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
[email protected]
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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