BoxCoxtransformation

Box-Cox transformation is a family of power transformations designed to stabilize variance and make data more normally distributed. For a positive-valued variable x and a parameter λ, the transformation is Tλ(x) = (x^λ - 1)/λ if λ ≠ 0, and T0(x) = log x. Because x must be positive, data with zeros or negatives commonly require a constant offset before applying the transformation.

The transformation is often applied to the dependent variable in parametric models (for example linear regression or ANOVA) to satisfy assumptions of normality of residuals and homoscedasticity. It can also linearize relationships between variables and improve model fit.

The value of λ is typically chosen to maximize the likelihood of the observed data under a normal model for the transformed response (MLE). In practice, software packages provide Box-Cox routines that estimate λ and apply the transformation to the data. Results on the transformed scale must be back-transformed for interpretation on the original scale, noting potential bias after back-transformation.

Box-Cox requires strictly positive data; with zero/negative values, a shift is needed or one can use extensions such as the Yeo–Johnson transformation, which handles non-positive data. The transformation is not a cure-all; interpretability and the quality of the model still depend on the context and data properties.