Detection of Hidden Additivity and Inference Under Model Uncertainty for Unreplicated Factorial Studies via Bayesian Model Selection and Averaging

Abstract

The two-way unreplicated layout remains a popular study design in the physical sciences. However, detection of statistical interaction and subsequent inference has been problematic in this class of designs. First, lack of replication precludes inclusion of standard interaction parameters. Second, while several restricted forms of interaction have been considered, existing approaches focus primarily on accept/reject decisions with respect to the presence of interaction. Approaches to estimate cell means and error variance are lacking when the possibility of interaction exists. For these reasons, we propose model selection and averaging-based approaches to facilitate statistical inference when the presence of interaction is uncertain. Hidden additivity, a recently proposed and intuitive form of interaction, is used to accommodate latent group-based nonadditive effects. The approaches are fully Bayesian and use the Zellner–Siow formulation of the mixture g-prior. The method is illustrated on three empirical datasets and simulated data. The estimates from the model averaging approach are compared with a customized regularization approach which shrinks interaction effects toward the additive model. The study concludes that Bayesian model selection is a fruitful approach to detect hidden additivity, and model averaging allows for inference on quantities of interest under model uncertainty with respect to interaction effects within the two-way unreplicated design.

Keywords:

• Hidden additivity
• Mixture g-priors
• Model averaging
• Model selection-priors
• Model selection
• Model averaging