Re: st: Non-linear regression: interpretation

["Justina Fischer" <JAVFischer@gmx.de>]

Hi

David is completely right.

I also would not focus on the marginal effects of the variables x and x2 separately when interpreting your findings.

Stata offers now a way to compute the total marginal effect (margins) - the first derivative of non-linear Y w.r.t x at certain value of x  - the user can specify this value herself (e.g. sample mean). 

Aside: as David said, in your model the first derivative changes its value depending on where you are on the non-linear Y-function. This is not the case for a regression model without a quadratic term, where the first derivative (slope) is a simple constant. 

Hope this helps

Justina 




-------- Original-Nachricht --------
> Datum: Tue, 08 Feb 2011 18:08:28 -0500
> Von: David Greenberg <dg4@nyu.edu>
> An: statalist@hsphsun2.harvard.edu
> Betreff: st: Non-linear regression: interpretation

> It is true that the quadratic term taken by itself can be hard to
> interpret. If the linear term is also in the equation, the coefficient for the
> quadratic term would seem to be an answer to a question that cannot have a
> meaningful answer, namely, how much the dependent variable changes in response
> to marginal change in the quadratic term, while holding the linear term
> constant. But it is impossible to hold x constant and allow x-squared to vary.
> However, the estimated coefficients of linear and quadratic terms together
> can be used to compute the estimated point at which the quadratic equation
> has a minimum or maximum, and that is something many researchers might
> want to know. One can also compute the value of the dependent variable at the
> minimum or maximum. David Greenberg, Sociology Department, New York
> University
> 
> ----- Original Message -----
> From: Maarten buis <maartenbuis@yahoo.co.uk>
> Date: Tuesday, February 8, 2011 4:55 am
> Subject: Re: st: Non-linear regression
> To: statalist@hsphsun2.harvard.edu
> 
> 
> > --- On Tue, 8/2/11, Hamizah Hassan wrote:
> > > I would like to run non-linear regression by including the
> > > linear and quadratic functions of the variable. 
> > 
> > Typically this is still refered to as a linear model, as the
> > model is still linear in the parameters.
> > 
> > > I just realize that if the variable is in percentage, the
> > > quadratic figure is higher than the linear figure. However,
> > > if it is in decimal, it would be the other way around and
> > > definitely it will effect on the meaning of the results. 
> > 
> > The models are mathematically equivalent. You can see that
> > by looking at the predictions. 
> > 
> > Generally, it is hard to give a substantive interpretation to
> > a quadritic term, regardless of how you scaled the original 
> > variable. If you care about interpreting the coefficients but 
> > still want to allow for non-linear effects, then your best 
> > guess is probably to use linear splines (which confusingly is 
> > actually a non-linear function...)
> > 
> > Consider the example below. The first part shows that the
> > two quadratic models result in the same predicted values. The
> > final part displays linear splines as an alternative. The final
> > graph shows that they result in fairly similar predictions, but
> > the spline terms can actually be interpreted: the parameter for
> > fuel_cons1 tells you that for cars with a fuel-consumption of 
> > less than 12 liters/100km an additional liter/100km leads to a 
> > non-significant price increase of 62$ (=.062*1000$). The 
> > parameter for fuel_cons2 tells you that for cars with a fuel 
> > consumption of more than 12 liters/100km an additional liter
> > per 100 kilometers will lead to a signinicant price increase of 
> > 1011$ (=1.011*1000$).
> > 
> > *----------------- begin example -----------------
> > //================================== first part
> > sysuse auto, clear
> > 
> > // since I am European and the question is about
> > // interpretation I first convert mpg from miles
> > // per gallon to liter / 100 km and price in 
> > // 1000 $
> > 
> > gen fuel_cons = 1/mpg * 3.78541178 / 1.609344 *100
> > label var fuel_cons "fuel consumption (l/100km)"
> > 
> > replace price = price / 1000
> > label var price "price (1000$)"
> > 
> > // create a "proportion-like" variable
> > sum fuel_cons , meanonly
> > gen prop = ( fuel_cons - r(min) ) / ( r(max) - r(min) )
> > 
> > // take a look at that new variable
> > spikeplot prop, ylab(0 1 2)
> > 
> > // turn it into percentages
> > gen perc = prop*100
> > spikeplot perc, ylab(0 1 2)
> > 
> > // add square terms using the new
> > // factor variable notation
> > reg price c.prop##c.prop
> > predict yhat_prop
> > 
> > reg price c.perc##c.perc
> > predict yhat_perc
> > 
> > // compare predicted values
> > twoway function identity = x,        ///
> >        range( 13 31 ) lcolor(gs8) || ///
> >        scatter yhat_prop yhat_perc,  ///
> >            aspect(1) msymbol(Oh)
> > 
> > //================================== final part           
> > // alternative with interpretable parameters
> > 
> > // create splines
> > mkspline fuel_cons1 12 fuel_cons2 = fuel_cons
> > 
> > reg price fuel_cons1 fuel_cons2
> > predict yhat_spline
> > 
> > twoway scatter price fuel_cons  ||           ///
> >        line yhat_prop yhat_spline fuel_cons, ///
> >        sort ytitle("price (1000 {c S|})")    ///
> >        legend(order( 1 "observations"        ///
> >                      2 "prediction,"         ///
> >                        "quadratric"          ///
> >                      3 "prediction,"         ///
> >                        "spline" ))       
> > *---------------- end example --------------
> > (For more on examples I sent to the Statalist see: 
> > http://www.maartenbuis.nl/example_faq )
> > 
> > Hope this helps,
> > Maarten
> > 
> > --------------------------
> > Maarten L. Buis
> > Institut fuer Soziologie
> > Universitaet Tuebingen
> > Wilhelmstrasse 36
> > 72074 Tuebingen
> > Germany
> > 
> > http://www.maartenbuis.nl
> > --------------------------
> > 
> > 
> >       
> > 
> > *
> > *   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/

-- 
Justina AV Fischer, PhD
Senior Researcher
Faculty of Economics
University of Mannheim

homepage: http://www.justinaavfischer.de/
e-mail: javfischer@gmx.de
papers: http://ideas.repec.org/e/pfi55.html


*
*   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/