Markov chain Monte Carlo methods for the Box-Behnken designs and centrally symmetric configurations

TL;DR

This paper develops Markov chain Monte Carlo methods for calculating p-values in count data models related to Box-Behnken designs, using algebraic structures like toric ideals and Groebner bases.

Contribution

It characterizes Markov bases for these models via generators of toric ideals for centrally symmetric configurations, linking algebraic geometry with statistical computation.

Findings

Markov bases are characterized as generators of toric ideals.

Groebner bases structure is elucidated for these configurations.

Numerical example demonstrates the method's application.

Abstract

We consider Markov chain Monte Carlo methods for calculating conditional p values of statistical models for count data arising in Box-Behnken designs. The statistical model we consider is a discrete version of the first-order model in the response surface methodology. For our models, the Markov basis, a key notion to construct a connected Markov chain on a given sample space, is characterized as generators of the toric ideals for the centrally symmetric configurations of root system D_n. We show the structure of the Groebner bases for these cases. A numerical example for an imaginary data set is given.