Connecting Tables with Allowing Negative Cell Counts

TL;DR

This paper explores connecting contingency tables via Markov bases when cell counts are allowed to be negative, proposing new methods and demonstrating their effectiveness through empirical data analysis.

Contribution

It introduces a novel approach to connect tables using subsets of Markov bases with negative cell counts and applies this to a no-three-way interaction model.

Findings

Subset of Markov basis connects all tables with a table of all ones

Basic moves of 2x2x2 minors can connect tables with -1 cell counts under certain models

Empirical MCMC experiments show the method's effectiveness

Abstract

It is well-known that computing a Markov basis for a discrete loglinear model is very hard in general. Thus, we focus on connecting tables in a fiber via a subset of a Markov basis and in this paper, we consider connecting tables if we allow cell counts in each tale to be $-1$. In this paper we show that if a subset of a Markov basis connects all tables in the fiber which contains a table with all ones, then moves in this subset connect tables in the fiber if we allow cell counts to be $-1$. In addition, we show that in some cases under the no-three-way interaction model, we can connect tables by all basic moves of $2 \times 2 \times 2$ minors with allowing $X_{ijk} \geq -1$. We then apply this Markov Chain Monte Carlo (MCMC) scheme to an empirical data on Naval officer and enlisted population. Our computational experiments show it works well and we end with the conjecture on the no-three-way interaction model.