Bayesian estimation of explained variance in ANOVA designs.

Area of Science:

• Statistics
• Experimental Design
• Psychometrics

Background:

• Effect size measures are crucial for interpreting the magnitude of experimental effects in statistical analyses.
• Traditional effect size measures may not fully capture the complexity of variance explained in analysis-of-variance (ANOVA) designs.
• Bayesian statistical methods offer a flexible framework for parameter estimation and uncertainty quantification.

Purpose of the Study:

• To propose the squared multiple correlation coefficient (R²) as a comprehensive effect size measure for experimental ANOVA.
• To develop Bayesian methods for estimating the posterior distribution of R² in ANOVA.
• To provide practical guidance and examples for applying these methods in psychological research.

Main Methods:

• Derivation of formulas for squared multiple, semipartial, and partial correlation coefficients for four common ANOVA designs.
• Application of Bayesian inference techniques to estimate the posterior distribution of the proposed effect size measure.
• Illustrative examples using real-world data to demonstrate the practical utility of the proposed approach.

Main Results:

• The squared multiple correlation coefficient (R²) is shown to be a suitable and informative effect size for ANOVA.
• Bayesian estimation provides a full posterior distribution for R², offering richer information than point estimates.
• The derived formulas and worked examples facilitate the application of R² in various ANOVA contexts.

Conclusions:

• The squared multiple correlation coefficient offers a valuable effect size metric for ANOVA, enhancing the interpretation of experimental findings.
• Bayesian estimation of R² provides a robust and informative approach to quantifying effect sizes.
• This work equips researchers with practical tools for more rigorous analysis and reporting of experimental results.