Box-Cox Transformation

Box-Cox transformation is a power transformation that involves logs and powers. It transforms data into normal distributions.

The Box-Cox transformation is defined as

$$ y_i^{(\lambda)} = \begin{cases} \lambda ^{-1} (y_i^\lambda - 1) & \quad \text{if } \lambda \neq 0\\ \log(y_i) & \quad \text{if } \lambda = 0. \end{cases} $$

By selecting a proper $\lambda$, we get a Guassian distributed data, with a variable mean. The transformation take $y$ to

$$ \rho(y^{(\lambda)}) =\frac{ \exp{\left( -(y^{(\lambda)} - \beta X)^{T} (y^{(\lambda)} - \beta X)/(2\sigma^2) \right) }}{(\sqrt{2\pi \sigma^2})^n} \prod_{i=1}^n \left\lvert \frac{d y_i^{(\lambda )}}{ dy_i } \right\rvert. $$

The term

$$ \prod_{i=1}^n \left\lvert \frac{d y_i^{(\lambda )}}{ dy_i } \right\rvert = \lvert J \rvert $$

is the Jacobian as we are establishing the relations between the pdf of the data before and after the transfomation.

Analysis of Variance

Note that in analysis of variance, the results is not affected by linear transformations, thus the analysis of variance results will be the same as

$$ y_i^{(\lambda)} = \begin{cases} y_i^\lambda & \quad \text{if } \lambda \neq 0\\ \log(y_i) & \quad \text{if } \lambda = 0. \end{cases} $$

To find the proper $\lambda$, we write down the likelihood and maximize it. The likelihood is determined by the Gaussian distribution

$$ L(\lambda, \beta, \sigma) = \rho(y^{(\lambda)}). $$

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LM (2021). 'Box-Cox Transformation', Datumorphism, 07 April. Available at: https://datumorphism.leima.is/cards/statistics/box-cox/.