Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- Frequently used notations and symbols
- 1 Algebraic and geometric methods in statistics
- Part I Contingency tables
- Part II Designed experiments
- 9 Generalised design: interpolation and statistical modelling over varieties
- 10 Design of experiments and biochemical network inference
- 11 Replicated measurements and algebraic statistics
- 12 Indicator function and sudoku designs
- 13 Markov basis for design of experiments with three-level factors
- Part III Information geometry
- Part IV Information geometry and algebraic statistics
- Part V On-line supplements
13 - Markov basis for design of experiments with three-level factors
from Part II - Designed experiments
Published online by Cambridge University Press: 27 May 2010
- Frontmatter
- Contents
- List of contributors
- Preface
- Frequently used notations and symbols
- 1 Algebraic and geometric methods in statistics
- Part I Contingency tables
- Part II Designed experiments
- 9 Generalised design: interpolation and statistical modelling over varieties
- 10 Design of experiments and biochemical network inference
- 11 Replicated measurements and algebraic statistics
- 12 Indicator function and sudoku designs
- 13 Markov basis for design of experiments with three-level factors
- Part III Information geometry
- Part IV Information geometry and algebraic statistics
- Part V On-line supplements
Summary
Abstract
We consider Markov bases arising from regular fractional factorial designs with three-level factors. They are used in a Markov chain Monte Carlo procedure to estimate p-values for various conditional tests. For designed experiments with a single observation for each run, we formulate a generalised linear model and consider a sample space with the same values of that sufficient statistic for the parameters under the null model as for the observed data. Each model is characterised by a covariate matrix, which is constructed from the main and the interaction effects. We investigate fractional factorial designs with 3p−q runs and underline a correspondence with models for 3p−q contingency tables.
Introduction
In the past decade, a new application of computational algebraic techniques to statistics has been developed rapidly. On one hand, (Diaconis and Sturmfels 1998) introduced the notion of Markov basis and presented a procedure for sampling from discrete conditional distributions by constructing a connected, aperiodic and reversible Markov chain on a given sample space. Since then, many works have been published on the topic of the Markov basis by both algebraists and statisticians. Contributions of the present authors on Markov bases can be found in (Aoki et al. 2008, Aoki and Takemura 2003, Aoki and Takemura 2005, Aoki and Takemura 2006, Aoki and Takemura 2008a, Aoki and Takemura 2008b Aoki et al. 2008, Hara et al. 2009, Takemura and Aoki 2004) and (Takemura and Aoki 2005).
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- Information
- Algebraic and Geometric Methods in Statistics , pp. 225 - 238Publisher: Cambridge University PressPrint publication year: 2009
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