A Bayesian Analysis of Unreplicated Two-Level Factorials Using Effects Sparsity, Hierarchy, and Heredity

ABSTRACT

This article proposes a Bayesian procedure to calculate posterior probabilities of active effects for unreplicated two-level factorials. The results from a literature survey are used to specify individual prior probabilities for the activity of effects and the posterior probabilities are then calculated in a three-step procedure where the principles of effects sparsity, hierarchy, and heredity are successively considered. We illustrate our approach by reanalyzing experiments found in the literature.

KEYWORDS:

• Bayesian analysis
• engineering judgments
• Markov chain Monte Carlo integration
• posterior probability of active effects
• prior information
• unreplicated factorials

ACKNOWLEDGMENTS

We gratefully acknowledge the financial support from the Swedish mining company LKAB, as well as the county administrative board under grant 303–02863-2008, and the Regional Development Fund of the European Union, grant 43206, that made this research possible. We thank the editor and the anonymous reviewers for the constructive comments that improved this article.

Notes

Column y is the number of responses in the experiment. Column k is the variance inflation factor according to Box and Meyer (1986).

Two of the experiments did not generate any active effects.

α for Round 1 is 0.2 for all effects. The posterior probabilities with values of 0.5 or larger are underlined.

*Effect larger than Lenth's ME.

**Effect larger than Lenth's SME. (Lenth's ME: 1.499, Lenth's SME 3.043, using the significance level 0.05).

is the average posterior probability for the contrast based on 200 randomly drawn single-replicate samples from the data in Table 6. s _P is the standard deviation for the 200 posterior probabilities for each round. The contrasts considered active in the original analysis are underlined and so are the average posterior probabilities with values of 0.5 or larger in the three rounds.