An Analysis of Transformations

TLDR

Data analysis typically assumes independent normal observations with constant variance and linear expectations in parameters θ, and this work relates its transformation methods to earlier procedures. The paper aims to relax the normality assumption by applying a suitable transformation to the data before fitting a linear model. The authors compute the likelihood and posterior distribution to infer the transformation and model parameters, separating the contributions of normality, homoscedasticity, and additivity. The proposed transformation methods are demonstrated through illustrative examples.

Abstract

Summary In the analysis of data it is often assumed that observations y 1, y 2, …, yn are independently normally distributed with constant variance and with expectations specified by a model linear in a set of parameters θ. In this paper we make the less restrictive assumption that such a normal, homoscedastic, linear model is appropriate after some suitable transformation has been applied to the y's. Inferences about the transformation and about the parameters of the linear model are made by computing the likelihood function and the relevant posterior distribution. The contributions of normality, homoscedasticity and additivity to the transformation are separated. The relation of the present methods to earlier procedures for finding transformations is discussed. The methods are illustrated with examples.