Markov chain Monte Carlo tests for designed experiments

Abstract

We consider conditional exact tests of factor effects in designed experiments for discrete response variables. Similarly to the analysis of contingency tables, a Markov chain Monte Carlo method can be used for performing exact tests, when large-sample approximations are poor and the enumeration of the conditional sample space is infeasible. For designed experiments with a single observation for each run, we formulate log-linear or logistic models and consider a connected Markov chain over an appropriate sample space. In particular, we investigate fractional factorial designs with $2^{p-q}$ runs, noting correspondences to the models for $2^{p-q}$ contingency tables.

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• Minimal Markov basis for tests of main effect models for $2^{p-1}$ fractional factorial designs of resolution $p$
• Markov chain Monte Carlo methods for the regular two-level fractional factorial designs and cut ideals
• Saturated fractions of two-factor designs
• Markov Chain Monte Carlo sampling for conditional tests: A link between permutation tests and algebraic statistics