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Re: st: RE: Interpretation of quadratic terms


From   Roger Newson <r.newson@imperial.ac.uk>
To   "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Subject   Re: st: RE: Interpretation of quadratic terms
Date   Wed, 10 Mar 2010 10:44:43 +0000

Of course, pre-emptive centering might be a good idea for other reasons. The intercept parameter is easy to explain when it is the fuel consumption (in gallons per mile) of a car of "average" weight (because weight has been pre-emptively centered), but less easy to explain when it is the fuel consumption of a fantasy car with zero weight (because weight has not been pre-emptively centered).

Roger


On 09/03/2010 21:02, Nick Cox wrote:
I think you're both right. In olden days, pre-emptive centring, as we
say in English, was a good idea in order to avoid numerical problems
with mediocre programs that did not handle near multicollinearity well.
Nowadays, decent programs including Stata take care that you get bitten
as little as possible by such problems. If course, if you really do have
multicollinearity, nothing much can help, except that Stata drops
predictors and flags the issue.

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

Rodolphe Desbordes

My point is that centering does not reduce multicollinearity. As you can
see in my example, the standard errors of the estimated marginal effects
at the mean of `mpg' are the same using uncentered or centered values of
`mpg'.

Rosie Chen

Thanks, Rodolphe, for this helpful demonstration. Agree that the major
purpose of centering seems to be that we make the interpretation of X
meaningful. I guess reducing multicollinearity is a bi-product of the
benefit.

Rodolphe Desbordes<rodolphe.desbordes@strath.ac.uk>

Centering will not affect your estimates and their uncertainty. However,
centering allows you to directly obtain the estimated effect of X on Y
for a meaningful value of X, i.e. the mean of X.

. sysuse auto.dta,clear
(1978 Automobile Data)

. gen double mpg2=mpg^2

. reg price mpg mpg2

       Source |       SS       df       MS              Number of obs =
74
-------------+------------------------------           F(  2,    71) =
18.28
        Model |   215835615     2   107917807           Prob>  F      =
0.0000
     Residual |   419229781    71  5904644.81           R-squared     =
0.3399
-------------+------------------------------           Adj R-squared =
0.3213
        Total |   635065396    73  8699525.97           Root MSE      =
2429.9

------------------------------------------------------------------------
------
        price |      Coef.   Std. Err.      t    P>|t|     [95% Conf.
Interval]
-------------+----------------------------------------------------------
------
          mpg |  -1265.194   289.5443    -4.37   0.000    -1842.529
-687.8593
         mpg2 |   21.36069   5.938885     3.60   0.001     9.518891
33.20249
        _cons |   22716.48   3366.577     6.75   0.000     16003.71
29429.24
------------------------------------------------------------------------
------

. sum mpg

     Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
          mpg |        74     21.2973    5.785503         12         41

. local m=r(mean)

. lincom _b[mpg]+2*_b[mpg2]*`m'

( 1)  mpg + 42.59459 mpg2 = 0

------------------------------------------------------------------------
------
        price |      Coef.   Std. Err.      t    P>|t|     [95% Conf.
Interval]
-------------+----------------------------------------------------------
------
          (1) |  -355.3442   58.86205    -6.04   0.000    -472.7118
-237.9766
------------------------------------------------------------------------
------

. gen double mpgm=mpg-`m'

. gen double mpgm2=mpgm^2

. reg price mpgm mpgm2

       Source |       SS       df       MS              Number of obs =
74
-------------+------------------------------           F(  2,    71) =
18.28
        Model |   215835615     2   107917807           Prob>  F      =
0.0000
     Residual |   419229781    71  5904644.81           R-squared     =
0.3399
-------------+------------------------------           Adj R-squared =
0.3213
        Total |   635065396    73  8699525.97           Root MSE      =
2429.9

------------------------------------------------------------------------
------
        price |      Coef.   Std. Err.      t    P>|t|     [95% Conf.
Interval]
-------------+----------------------------------------------------------
------
         mpgm |  -355.3442   58.86205    -6.04   0.000    -472.7118
-237.9766
        mpgm2 |   21.36069   5.938885     3.60   0.001     9.518891
33.20249
        _cons |   5459.933   343.8718    15.88   0.000     4774.272
6145.594
------------------------------------------------------------------------
------


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--
Roger B Newson BSc MSc DPhil
Lecturer in Medical Statistics
Respiratory Epidemiology and Public Health Group
National Heart and Lung Institute
Imperial College London
Royal Brompton Campus
Room 33, Emmanuel Kaye Building
1B Manresa Road
London SW3 6LR
UNITED KINGDOM
Tel: +44 (0)20 7352 8121 ext 3381
Fax: +44 (0)20 7351 8322
Email: r.newson@imperial.ac.uk
Web page: http://www.imperial.ac.uk/nhli/r.newson/
Departmental Web page:
http://www1.imperial.ac.uk/medicine/about/divisions/nhli/respiration/popgenetics/reph/

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