Estimating a transformation and its effect on Box-Cox T-ratio

This article concerns i) the stochastic behavior of the Box-Cox transformation estimator and ii) the effect of estimating a transformation on the Box-Cox T-ratio used for the post-transformation analysis. It is shown that the transformation estimator depends on three factors: the model structure, the mean-spread and the error standard deviation sigma(0). In general, a structured model is able to estimate the transformation very well; an unstructured model can do well also unless the mean-spread and sigma(0) are both small; and a one-mean mode can give a poor estimate if sigma(0) is small. When the sample is not large, it is shown that the unconditional effect of estimating a transformation on the Box-Cox T-ratio is generally small, and the "conditional" effect is also negligible in most of the situations except the case of one-way ANOVA with small sigma(0). Extensive Monte Carlo simulations are performed to support the theoretical findings.