A Markov Basis for Conditional Test of Common Diagonal Effect in Quasi-Independence Model for Square Contingency Tables

TL;DR

This paper develops a Markov basis for conducting conditional tests on the common diagonal effect in quasi-independence models for square contingency tables, enabling efficient MCMC-based goodness-of-fit testing.

Contribution

It derives an explicit Markov basis for the common diagonal effect test, facilitating algebraic statistical methods for this specific model.

Findings

Markov basis enables exact conditional testing

MCMC method applied successfully to real data

Simplifies testing for common diagonal effects

Abstract

In two-way contingency tables we sometimes find that frequencies along the diagonal cells are relatively larger(or smaller) compared to off-diagonal cells, particularly in square tables with the common categories for the rows and the columns. In this case the quasi-independence model with an additional parameter for each of the diagonal cells is usually fitted to the data. A simpler model than the quasi-independence model is to assume a common additional parameter for all the diagonal cells. We consider testing the goodness of fit of the common diagonal effect by Markov chain Monte Carlo (MCMC) method. We derive an explicit form of a Markov basis for performing the conditional test of the common diagonal effect. Once a Markov basis is given, MCMC procedure can be easily implemented by techniques of algebraic statistics. We illustrate the procedure with some real data sets.