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Re: st: RE: Rescaling


From   Roger Newson <roger.newson@kcl.ac.uk>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: RE: Rescaling
Date   Mon, 09 Aug 2004 18:25:51 +0100

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.

I hope this helps.

Roger


--
Roger Newson
Lecturer in Medical Statistics
Department of Public Health Sciences
King's College London
5th Floor, Capital House
42 Weston Street
London SE1 3QD
United Kingdom

Tel: 020 7848 6648 International +44 20 7848 6648
Fax: 020 7848 6620 International +44 20 7848 6620
or 020 7848 6605 International +44 20 7848 6605
Email: roger.newson@kcl.ac.uk
Website: http://www.kcl-phs.org.uk/rogernewson

Opinions expressed are those of the author, not the institution.

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