In the book, Linear Models with R by Julian Faraway, it says that if max(y)/min(y) is small, then Box Cox won't do anything because power transformations are well approximated by linear transformations over short intervals. Why does it say that power transformation is approximated by linear transformation over short intervals? And if so, why when max(y)/min(y) is small, Box Cox fails?
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$\begingroup$ $(m+h)^r\approx m^r(1+r\frac h m)$ for small $\frac h m\le\frac M m-1$ $\endgroup$– A.S.Commented Dec 17, 2015 at 21:59
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2$\begingroup$ I don't think Box-Cox fails here. It just tells you that keeping the existing scale is fine, or that using a transformed scale doesn't help much. $\endgroup$– Nick CoxCommented Dec 17, 2015 at 22:44
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(mostly copied from the comment by Nick Cox) The Box-Cox transform does not really fail: it is more that it is unnecessary, as there will be no need of transformation if max/min is small. Mostly, with max/min small all the observations are away from zero (relatively), so the power transform will be well approximated linearly over a short interval!
answered Feb 25, 2017 at 14:18