Markov basis for design of experiments with three-level factors

TL;DR

This paper develops Markov bases for fractional factorial designs with three-level factors, enabling conditional tests via Markov chain Monte Carlo for experiments with count data.

Contribution

It introduces a method to construct Markov bases for three-level factorial designs, linking experimental design with contingency table models.

Findings

Markov bases are derived for specific fractional factorial designs.

The approach allows for exact conditional inference in count-based experiments.

Connections are established between experimental designs and contingency table models.

Abstract

We consider Markov basis arising from fractional factorial designs with three-level factors. Once we have a Markov basis, $p$ values for various conditional tests are estimated by the Markov chain Monte Carlo procedure. For designed experiments with a single count observation for each run, we formulate a generalized linear model and consider a sample space with the same sufficient statistics to the observed data. Each model is characterized by a covariate matrix, which is constructed from the main and the interaction effects we intend to measure. We investigate fractional factorial designs with $3^{p-q}$ runs noting correspondences to the models for $3^{p-q}$ contingency tables.