Explicit agreement extremes for a $2\times2$ table with given marginals

TL;DR

This paper provides an explicit analytical solution for maximizing or minimizing agreement between two 2x2 clusterings with fixed marginals, advancing beyond previous reliance on numerical algorithms.

Contribution

It introduces a closed-form solution for agreement extremes in 2x2 clustering tables, simplifying analysis and computation.

Findings

Explicit formula for agreement extremes in 2x2 tables

Reduces reliance on numerical algorithms

Facilitates faster analysis of clustering agreement

Abstract

The problem of maximizing (or minimizing) the agreement between clusterings, subject to given marginals, can be formally posed under a common framework for several agreement measures. Until now, it was possible to find its solution only through numerical algorithms. Here, an explicit solution is shown for the case where the two clusterings have two clusters each.