Conjugate distributions in hierarchical Bayesian ANOVA for computational efficiency and assessments of both practical and statistical significance

TL;DR

This paper introduces a hierarchical Bayesian ANOVA framework using conjugate distributions that unifies various models, enhances computational efficiency, and emphasizes practical significance over purely statistical measures.

Contribution

It presents a comprehensive Bayesian ANOVA method that unifies fixed, random, and mixed effects models with straightforward computational procedures and addresses parameter identifiability.

Findings

Provides a unified Bayesian ANOVA framework

Offers computationally efficient inferential procedures

Focuses on practical significance in analysis

Abstract

Assessing variability according to distinct factors in data is a fundamental technique of statistics. The method commonly regarded to as analysis of variance (ANOVA) is, however, typically confined to the case where all levels of a factor are present in the data (i.e. the population of factor levels has been exhausted). Random and mixed effects models are used for more elaborate cases, but require distinct nomenclature, concepts and theory, as well as distinct inferential procedures. Following a hierarchical Bayesian approach, a comprehensive ANOVA framework is shown, which unifies the above statistical models, emphasizes practical rather than statistical significance, addresses issues of parameter identifiability for random effects, and provides straightforward computational procedures for inferential steps. Although this is done in a rigorous manner the contents herein can be seen as ideological in supporting a shift in the approach taken towards analysis of variance.