial autocorrelation while still retaining the appropriate temporal relationships between the placement of new TRIFs and the existing population at the time of placement.

5 This material follows Hamilton’s (1995) focus on potential compensation costs. Because our data are for 1990–2000, any changes in law since then do not invalidate these concerns at that time.

6 The Bayesian regression models are estimated using WinBUGS software. Posterior distributions are constructed with 100,000 samples by eliminating the initial 5,000 samples to disregard samples from unstabilized chains (the “burn-in” process) and taking every 10th sample.

7 When the offset term is dropped, Poisson regression fits to raw count data rather than rate data. See McCullagh and Nelder (1989) for a detailed discussion of Poisson regression.