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

To |
"statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu> |

Subject |
RE: st: RE: Interpretation of quadratic terms |

Date |
Wed, 10 Mar 2010 12:34:51 +0000 |

Dear Rosie, Nick and Roger, To conclude this thread and summarise the main arguments put forward by Nick, Roger and myself: A) There can be some good reasons for "pre-emptive centering": a) to avoid computational issues, which are unlikely to arise with modern econometric softwares such as Stata; b) to provide substantive interpretation. However Rosie wrote in her first message " To avoid multicollinearity problem with the original variable and its quadratic term, I centered the variable first (X) and then created the square term (Xsq). The model with the quadratic term (Xsq) was proved to be significantly better." B) Centering will not magically improve the precision/accuracy of the estimates. After centering, the estimates and their associated standard errors may differ from those obtained with the noncentered data, but that does not mean that the "centered model" performs better than the "noncentered model". Rodolphe ________________________________________ From: owner-statalist@hsphsun2.harvard.edu [owner-statalist@hsphsun2.harvard.edu] On Behalf Of Roger Newson [r.newson@imperial.ac.uk] Sent: 10 March 2010 10:44 To: statalist@hsphsun2.harvard.edu Subject: Re: st: RE: Interpretation of quadratic terms 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 > ------------------------------------------------------------------------ > ------ > > > * > * For searches and help try: > * http://www.stata.com/help.cgi?search > * http://www.stata.com/support/statalist/faq > * http://www.ats.ucla.edu/stat/stata/ -- 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. * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/ * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**References**:**Re: st: RE: Interpretation of quadratic terms***From:*Rosie Chen <jiarongchen2002@yahoo.com>

**RE: st: RE: Interpretation of quadratic terms***From:*Rodolphe Desbordes <rodolphe.desbordes@strath.ac.uk>

**RE: st: RE: Interpretation of quadratic terms***From:*"Nick Cox" <n.j.cox@durham.ac.uk>

**Re: st: RE: Interpretation of quadratic terms***From:*Roger Newson <r.newson@imperial.ac.uk>

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