Power transform

A power transformation replaces positive data by a member of a parameterized family of powers, commonly to improve the fit of a normal linear model.

For $y>0$, the Box–Cox transformation isThe value of $\lambda$ can be selected by maximizing its profile likelihood; $\lambda =1$ corresponds to an affine transformation of the original response, while $\lambda =0$ gives a logarithmic transformation.

A transformed-response regression estimates the conditional mean $m$ on the transformed scale. Inverting that mean does not generally estimate the original-scale mean, because nonlinear transformations do not commute with expected values. For a smooth inverse $g$ and centered transformed error of variance ${\sigma }^2$, a second-order Taylor expansion gives the displayed local correction. For $g(z)=z^q$ at positive $m$, it is $m^q+\frac{1}{2}q(q-1)m^{q-2}{\sigma }^2$. The variance is the response's residual variance, not the sampling variance of the estimated mean. This approximation requires errors small enough that the inverse is meaningful over the relevant range; it is not an exact moment formula for arbitrary transformed noise.