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RE: st: RE: Rescaling


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: RE: Rescaling
Date   Mon, 9 Aug 2004 18:32:46 +0100

This reminds me that Sir David Cox  
in various places emphasises treating regression 
of y on x as 

y = mean of y + b(x - mean of x) 

Nick 
n.j.cox@durham.ac.uk 

Roger Newson
 
> At 18:03 09/08/2004, Nick Cox wrote (in reply to Cordula Stolberg):
> >The units of the intercept are the same
> >as those of the response. As I understand
> >it, you can restate in other units exactly
> >as convenience or whim dictates. No
> >statistical issue arises.
> 
> I think what Cordula really wants might be centring, rather 
> than scaling. 
> If you extract a constant X_0 from an X-variate before fitting the 
> regression model, and therefore regress Y with respect to 
> X-X_0, then the 
> intercept will be the expected value of Y if X==X_0, instead of the 
> expected value of Y if X==0. This often causes the intercept 
> to make more 
> sense, although, as Nick says, the intercept is still 
> expressed in Y-units.
> 
> For instance, in the -auto- data we might do the example:
> . sysuse auto, clear
> (1978 Automobile Data)
> 
> . replace weight=weight-2000
> (74 real changes made)
> 
> . regress mpg weight foreign
> 
>        Source |       SS       df       MS              
> Number of obs =      74
> -------------+------------------------------           F(  2, 
>    71) =   69.75
>         Model |   1619.2877     2  809.643849           Prob 
> > F      =  0.0000
>      Residual |  824.171761    71   11.608053           
> R-squared     =  0.6627
> -------------+------------------------------           Adj 
> R-squared =  0.6532
>         Total |  2443.45946    73  33.4720474           Root 
> MSE      =  3.4071
> 
> --------------------------------------------------------------
> ----------------
>           mpg |      Coef.   Std. Err.      t    P>|t|     
> [95% Conf. Interval]
> -------------+------------------------------------------------
> ----------------
>        weight |  -.0065879   .0006371   -10.34   0.000    
> -.0078583   -.0053175
>       foreign |  -1.650029   1.075994    -1.53   0.130      
> -3.7955    .4954422
>         _cons |   28.50393   .9630195    29.60   0.000     
> 26.58372    30.42414
> --------------------------------------------------------------
> ----------------
> 
> .
> 
> The intercept is then the miles per gallon expected in a 
> realistic US-made 
> car weighing 2000 pounds (1 US ton), instead of the miles per gallon 
> expected in a fantasy US-made car with zero weight, and the 
> standard error 
> will be reduced because the line is not being extrapolated 
> off the edge of 
> the paper.
> 
> If we typed our -replace- statement as
> 
> . replace weight=(weight-2000)/2000
> 
> then we would have computed a regression coefficient for 
> -weight- equal to 
> a decrease in mileage per incremental US ton, which might be 
> easier to 
> explain than a decrease in mileage per incremental pound.

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