Interpretation of probit post-estimation test

[roamiefun]

Hi Statalisters,

I have a question about interpreting post-estimation tests in Stata. Suppose I estimate a Probit regression to predict whether someone passes a test as a function of their sex (sex = {1,2}) and their level of education (educ={1,2,3}):

probit pass i.sex##i.educ

I then wish to test whether the predicted probability of passing the test is the same between sexes within each education group:

margins r.sex , at(educ=(1(1)3)) asobserved

The output in Stata includes a chi2 statistic and a p-value. My understanding is that this is asking Stata to test a linear restriction of model parameters, which can be done with, e.g., Wald or LR tests, both of which have associated test statistics that are chi-square distributed under the null of equality of the parameters. Which test is Stata doing here to generate the p-value? Or perhaps I am misinterpreting the output somehow? Any thoughts would be very helpful.

Thank you in advance!

- Roamie

[Andrew Musau]

It has been a while since you last posted and there you had indicated that you were in the process of requesting that your name be changed.

Okay, this is very helpful. Thanks. And I had tried to change my name the first time you suggested I do so, Ben, but couldn't see how. I will now, though.

Please click on the "CONTACT US" link at the bottom right-hand side of the forum and do so. Also, when asking questions, post the actual Stata output. Here, we have no clue what output you are referring to, but the commands themselves are helpful (See FAQ Advice #12 for tips on how to ask questions).

margins r.sex , at(educ=(1(1)3)) asobserved

The output in Stata includes a chi2 statistic and a p-value. My understanding is that this is asking Stata to test a linear restriction of model parameters, which can be done with, e.g., Wald or LR tests, both of which have associated test statistics that are chi-square distributed under the null of equality of the parameters. Which test is Stata doing here to generate the p-value?

Consider a reproducible example:

webuse lbw, clear
probit low i.smoke##i.race
margins r.smoke , at(race=(1/3)) asobserved

Res.:

. margins r.smoke , at(race=(1/3)) asobserved

Contrasts of adjusted predictions               Number of obs     =        189
Model VCE    : OIM

Expression   : Pr(low), predict()

1._at        : race            =           1

2._at        : race            =           2

3._at        : race            =           3

------------------------------------------------------------
                         |         df        chi2     P>chi2
-------------------------+----------------------------------
               smoke@_at |
(smoker vs nonsmoker) 1  |          1       11.89     0.0006
(smoker vs nonsmoker) 2  |          1        2.21     0.1373
(smoker vs nonsmoker) 3  |          1        0.11     0.7346
                  Joint  |          3       14.21     0.0026
------------------------------------------------------------

--------------------------------------------------------------------------
                         |            Delta-method
                         |   Contrast   Std. Err.     [95% Conf. Interval]
-------------------------+------------------------------------------------
               smoke@_at |
(smoker vs nonsmoker) 1  |   .2744755   .0796084       .118446    .4305051
(smoker vs nonsmoker) 2  |      .2875   .1934625     -.0916795    .6666795
(smoker vs nonsmoker) 3  |   .0530303   .1564033     -.2535145    .3595751
--------------------------------------------------------------------------

These are just Wald tests which you can replicate using the test command. In blue, the test is whether the difference in the coefficients of black smokers and black nonsmokers is significantly different from zero. The joint test is a simultaneous test of whether the differences in the coefficients of each of the groups are different from zero (i.e. between smokers and nonsmokers within a group). Using test:

webuse lbw, clear
probit low i.smoke##i.race
margins i.smoke#i.race, post
test 0.smoke#2.race= 1.smoke#2.race
test (0.smoke#1.race= 1.smoke#1.race)  (0.smoke#2.race =1.smoke#2.race)  (0.smoke#3.race= 1.smoke#3.race)

Res.:

. margins i.smoke#i.race, post

Adjusted predictions                                       Number of obs = 189
Model VCE: OIM

Expression: Pr(low), predict()

----------------------------------------------------------------------------------
                 |            Delta-method
                 |     Margin   std. err.      z    P>|z|     [95% conf. interval]
-----------------+----------------------------------------------------------------
      smoke#race |
nonsmoker#white  |   .0909091   .0433392     2.10   0.036     .0059658    .1758524
nonsmoker#black  |      .3125   .1158781     2.70   0.007     .0853831    .5396169
nonsmoker#other  |   .3636364   .0648642     5.61   0.000     .2365049    .4907678
   smoker#white  |   .3653846   .0667773     5.47   0.000     .2345035    .4962657
   smoker#black  |         .6   .1549193     3.87   0.000     .2963637    .9036363
   smoker#other  |   .4166667   .1423188     2.93   0.003      .137727    .6956063
----------------------------------------------------------------------------------



. test 0.smoke#2.race= 1.smoke#2.race

 ( 1)  0bn.smoke#2.race - 1.smoke#2.race = 0

           chi2(  1) =    2.21
         Prob > chi2 =    0.1373

.
. test (0.smoke#1.race= 1.smoke#1.race)  (0.smoke#2.race =1.smoke#2.race)  (0.smoke#3.race= 1.smoke#3.race)

 ( 1)  0bn.smoke#1bn.race - 1.smoke#1bn.race = 0
 ( 2)  0bn.smoke#2.race - 1.smoke#2.race = 0
 ( 3)  0bn.smoke#3.race - 1.smoke#3.race = 0

           chi2(  3) =   14.21
         Prob > chi2 =    0.0026