An Analysis of Transformations Revisited

Abstract Following Box and Cox (1964), we assume that a transform Z i = h (Yi, λ) of our original data Yi satisfies a linear model. Consistency properties of the Box-Cox estimates (MLE's) of λ and the parameters in the linear model, as well as the asymptotic variances of these estimates, are considered. We find that in some structured models such as transformed linear regression with small to moderate error variances, the asymptotic variances of the estimates of the parameters in the linear model are much larger when the transformation parameter λ is unknown than when it is known. In some unstructured models such as transformed one-way analysis of variance with moderate to large error variances, the cost of not knowing λ is moderate to small. The case where the error distribution in the linear model is not normal but actually unknown is considered, and robust methods in the presence of transformations are introduced for this case. Asymptotics and simulation results for the transformed additive two-way layout show that much is gained by this robustification of the Box-Cox methods when the ratios of the means to the error standard deviation are moderate to large. However, the performance of all Box-Cox type procedures is unstable and highly dependent on the parameters of the model in structured models with small to moderate error variances.