# Separating effect from significance in Markov chain tests

## TL;DR

This paper improves the theoretical foundations for significance testing in Markov Chains, aiming to make claims about political districtings more rigorous and interpretable by distinguishing effect from significance.

## Contribution

It provides new theorems that enhance the ability to demonstrate the extremeness of Markov Chain states, especially in the context of political gerrymandering analysis.

## Key findings

- Enhanced theorems for significance testing in Markov Chains.
- Ability to demonstrate extreme unlikelihood of states at desired significance levels.
- Specialized results leveraging product structures in probability spaces.

## Abstract

We give qualitative and quantitative improvements to theorems which enable significance testing in Markov Chains, with a particular eye toward the goal of enabling strong, interpretable, and statistically rigorous claims of political gerrymandering. Our results can be used to demonstrate at a desired significance level that a given Markov Chain state (e.g., a districting) is extremely unusual (rather than just atypical) with respect to the fragility of its characteristics in the chain. We also provide theorems specialized to leverage quantitative improvements when there is a product structure in the underlying probability space, as can occur due to geographical constraints on districtings.