2
$\begingroup$

I have come across an industry example of a simple linear regression ($y=a+bx+\epsilon$) where the slope coefficient has been adjusted by the mean of $y$ ($b/\text{mean}(y)$) and described as a "slope impact". The exact interpretation that they use is that this slope impact represents "the percentage change in $y$ for every unit above the average value of $x$".

I am not sure this interpretation is correct but would appreciate any feedback you could give on this calculation.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

It is not correct. Think of the example where mean(y)=0. You cannot in general use a mean like this. If you needed a standardized impact measure (highly discouraged; reporting in real units is much more helpful and relevant) you almost always need the normalizing factor to be a measure of dispersion that cannot be zero unless all values are identical, for example, Gini's mean difference (mean absolute difference between any two y-values) or standard deviation.

By in my humble opinion the desire for unitless impact measures of the type you read about is often caused by avoidance of thinking.

Improve this answer
$\endgroup$
5
  • $\begingroup$ Thanks for swift response. The scary part is that this is being used in an energy demand forecast model! $\endgroup$
    – barryq
    Commented Aug 2, 2013 at 13:03
  • $\begingroup$ Yes, and most people doing this kind of scaling don't recognize the estimation error in the thing used in scaling (here, the mean). $\endgroup$
    – Frank Harrell
    Commented Aug 2, 2013 at 15:58
  • $\begingroup$ Frank, just to clarify, if a measure of dispersion such as the SD is used, what is the appropriate interpretation for the Beta coefficient ? $\endgroup$
    – barryq
    Commented Aug 5, 2013 at 6:57
  • 2
    $\begingroup$ I don't like scaling on $Y$ but rather finding changes in $Y$ per meaningful changes in $X$. We condition on $X$ anyway so computing changes in $X$ doesn't cause estimation error in the usual sense. I like to compute inter-quartile-range effects for each $X$, i.e., keep other $X$s constant and vary one $X$ from its first to its third quartile, computing two predicted values, then subtracting one from another. That is the default in my R rms package's summary function. $\endgroup$
    – Frank Harrell
    Commented Aug 5, 2013 at 12:40
  • $\begingroup$ That does seem much more intuitive. Thanks i will take a detailed look at your r package. $\endgroup$
    – barryq
    Commented Aug 6, 2013 at 13:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.