How do I calculate marginal effects for an interaction term in an ordinal logistic regression?

[Derek Tharp]

I am trying to use mchange within Long and Freese's spost13 package (-findit spost13_ado-) in Stata 15.1.

Here is my code:

mlogit PaidUnpaidSelf12Cats2 c.WantToBePrimaryEarner#Male

Then:

mchange

The output from mlogit contains the interaction term:

. mlogit PaidUnpaidSelf12Cats2 c.WantToBePrimaryEarner#Male, base(1)

Iteration 0:   log likelihood = -76.896294  
Iteration 1:   log likelihood = -71.073489  
Iteration 2:   log likelihood = -70.583766  
Iteration 3:   log likelihood = -70.579145  
Iteration 4:   log likelihood = -70.579144  

Multinomial logistic regression                 Number of obs     =         82
                                                LR chi2(4)        =      12.63
                                                Prob > chi2       =     0.0132
Log likelihood = -70.579144                     Pseudo R2         =     0.0822

----------------------------------------------------------------------------------------------
       PaidUnpaidSelf12Cats2 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-----------------------------+----------------------------------------------------------------
Long                         |  (base outcome)
-----------------------------+----------------------------------------------------------------
SensitiveToPay               |
Male#c.WantToBePrimaryEarner |
                          0  |   .2905134   .1891178     1.54   0.1245    -.0801506    .6611774
                          1  |   .3010523   .1433273     2.10   0.0357     .0201359    .5819687
                             |
                       _cons |  -.6760504   .7109363    -0.95   0.3416     -2.06946    .7173592
-----------------------------+----------------------------------------------------------------
Short                        |
Male#c.WantToBePrimaryEarner |
                          0  |   .4704996   .3818554     1.23   0.2179    -.2779232    1.218922
                          1  |    .697739   .2824248     2.47   0.0135     .1441966    1.251281
                             |
                       _cons |  -4.124606   1.682924    -2.45   0.0143    -7.423075   -.8261363
----------------------------------------------------------------------------------------------

But the output from the mchange command does not:

. mchange

mlogit: Changes in Pr(y) | Number of obs = 82

Expression: Pr(PaidUnpaidSelf12Cats2), predict(outcome())

             |      Long  Sensiti~y      Short 
-------------+---------------------------------
Male         |                                 
      1 vs 0 |    -0.034     -0.066      0.100 
     p-value |     0.745      0.561      0.134 
WantToBePr~r |                                 
          +1 |    -0.062      0.016      0.046 
     p-value |     0.012      0.661      0.135 
         +SD |    -0.108      0.017      0.092 
     p-value |     0.008      0.810      0.155 
    Marginal |    -0.066      0.024      0.042 
     p-value |     0.017      0.488      0.101

Any suggestions would be greatly appreciated!

[Clyde Schechter]

I am not familiar with the -mchange- command, so I can't make a specific comment about this. However, I infer from the title of your post and the general content that you are trying to calculate marginal effects after fitting a model containing an interaction term and you are puzzled that you are not getting output for the marginal effect of the interaction term.

That is because there is no such thing as the marginal effect of an interaction term. Marginal effects are definable only for main effects in the model.

To see why, consider the definition of a marginal effect of a predictor. It is the expected change in the outcome variable per unit change in the predictor, all else held constant. But it is in principle impossible for an interaction to change while its constituent ("main") effects are held constant. If you try to get around this by saying, OK, we'll let the constituents change in such a way that they result in a unit change in the interaction term, you run into the problem that there are many different combinations of changes in the constituent effects that will together produce a unit change in the interaction term, but those result in different changes in the outcome. So there is no coherent way to define a marginal effect for an interaction term.

[Derek Tharp]

Ah, yes. Thank you. I wasn't seeing the forest for the trees.