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I was arguing with my Stats teacher today in class because we were stuck on the topic of re-expression. She decided to give me extra credit if I could prove to her, using the internet, something related to unit conversions during re-expression. When you perform a power transformation with a power of $-1/2$ or $-1$, you're supposed to change the units and the value to negative (ie. for a power of $-1$ you might get $-32.8$ -gallon/mile). Are you supposed to change it back to $32.8$ gallon/mile?

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    $\begingroup$ When you say "When you perform -1/2 and -1 degree of re-expression", what are you talking about? Are you trying to discuss power transformations? Box-Cox transformations? Something else? $\endgroup$ – Glen_b Nov 27 '12 at 7:33
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    $\begingroup$ When you do y^-1/2 or y^-1 $\endgroup$ – user17211 Nov 29 '12 at 0:29
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    $\begingroup$ so yes power transformations $\endgroup$ – user17211 Nov 29 '12 at 0:29
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The answer is no, you leave it negative so the observations are in the original order. The point of negating after a negative power transformation is because the negative power reverse the order of the observations, and negating them returns the order to the same as the original order.

If you cancelled the negative out with the negative in the units, you'd end up as if you'd done nothing - what would be the point in negating it only to immediately un-negate it again?

(If you use Box-Cox, the denominator takes care of the ordering automatically.)

The source for this is Tukey himself; I think the reasoning for the negative is explained either in EDA or UREDA ('77 and '83 respectively)

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