Re: st: question on margins and interaction between factor and continuous var

[jpitblado@stata.com (Jeff Pitblado, StataCorp LP)]

Robert Duval <rduval@gmail.com> is using the -margins- postestimation command
with a model fit containing the interaction of a factor variable with a
continuous variable:

> I am a bit confused about the syntax of margins.
> 
> I have a model where I interact a categorical variable, say cat, with
> a continuous variable say cont. For simplicity assume cat has 3
> levels.
> 
> The command would generically look like
> 
> command depvar i.cat cont i.cat#c.cont
> 
> I want to obtain the marginal effects of
> 
> cat2
> cat3
> cont
> cat2#cont
> cat3#cont

To make things a little more concrete lets use a similar setup with the auto
data:

	. sysuse auto
	. keep if inlist(rep78,3,4,5)
	. gen cat = rep78 - 2
	. rename turn cont
	. rename mpg depvar
	. regress depvar cat##c.cont

> However, when i run margins as
> 
> margins , dydx(cat cont)
> 
> I obtain the overall average marginal effects for each variable
> (averaging out the interaction)

Using our dataset, here is what Robert would get:

***** BEGIN:
. margins, dydx(cat cont)

Average marginal effects                          Number of obs   =         59
Model VCE    : OLS

Expression   : Linear prediction, predict()
dy/dx w.r.t. : 2.cat 3.cat cont

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         cat |
          2  |   .1310132   1.218829     0.11   0.914    -2.257848    2.519874
          3  |  -4.835326   3.448432    -1.40   0.161    -11.59413    1.923476
             |
        cont |  -1.266052    .193393    -6.55   0.000    -1.645095   -.8870083
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.
***** END:

Given this simple model, we can verify the above results by computing these
average marginal effects by hand without too much trouble.

***** BEGIN:
* individual predictions for the levels of 'cat', then get the differences
* from the base level of '1'
gen p_cat1 = _b[_cons]+_b[cont]*cont
gen p_cat2 = _b[_cons]+_b[cont]*cont+_b[2.cat]+_b[2.cat#cont]*cont
gen p_cat3 = _b[_cons]+_b[cont]*cont+_b[3.cat]+_b[3.cat#cont]*cont
gen d_cat2 = p_cat2 - p_cat1
gen d_cat3 = p_cat3 - p_cat1
* derivative of the linear prediction with respect to 'cont'
gen d_cont = _b[cont]+_b[2.cat#cont]*2.cat+_b[3.cat#cont]*3.cat
* compute the mean for each of these new predictions
sum d_cat2 d_cat3 d_cont
***** END:

Next Robert mentions the following:

> if i run
> 
> margins cat  , dydx(cont)
> 
> i obtain what it seems to be the average marginal effects for
> different levels of cat, i.e. cat2#cont cat3#cont

***** BEGIN:
. margins cat, dydx(cont)

Average marginal effects                          Number of obs   =         59
Model VCE    : OLS

Expression   : Linear prediction, predict()
dy/dx w.r.t. : cont

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
cont         |
         cat |
          1  |  -.7499364   .1713144    -4.38   0.000    -1.085706   -.4141664
          2  |  -.9800797   .2020792    -4.85   0.000    -1.376148   -.5840117
          3  |  -3.141593   .8650601    -3.63   0.000     -4.83708   -1.446106
------------------------------------------------------------------------------
***** END:

Let's reproduce these calculations by hand.

***** BEGIN:
* derivative of the linear prediction with respect to 'cont', assuming 'cat'
* is equal to each of its levels
gen dd_cat1 = _b[cont]
gen dd_cat2 = _b[cont]+_b[2.cat#cont]
gen dd_cat3 = _b[cont]+_b[3.cat#cont]
* compute the mean (NOTE: these variables are constant)
sum dd_cat1 dd_cat2 dd_cat3
***** END:

Finally, Robert asks

> But then how to obtain the marginal effects for cat2 and cat2 (i.e.
> when cont==0)?

The -at()- option allows you to set the value of predictors (both continuous
and factor) at fixed values.  In this case, Robert probably wants to use
-at(cont=0)-.  Here is result using our dataset.

***** BEGIN:
. margins, dydx(cat) at(cont=0)

Conditional marginal effects                      Number of obs   =         59
Model VCE    : OLS

Expression   : Linear prediction, predict()
dy/dx w.r.t. : 2.cat 3.cat
at           : cont            =           0

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         cat |
          2  |   9.169014   10.55421     0.87   0.385    -11.51686    29.85489
          3  |   89.08786   31.65038     2.81   0.005     27.05426    151.1215
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.
***** END:

The following reproduces these calculations by hand.

***** BEGIN:
* individual predictions for the levels of 'cat', assuming 'cont' is zero,
* then get the differences from the base level of '1'
gen p0_cat1 = _b[_cons]
gen p0_cat2 = _b[_cons]+_b[2.cat]
gen p0_cat3 = _b[_cons]+_b[3.cat]
gen d0_cat2 = p0_cat2 - p0_cat1
gen d0_cat3 = p0_cat3 - p0_cat1
* compute the mean (NOTE: these variables are constant)
sum d0_cat2 d0_cat3
***** END:

--Jeff
jpitblado@stata.com
*
*   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/