Discussion
Which of the following statements is not true in the context of the transformation of a response variable under a regression set-up ?
Answer: D
The statement that is not true is The Box-Cox transformation is a discontinuous function of \(\lambda\).
The Box-Cox transformation is \(g(Y, \lambda) = (Y^{\lambda} - 1)/\lambda\) for \(\lambda \ne 0\) and \(\log Y\) for \(\lambda = 0\). The \(-1\) and the division by \(\lambda\) are there precisely to make it continuous in \(\lambda\): as \(\lambda \to 0\), \((Y^{\lambda} - 1)/\lambda \to \log Y\).
The other statements are the standard variance-stabilising rules. When the variance equals the mean (Poisson-like data), \(\sqrt{Y}\) stabilises it (A). When the standard deviation is proportional to the mean, \(\log Y\) does (B). And a positive change of scale does not change the sign of the association (C).
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