2

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.

Sven Hohenstein
  • 6,833
barryq
  • 123

1 Answers1

4

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.

Frank Harrell
  • 91,879
  • 6
  • 178
  • 397
  • Thanks for swift response. The scary part is that this is being used in an energy demand forecast model! – barryq Aug 02 '13 at 13:03
  • Yes, and most people doing this kind of scaling don't recognize the estimation error in the thing used in scaling (here, the mean). – Frank Harrell Aug 02 '13 at 15:58
  • Frank, just to clarify, if a measure of dispersion such as the SD is used, what is the appropriate interpretation for the Beta coefficient ? – barryq Aug 05 '13 at 06:57
  • 2
    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. – Frank Harrell Aug 05 '13 at 12:40
  • That does seem much more intuitive. Thanks i will take a detailed look at your r package. – barryq Aug 06 '13 at 13:16