# Markov basis and Groebner basis of Segre-Veronese configuration for testing independence in group-wise selections

## TL;DR

This paper develops a method using Markov and Groebner bases for efficient conditional testing of independence in complex group-wise selection models, with applications to educational testing and genetics.

## Contribution

It provides explicit Groebner bases for Segre-Veronese configurations, enabling practical MCMC-based independence tests in restricted selection models.

## Key findings

- Explicit degree-two Groebner bases for Segre-Veronese configurations.
- Application to Japanese university entrance exam data.
- Testing independence in genetic data involving multiple loci.

## Abstract

We consider testing independence in group-wise selections with some restrictions on combinations of choices. We present models for frequency data of selections for which it is easy to perform conditional tests by Markov chain Monte Carlo (MCMC) methods. When the restrictions on the combinations can be described in terms of a Segre-Veronese configuration, an explicit form of a Gr\"obner basis consisting of moves of degree two is readily available for performing a Markov chain. We illustrate our setting with the National Center Test for university entrance examinations in Japan. We also apply our method to testing independence hypotheses involving genotypes at more than one locus or haplotypes of alleles on the same chromosome.