Logit Marginal Effects at means with interaction terms

[Laura Stortz]

Hi all,

I am working on my MSc. and I have data for 201 participants who each made 457 choices (total choices ~ 91,000 choices). I am using a binary logit model of choice 1 and choice 2 groupings. I am using STATA 15. I have no problems running my logit regression (see example below) but I am having trouble with the interaction terms showing up when I do the marginal effects. The marginal effects for the interaction term disappears so I can't estimate the effect size for the interaction term. I have been trying to find the answer to this for a few days and I have read a lot about how to operationalize the commands (Thank you, Richard Williams & Clyde Schechter for all your answers especially). Obviously, I am either trying to do something I shouldn't be doing or going about it the wrong way. Any help would be appreciated greatly. I am clustering the standard errors by ID (of each person)-- also unsure if I should be doing that but it isn't the focus of this question.

Background: anything with i. is a factor variable (0,1) and otherwise, it is continuous.

. logit buy i.treatment##i.income_low i.female i.young i.fulltime i.children i.recentimmigrin
> t i.highschool_or_below price nvs_score_total, cluster(ID)

Iteration 0: log pseudolikelihood = -4419.9312
Iteration 1: log pseudolikelihood = -4386.08
Iteration 2: log pseudolikelihood = -4384.7678
Iteration 3: log pseudolikelihood = -4384.765
Iteration 4: log pseudolikelihood = -4384.765

Logistic regression Number of obs = 91,857
Wald chi2(11) = 59.14
Prob > chi2 = 0.0000
Log pseudolikelihood = -4384.765 Pseudo R2 = 0.0080

(Std. Err. adjusted for 201 clusters in ID)
---------------------------------------------------------------------------------------
| Robust
buy | Coef. Std. Err. z P>|z| [95% Conf. Interval]
----------------------+----------------------------------------------------------------
1.treatment | -.2627016 .0917476 -2.86 0.004 -.4425235 -.0828797
1.income_low | -.2187479 .1241379 -1.76 0.078 -.4620536 .0245579
|
treatment#income_low |
1 1 | .4124181 .217789 1.89 0.058 -.0144404 .8392766
|
1.female | .0782018 .084832 0.92 0.357 -.0880658 .2444695
1.young | -.2152019 .099856 -2.16 0.031 -.4109162 -.0194877
1.fulltime | .0954429 .0888683 1.07 0.283 -.0787357 .2696215
1.children | .0427962 .1040897 0.41 0.681 -.1612158 .2468082
1.recentimmigrint | -.2436619 .1236873 -1.97 0.049 -.4860846 -.0012392
1.highschool_or_below | .2040199 .1194476 1.71 0.088 -.030093 .4381329
price | -.1089045 .0244309 -4.46 0.000 -.1567883 -.0610208
n_score_total | .0225023 .0299007 0.75 0.452 -.0361019 .0811065
_cons | -4.385361 .1775846 -24.69 0.000 -4.73342 -4.037301
---------------------------------------------------------------------------------------

margins, dydx(*) atmeans

---------------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
----------------------+----------------------------------------------------------------
1.treatment | -.0014183 .0006661 -2.13 0.033 -.0027239 -.0001127
1.income_low | -.0000747 .0009576 -0.08 0.938 -.0019515 .001802
1.female | .0006153 .000663 0.93 0.353 -.0006841 .0019147
1.young | -.0016644 .0007549 -2.20 0.027 -.0031441 -.0001847
1.fulltime | .000764 .000719 1.06 0.288 -.0006453 .0021733
1.children | .0003424 .0008402 0.41 0.684 -.0013044 .0019891
1.recentimmigrint | -.0018043 .0008555 -2.11 0.035 -.0034811 -.0001276
1.highschool_or_below | .0017283 .0010859 1.59 0.111 -.0004 .0038566
price | -.0008627 .0001912 -4.51 0.000 -.0012374 -.0004881
nvs_score_total | .0001783 .0002363 0.75 0.451 -.0002848 .0006413
---------------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

[FernandoRios]

Hi Laura,
this is actually a very common mistake most people do when thinking about marginal effects.
A good source for a better understanding of how margins work is Williams (2012) paper.
the bottom line is that margins for interactions cannot be directly estimated with margins. Why?
Margins can only estimate first order effects (first derivatives) (dydx). So it can comfortably estimate the effects of "treatment" or of income_low. It cannot estimate the marginal effects of interaction because that implies a second derivative.
What you can do alternatively is to use something like this:

margins r.treatment t.income_low 
margins r.treatment#t.income_low

The second one will give you something like the "treatment effect" that comes from the interaction.
HTH
Fernando

[Laura Stortz]

Hi Laura,
this is actually a very common mistake most people do when thinking about marginal effects.
A good source for a better understanding of how margins work is Williams (2012) paper.
the bottom line is that margins for interactions cannot be directly estimated with margins. Why?
Margins can only estimate first order effects (first derivatives) (dydx). So it can comfortably estimate the effects of "treatment" or of income_low. It cannot estimate the marginal effects of interaction because that implies a second derivative.
What you can do alternatively is to use something like this:

margins r.treatment t.income_low
margins r.treatment#t.income_low

The second one will give you something like the "treatment effect" that comes from the interaction.
HTH
Fernando

Hi Fernando,

Thank you for your response, that was really helpful to me. I have a follow-up question to your code:

Should I write
margins r.treatment t.income_low, dydx(*) atmeans
margins r.treatment#t.income_low, dydx(*) atmeans

Or is that unnecessary? Also I didn't find in the margins documentation what r. and t. mean? Sorry for my many follow up questions, I just want to make sure I understand before I use it.

[FernandoRios]

That was a typo
it should be r in both cases
and the dydx is not necessary.
I haven’t tried it with atmeans option

also I did provide you with both codes so you could see how the dif in dif worked.

[Laura Stortz]

Hi Fernando,

Thank you very much for your response! It has helped me very much. I will read up on it.

Cheers,
Laura