The Effects of Spatial Population Distributions and Political Districting on Minority Representation

Rogerson, Peter A.; Yang, Zongxiang

doi:10.1177/089443939901700104

Cited by 3 publications

(3 citation statements)

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“…Some work has been done in which the properties of a "valid" districting are defined (which may be required to have have roughly equal populations among districts, have districts with reasonable boundaries, etc.) so that the characteristics of a given districting can be compared with what would be "typical" for a valid districting of the state in question, by using computers to generate random districtings [15,16]; see also [13] for discussion. However, much of this work has relied on heuristic sampling procedures which do not have the property of selecting districtings with equal probability (and, more generally, whose distributions are not well-characterized), undermining rigorous statistical claims about the properties of typical districts.…”

Section: Detecting Bias In Political Districtingmentioning

confidence: 99%

Assessing significance in a Markov chain without mixing

Chikina

Frieze

Pegden

2017

Proc. Natl. Acad. Sci. U.S.A.

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We present a statistical test to detect that a presented state of a reversible Markov chain was not chosen from a stationary distribution. In particular, given a value function for the states of the Markov chain, we would like to show rigorously that the presented state is an outlier with respect to the values, by establishing a p value under the null hypothesis that it was chosen from a stationary distribution of the chain. A simple heuristic used in practice is to sample ranks of states from long random trajectories on the Markov chain and compare these with the rank of the presented state; if the presented state is a 0.1% outlier compared with the sampled ranks (its rank is in the bottom 0.1% of sampled ranks), then this observation should correspond to a p value of 0.001. This significance is not rigorous, however, without good bounds on the mixing time of the Markov chain. Our test is the following: Given the presented state in the Markov chain, take a random walk from the presented state for any number of steps. We prove that observing that the presented state is an ε-outlier on the walk is significant at p = √ 2ε under the null hypothesis that the state was chosen from a stationary distribution. We assume nothing about the Markov chain beyond reversibility and show that significance at p ≈ √ ε is best possible in general. We illustrate the use of our test with a potential application to the rigorous detection of gerrymandering in Congressional districting.Markov chain | mixing time | gerrymandering | outlier | p value T he essential problem in statistics is to bound the probability of a surprising observation under a null hypothesis that observations are being drawn from some unbiased probability distribution. This calculation can fail to be straightforward for a number of reasons. On the one hand, defining the way in which the outcome is surprising requires care; for example, intricate techniques have been developed to allow sophisticated analysis of cases where multiple hypotheses are being tested. On the other hand, the correct choice of the unbiased distribution implied by the null hypothesis is often not immediately clear; classical tools like the t test are often applied by making simplifying assumptions about the distribution in such cases. If the distribution is well-defined but is not be amenable to mathematical analysis, a p value can still be calculated using bootstrapping if test samples can be drawn from the distribution.A third way for p value calculations to be nontrivial occurs when the observation is surprising in a simple way and the null hypothesis distribution is known but where there is no simple algorithm to draw samples from this distribution. In these cases, the best candidate method to sample from the null hypothesis is often through a Markov chain, which essentially takes a long random walk on the possible values of the distribution. Under suitable conditions, theorems are available that guarantee that the chain converges to its stationary distribution, allowing a random sample t...

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Section: Detecting Bias In Political Districtingmentioning

confidence: 99%

Assessing significance in a Markov chain without mixing

Chikina

Frieze

Pegden

2017

Proc. Natl. Acad. Sci. U.S.A.

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“…For 'random' redistricting analysis, such as to probe the characteristics of arbitrary redistricting plans (see Engstrom and Wildgen 1977;Rossiter and Johnston 1981;O'Loughlin 1982;Grofman 1982;Cirincione et al 2000), generation may be used without refinement, or with refinement only to absolute legal requirements. Analysis of legal constraints on redistricting, as in Altman (1997) and Rogerson and Yang (1999), may involve repeated generation and refinement. BARD further supports automated reweighting of a score function to generate a profile of how one redistricting criterion changes as another is optimized and in this manner it will automatically generate profiles of plans that explore tradeoffs among redistricting criteria.…”

Section: Redistricting With Bardmentioning

confidence: 99%

“…Scholars have proposed methods other than full enumeration to overcome the computational limitations of enumeration on even modestly redistricting problems, with as little as 50 units to assign to districts. Another way of using automated redistricting for analysis is taken by Altman (1997) and Rogerson and Yang (1999). These authors propose automated refinement of redistricting plans to simulate the effects of additional legal constraints on redistricting outcomes.…”

Section: Use Of Automated Redistricting For Plan Analysismentioning

confidence: 99%

BARD: Better Automated Redistricting

Altman¹,

McDonald²

2011

J. Stat. Soft.

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BARD is the first (and at time of writing, only) open source software package for general redistricting and redistricting analysis. BARD provides methods to create, display, compare, edit, automatically refine, evaluate, and profile political districting plans. BARD aims to provide a framework for scientific analysis of redistricting plans and to facilitate wider public participation in the creation of new plans. BARD facilitates map creation and refinement through command-line, graphical user interface, and automatic methods. Since redistricting is a computationally complex partitioning problem not amenable to an exact optimization solution, BARD implements a variety of selectable metaheuristics that can be used to refine existing or randomlygenerated redistricting plans based on user-determined criteria. Furthermore, BARD supports automated generation of redistricting plans and profiling of plans by assigning different weights to various criteria, such as district compactness or equality of population. This functionality permits exploration of trade-offs among criteria. The intent of a redistricting authority may be explored by examining these trade-offs and inferring which reasonably observable plans were not adopted. Redistricting is a computationally-intensive problem for even modest-sized states. Performance is thus an important consideration in BARD's design and implementation. The program implements performance enhancements such as evaluation caching, explicit memory management, and distributed computing across snow clusters.

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The Limitations of Quantitative Methods for Analyzing Gerrymanders: Indicia, Algorithms, Statistics and Revealed Preference

Altman¹,

McDonald

2007

SSRN Journal

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The Effects of Spatial Population Distributions and Political Districting on Minority Representation

Cited by 3 publications

References 4 publications

Assessing significance in a Markov chain without mixing

Assessing significance in a Markov chain without mixing

BARD: Better Automated Redistricting

The Limitations of Quantitative Methods for Analyzing Gerrymanders: Indicia, Algorithms, Statistics and Revealed Preference

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