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# Re: st: linear, quadratic terms and centering

 From "Michael N. Mitchell" To statalist@hsphsun2.harvard.edu Subject Re: st: linear, quadratic terms and centering Date Sat, 21 Aug 2010 12:43:29 -0700

```Dear Fabio

```
I created an example using the -auto- dataset myself and see the same kind of behavior you are describing in terms of VIF values, even after centering both -x1- and -x2-. In my example, I use -price- as -x1- and -weight- as -x2-. I center them both, and then run the model predicting mpg from c.centprice##c.centprice##c.centweight.
```
--- snip ---
clear
sysuse auto

summarize price
generate centprice = price - `r(mean)'
summarize weight
generate centweight = weight - `r(mean)'
summ cent*

regress mpg c.centprice##c.centprice##c.centweight
vif
--- snip ---

```
As you can see in my example, the last VIF is about 12. The coefficients and standard errors are extremely tiny for the last term, but I think this is an issue of scaling (see my next example).
```
. regress mpg c.centprice##c.centprice##c.centweight

Source |       SS       df       MS              Number of obs =      74
-------------+------------------------------           F(  5,    68) =   31.48
Model |  1706.23426     5  341.246852           Prob > F      =  0.0000
Residual |    737.2252    68  10.8415471           R-squared     =  0.6983
Total |  2443.45946    73  33.4720474           Root MSE      =  3.2927

------------------------------------------------------------------------------
mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
centprice |  -.0004901   .0002641    -1.86   0.068     -.001017    .0000368
|
c.centprice#|
c.centprice |  -1.55e-08   6.76e-08    -0.23   0.819    -1.50e-07    1.19e-07
|
centweight |  -.0058595   .0007058    -8.30   0.000    -.0072678   -.0044511
|
c.centprice#|
c.centweight |   5.53e-07   3.16e-07     1.75   0.085    -7.81e-08    1.18e-06
|
c.centprice#|
c.centprice#|
c.centweight |   2.13e-11   6.39e-11     0.33   0.740    -1.06e-10    1.49e-10
|
_cons |   20.63903   .5813351    35.50   0.000       19.479    21.79907
------------------------------------------------------------------------------

. vif

Variable |       VIF       1/VIF
-------------+----------------------
centprice |      4.08    0.244850
c.centprice#|
c.centprice |      8.78    0.113918
centweight |      2.03    0.493619
c.centprice#|
c.centweight |      5.13    0.194814
c.centprice#|
c.centprice#|
c.centweight |     12.01    0.083255
-------------+----------------------
Mean VIF |      6.41

```
Now, this time I created z scores for the variables, to try and change the scaling of the variables so the coefficients and standard errors would not be so close to 0.
```
--- snip ---
summarize price
generate zprice = (price - `r(mean)')/`r(sd)'
summarize weight
generate zweight = (weight - `r(mean)')/`r(sd)'
summ z*

regress mpg c.zprice##c.zprice##c.zweight
vif
--- snip ----

```
In the output below the p values are the same, and the VIF values are the same, but the coefficients and standard errors are not super close to 0. Even though the last term has a VIF of 12, its standard error does not seem outrageously inflated.
```
. regress mpg c.zprice##c.zprice##c.zweight

Source |       SS       df       MS              Number of obs =      74
-------------+------------------------------           F(  5,    68) =   31.48
Model |  1706.23426     5  341.246852           Prob > F      =  0.0000
Residual |  737.225201    68  10.8415471           R-squared     =  0.6983
Total |  2443.45946    73  33.4720474           Root MSE      =  3.2927

------------------------------------------------------------------------------
mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
zprice |  -1.445663   .7788145    -1.86   0.068    -2.999763     .108437
|
c.zprice#|
c.zprice |  -.1348958   .5882425    -0.23   0.819    -1.308715    1.038924
|
zweight |  -4.553946   .5485145    -8.30   0.000    -5.648489   -3.459402
|
c.zprice#|
c.zweight |   1.266699   .7245552     1.75   0.085    -.1791279    2.712527
|
c.zprice#|
c.zprice#|
c.zweight |   .1437903   .4317442     0.33   0.740    -.7177418    1.005322
|
_cons |   20.63903   .5813351    35.50   0.000       19.479    21.79907
------------------------------------------------------------------------------

. vif

Variable |       VIF       1/VIF
-------------+----------------------
zprice |      4.08    0.244850
c.zprice#|
c.zprice |      8.78    0.113918
zweight |      2.03    0.493619
c.zprice#|
c.zweight |      5.13    0.194814
c.zprice#|
c.zprice#|
c.zweight |     12.01    0.083255
-------------+----------------------
Mean VIF |      6.41

```
My thinking is that these kinds of VIF values are, perhaps, the nature of this kind of model. Perhaps others have differing thoughts to share?
```
Lastly, I would recommend seeing...

http://www.ats.ucla.edu/stat/sas/faq/spplot/reg_int_cont.htm

```
which shows some 3-d visualizations of what these models look like. There are free 3d graphing programs you can search for and use over the internet to create 3d graphs of your results to help visualize them.
```
I fret I will be off email the rest of the weekend, but I hope this helps!

Best luck,

Michael N. Mitchell
Data Management Using Stata      - http://www.stata.com/bookstore/dmus.html
A Visual Guide to Stata Graphics - http://www.stata.com/bookstore/vgsg.html
Stata tidbit of the week         - http://www.MichaelNormanMitchell.com

On 2010-08-21 2.20 AM, Fabio Zona wrote:
```
```Thank you very much Michael!

One more information: I have to make a complex interaction, whereby a linear and a square term are both interacted with a linear (continuous) variable x2:

y = a +   x1 + x1_Square  + x2  +   x2 x1  +   x2 x1_Square

One of my VIF ( the one related to  x2 x1_Square  ) is just above 10 ( it is 10,949),  while the maximum condition index reaches the value of  19,782 (for the same  interaction term  x2  x1_Square ).

Do I have a problem of multicollinearity with this value just above the threshold of 10 ? How can I fix this problem ?

I know there is this command in STATA orthog; how does this command work? I though that I had to calculate the quadratic term and then use orthog between the linear and the quadratic term. However, since you said that (for centering variables) I first center the linear term and then calculate the quadratic term, I get confused also about orthog!
Thanks

----- Messaggio originale -----
Da: "Michael N. Mitchell"<Michael.Norman.Mitchell@gmail.com>
A: statalist@hsphsun2.harvard.edu
Inviato: Sabato, 21 agosto 2010 10:43:12 GMT +01:00 Amsterdam/Berlino/Berna/Roma/Stoccolma/Vienna
Oggetto: Re: st: linear, quadratic terms and centering

Dear Fabio

You were correct when you wrote...

Or do I first center the linear term and calculate the square term on the basis of the
"centered" linear term ?

Or, in Stata, you can do this (say you want to center -x- around 100)...

generate xcentered = x - 100
regress y c.xccentered##c.xccentered

That -regress- command will include -xcentered- as well as the squared term.

Hope this helps,

Michael N. Mitchell
Data Management Using Stata      - http://www.stata.com/bookstore/dmus.html
A Visual Guide to Stata Graphics - http://www.stata.com/bookstore/vgsg.html
Stata tidbit of the week         - http://www.MichaelNormanMitchell.com

On 2010-08-21 12.40 AM, Fabio Zona wrote:
```
```Dear Statalist,

a very simple question: I have

y = c + x0 + x1 + x1square

In order to center x1 and x1square, do I first need to calculate the square term and afterwards to center both the linear and the quadratic term around their respective means?
Or do I first center the linear term and calculate the square term on the basis of the "centered" linear term ?

I guess that the first alternative works.

Thanks

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```