Bayesian Analysis of ANOVA and Mixed Models on the Log-Transformed Response Variable

Area of Science:

• Psychology
• Statistics
• Computational Statistics

Background:

• Analysis of variance (ANOVA) and mixed models are widely used in psychological research.
• Bayesian inference offers advantages for complex experimental designs and data structures.
• Log-transforming response variables can lead to issues with standard priors on variance parameters, causing non-existent posterior moments.

Purpose of the Study:

• To address inferential challenges with standard priors on variance parameters in Bayesian mixed models when using log-transformed response variables.
• To propose a robust and generalizable prior distribution for variance parameters.
• To facilitate accurate Bayesian inference on the original data scale.

Main Methods:

• Utilizing generalized inverse Gaussian distributions as priors for variance parameters.
• Careful selection of hyper-parameters for the proposed priors.
• Theoretical derivations and simulation studies to validate the approach.
• Development of accompanying software for implementation.

Main Results:

• The proposed generalized inverse Gaussian priors resolve issues of non-existent posterior moments and predictive distributions in the original data scale.
• Theoretical and simulation results confirm the effectiveness of the proposed method.
• A practical software package is available for applying the analysis.

Conclusions:

• Generalized inverse Gaussian distributions provide a suitable and general solution for priors on variance parameters in Bayesian mixed models with log-transformed data.
• This approach enables valid posterior inference and prediction on the original response scale, particularly relevant for response time data.
• The developed methodology and software enhance the application of Bayesian mixed models in psychological research.