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Re: st: RE: Interpretation of quadratic terms


From   Rosie Chen <jiarongchen2002@yahoo.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: RE: Interpretation of quadratic terms
Date   Tue, 9 Mar 2010 12:30:16 -0800 (PST)

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.  


Rosie

----- Original Message ----
From: Rodolphe Desbordes <rodolphe.desbordes@strath.ac.uk>
To: "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Sent: Mon, March 8, 2010 7:15:12 PM
Subject: RE: st: RE: Interpretation of quadratic terms

Dear Rosie,

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.

Rodolphe


. 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
------------------------------------------------------------------------------

.
end of do-file



________________________________________
From: owner-statalist@hsphsun2.harvard.edu [owner-statalist@hsphsun2.harvard.edu] On Behalf Of Rosie Chen [jiarongchen2002@yahoo.com]
Sent: 08 March 2010 21:15
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: RE: Interpretation of quadratic terms

Thanks a lot for the advice, Rodolphe. I found several resources that suggest centering before creating quadratic terms. Below is one example.

http://www.ats.ucla.edu/stat/mult_pkg/faq/general/curves.htm

Rosie



----- Original Message ----
From: Rodolphe Desbordes <rodolphe.desbordes@strath.ac.uk>
To: "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Sent: Mon, March 8, 2010 1:42:31 PM
Subject: st: RE: Interpretation of quadratic terms

Dear Rosie,

If the coefficient on X is positive and the coefficient on X^2 is negative, that suggests that X has a positive effect on Y until a turning point is reached, e.g. 1.3/(2*0.2)=3.25. Beyond that value, X has a negative impact on Y.

Rodolphe

PS: I am not sure that `centering' reduces multicollinearity.

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Rosie Chen
Sent: lundi 8 mars 2010 18:28
To: statalist@hsphsun2.harvard.edu
Subject: st: Interpretation of quadratic terms

Dear all,

    I have a question regarding how to interpret quadratic terms in regression, and would appreciate your help very much.

    Because the non-linear nature of the relationship between X and Y; I need to include quadratic terms in the model. 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. Suppose the output is like the following (both coefficients are significant), how to interpret the results? The two signs are opposite. Could anyone provide some insight? Thank you very much in advance!  --Rosie

y= a + 1.3*X - 0.2*Xsq + e



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