Interpretation of average marginal effects after multinomial logit

[Luis Ortiz]

Dear members of the list,

For a sample of undergraduate students, I am running a multinomial logit model where the dependent variable ('expect3', below) captures the student's preference upon graduation (what they want to do after graduation). The options are ‘Doing another BA degree’, ‘Doing a MA or PhD’, ‘Other studies’, ‘Looking for a job’, ‘Enjoying a sabbatical’ or ‘Not decided yet’. The option or preference that I take as reference category in the dependent variable is 'Looking for a job' (baseoutcome(4); that is, to straightaway enter into the labour market upon graduation

mlogit expect3 ib2.edu_parents female progresion i.codi_a2 rendimiento, baseoutcome(4)

After running this model, I generate the average marginal effect of all the independent variables in the model. Yet, I am particularly interested in their average marginal effect on one of the outcomes, which is enrolling in a master degree ('MA/PhD')

margins, dydx(*) predict(outcome(MA_PhD))

The results are as follows

. margins, dydx(*) predict(outcome(MA_PhD))

Average marginal effects                                Number of obs = 34,454
Model VCE: OIM

Expression: Pr(expect3==MA_PhD), predict(outcome(MA_PhD))
dy/dx wrt:  1.edu_parents 3.edu_parents 4.edu_parents 5.edu_parents 6.edu_parents 7.edu_parents female progresion rendimiento

--------------------------------------------------------------------------------------
                     |            Delta-method
                     |      dy/dx   std. err.      z    P>|z|     [95% conf. interval]
---------------------+----------------------------------------------------------------
         edu_parents |
         No studies  |   .0027992   .0243434     0.11   0.908     -.044913    .0505113
    Lower secondary  |   .0598721   .0127219     4.71   0.000     .0349376    .0848066
Upper sec/lower voc  |   .0821971   .0120098     6.84   0.000     .0586583    .1057359
          Upper voc  |   .0976009   .0130215     7.50   0.000     .0720792    .1231226
        Univ(BA/MA)  |   .1218801   .0114611    10.63   0.000     .0994168    .1443434
          Univ(PhD)  |   .1467838    .014754     9.95   0.000     .1178665     .175701
                     |
              female |  -.0156269   .0055045    -2.84   0.005    -.0264155   -.0048383
          progresion |  -.0004666    .000035   -13.32   0.000    -.0005352    -.000398
         rendimiento |   .0285021   .0026697    10.68   0.000     .0232697    .0337346
--------------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

The first independent variable in the outcome above is parental education. The reference category in this variable is 'Primary studies'.

Shall I understand that, for instance, undergraduate students with at least one parent with a PhD degree ('Univ(PhD)') are 14% more likely to express the intention of doing a master degree instead of 'Work' (reference cat. in the dependent variable) than undergraduate students whose parents have 'primary studies' at most (reference category in the independent variable)? Is this the right interpretation of these average marginal effects?

Thanks for your help or assistance with this

L.O

[Clyde Schechter]

Yes, that is correct, except that it should be 14 percentage points, not 14%. The marginal effect is an additive difference in outcome probability, not multiplicative.

[Luis Ortiz]

That's awfully helpful, Clyde; many thanks.

The interpretation of the average marginal effects after a multinomial logit model (margins, dydx(*) should be referred to the category that has been taken in the dependent variable (mlogit); right?

My apologies if this is insisting too much, Clyde. Quite unfortunately, although I have tried hard, it has been difficult for me to find an explicit answer, either in Statalist or somewhere else. Your contribution in this respect is very much appreciated.

All the best

Luis

[Clyde Schechter]

Oh, I think I misunderstood your original question. No, the marginal effect is not relative to a reference category in the dependent variable. It is relative to the reference category in the explanatory (independent) variable. Here's how I would interpret the number you showed in red in #1:

After adjustment for other explanatory variables, a student whose parents have a PhD degree are 14 percentage points more likely to expect to undertake MA/PhD studies than a student whose parents have only completed primary studies.

There is no inferred comparison to the probability that either student will look for a job.

[Luis Ortiz]

Many thanks again, Clyde.

Then, we should understand that, unlike coefficients in multinomial logit models, which I understand that should be interpreted taking the reference category in the dependent variable into account, this is not the case with average marginal effects; am I right? It does not matter which ihe reference category in the dependent variable is.

Best

L.O.

[Clyde Schechter]

Then, we should understand that, unlike coefficients in multinomial logit models, which I understand that should be interpreted taking the reference category in the dependent variable into account, this is not the case with average marginal effects; am I right? It does not matter which ihe reference category in the dependent variable is.

Correct.

[Richard Williams]

This handout might help:

https://www3.nd.edu/~rwilliam/xsoc73994/Margins05.pdf

[Luis Ortiz]

Many thanks to you both, Clyde and Richard...¡¡

All the best

Luis Ortiz

[Luis Ortiz]

Dear Clyde and Richard,

This is an additional post quite a while after your help to me regarding the way of interpreting predicted probabilities and average marginal effects in multinomial logit models, which I very much appreciate. It has been very valuable to revise my work and my results.

I wonder if, besides the material that Richard kindly provided, you could share with me any reference or example that I could provide for the papers and the communication with the anonymous reviewers who have reviewed my work. It would be great to be able to refer in this way the interpretation that you have provided me with. In particular, it would be great to properly ground in previous methodological work or examples of the use of AME for multinomial logit what Clyde provides as an explanation in #4 above.

At any rate, thanks for your attention

And my apologies for this addenda to my post

Kind regards

Luis Ortiz