Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 206 / 2 / d

The Box–Cox transformation searches over transformations of the positive response,The plotted horizontal coordinate is $\lambda$ and the vertical coordinate is its maximized profile likelihood on the log scale. The likelihood includes the transformation Jacobian determinant; omitting it would compare response scales incorrectly. For the Gaussian regression it is, up to constants,The maximum occurs around $\lambda =2$. The horizontal 95% line is approximately ${\ell }_p(\hat{\lambda })-{\chi }_{1,0.95}^2/2$, and its intersections give an approximate interval roughly $1.2$ to $2.8$, read visually. In particular, $\lambda =1$ is below the line, and $\lambda =0$ is far below it.

Square the response: a convenient transformation is $Y_i^2$, equivalent up to affine rescaling to $T_2(Y_i)=(Y_i^2-1)/2$. Transform energy, not the sunny-days or latitude predictors. Refit and check the normal linear model on the transformed scale; the plot is evidence for a transformation, not proof that the transformed residual assumptions hold. Inference on transformed means cannot be inverted naively to obtain original-scale conditional means.