BoxCox

The Box-Cox transformation is a family of power transformations used to stabilize variance and improve normality of data for statistical modeling. It was introduced by George Box and David Cox in 1964. The transformation is defined for a positive response variable y as T(y; λ) = (y^λ − 1)/λ for λ ≠ 0, and T(y; 0) = log(y).

In practice, the parameter λ is chosen to optimize a criterion, typically maximizing the likelihood under the assumption that the transformed data follow a normal distribution with constant variance. This approach often improves the applicability of linear models, ANOVA, regression diagnostics, and time-series modeling by mitigating heteroscedasticity and skewness.

The transformation requires positive data, since y^λ is defined for positive y; common practice is to shift data by a constant to render it positive if necessary. The inverse transformation is y = (λ T + 1)^{1/λ} for λ ≠ 0, and y = exp(T) when λ = 0.

Extensions of the method include the Box-Cox profile, which plots the log-likelihood as a function of λ to help select the best value. If data include zero or negative values, alternatives such as the Yeo–Johnson transformation extend the Box-Cox family to non-positive data.

Applications of the Box-Cox transformation span regression, ANOVA, and various time-series analyses, where stabilizing variance and approximating normality can lead to better model assumptions, improved inference, and more reliable predictions.