Mixed Model Methodology and the Box-Cox Theory of Transformations: A Bayesian Approach

TLDR: In this article, an extension of the Box-Cox theory of transformations to univariate mixed linear models is presented, including estimation of the transformation and of the required variance components, including computing algorithms.

Abstract: It is often assumed in animal breeding theory that models used for data analysis are “correct” with respect to functional form and distributional assumptions. However, a transformation may be needed to achieve this. An extension of the Box-Cox theory of transformations to univariate mixed linear models is presented. The discussion includes estimation of the transformation and of the required variance components, including computing algorithms. An analysis of fixed effects and breeding values after the transformation involves the following steps: (1) estimate ratios of variance components and the transformation parameter from their joint posterior distribution; (2) conditionally on these values, integrate out the residual variance (σ e 2 ) from the joint posterior distribution of fixed, random effects and σ e 2 , and (3) complete inferences using a multivariate-t distribution.