Spatial Autocorrelation and Political Redistricting: A Task for the Uniform Distribution

Abstract

An important quantitative geography application area derives from the periodically repeating apportionment process of redrawing legislative district boundaries to guarantee equal voter representation through identical or equivalent population counts, a situation confronting the U.S. Supreme Court with gerrymandering rulings on a decennial basis. This remains a small geographic sample size domain, with states having between one and fifty-two (i.e., California) congressional precincts in 2023. Texas constitutes an informative case study because its sample size of thirty-eight exceeds the frequently touted minimum of thirty, and its 2020 increase was two seats, more than any other state, some of which experienced decreases. Hence, although all fifty states also have undergone internal population redistribution, the Texas redrawn boundaries offer the greatest opportunity for dramatic borderline changes. This article appraises outcomes of this Texas gain within its legal constraints as they pertain to a novel uniform distribution explicitly embracing spatial autocorrelation, a fundamental georeferenced data property. Its name is sui-uniform to differentiate it from the prevalent auto- model convention; this article furnishes its first empirical application. After accounting for spatial autocorrelation, the inferential conclusion for Texas is as preferred: Superpopulation geotagged voting-age populations conform to a uniform distribution, whereas racial and ethnic subpopulations do not.

Key Words:

• gerrymandering
• political redistricting
• spatial autocorrelation
• sui-uniform distribution
• Texas

Disclosure Statement

No potential conflict of interest was reported by the author.

Notes

1 Just as for the standard beta-binomial parametric mixture distribution, the beta here is labeled a (conjugate) prior, whether the employed statistical approach is frequentist or Bayesian. In other words, frequentist, like Bayesian, statistics considers this beta to be a prior distribution, although parameter estimation is typically done using maximum likelihood or other frequentist techniques. In contrast, Bayesian statistics uses this beta to model a posterior distribution, given observed data; this beta furnishes a prior for the probability p variable, whereas posterior distribution updates involve observed data using Bayes’s theorem. Therefore, both the frequentist and Bayesian approaches call and use this beta distribution as a prior; however, its interpretation and parameter estimation methods might differ between these two frameworks. Its manifestation also differs: The frequentist mixture has n single values of a given variable (e.g., a varying intercept), whereas the Bayesian posterior distribution has a separate full distribution for each of these n individuals.

2 The practice of manipulating electoral district boundaries to bias political outcomes. It involves intentionally drawing district lines in a way that gives an unfair advantage to one political party, incumbent, or group over others in elections. This manipulation creates geographic distributions of unbalanced voting power, a situation appropriate uniform distribution analysis is capable of informing.

3 The Geary ratio exhibits its sensitivities to outliers, which y is, and deviant numbers of neighbors (i.e., deviations from the asymptotic average), with limiting values across the range of possibilities; only those cases having the asymptotic average number of neighbors yields a zero difference in the limiting mathematical expression.

4 For n = 30, a fully connected planar situation (i.e., b = 6(n‒2)), and a typical areal unit connection (i.e., a = 5), the absolute value of the MC difference is < 0.01, whereas the absolute value of the GR difference is roughly 0.10, neither of which is close to being statistically significant.

5 The difference is in scope. A mixed model random effect has n single estimates, whereas its Bayesian counterpart has a probability distribution for each of these n individual estimates.

6 Empirical data simply need to be converted to cumulative distribution percentages (e.g., see Blom 1954), and then estimated with a GLM beta regression technique to construct an ESF accounting for SA. One instant sui-uniform random variable diagnostic is that its intercept-only scale estimate should be $\widehat{\phi }$ = $\widehat{\alpha }$ + $\widehat{\beta }$ ≈ 2. This formulation requires both endpoints of the empirical cumulative distribution function to be adjusted by a small amount that moves them away from zero and one, similar to the Blom adjustments of (r – 1/8) and (n + 1/4) that accordingly shrink its lower and upper cumulative distribution function values from the closed interval [0, 1] to inside the open interval (0, 1).

7 Also see Van Hala et al. (2017), who demonstrate that “maximum empirical likelihood parameter estimator. exhibits surprisingly non-standard asymptotic properties for irregularly located spatial data” (109).

Additional information

Notes on contributors

Daniel A. Griffith

DANIEL A. GRIFFITH is Ashbel Smith Professor of Geospatial Information Sciences, School of Economic, Political, and Policy Sciences, University of Texas at Dallas, Richardson, TX 75080. E-mail: [email protected]. His current research interests include mathematics of eigenfunctions, spatial statistics, regional and urban spatial economics, and urban public health.