Interpret marginal effects reported by the command margins when interaction terms are included in the model

[Meng Song]

Dear all,
I'd like to ask, when there is an interaction term included in a model, how one should interpret the reported marginal effects reported by "margins, dydx(*) predict(pu0)" after using xtlogit. "Margins" doesn't report the marginal effect of the interaction term, because the value of the interaction term can’t change independently of the values of the component terms, so one can’t estimate a separate effect for the interaction. But I want to know how to interpret the reported marginal effects then?
As an example, if the model is like this:
y = a0 + a2*x + a3*z + a4*x*z + error
If estimating this using "xtlogit y c.x##c.z, re", then get marginal effects using "margins, dydx(*) predict(pu0)", how shall I interpret the marginal effects? How are such marginal effects calculated when an interaction term "c.x##c.z" is included in the model but not reported by margins?
Thank you very much for your insights.

[Clyde Schechter]

The -margins, dydx()- output for x, represents the average marginal return of x, adjusted to the distributions of both x and z in the estimation sample. There is an analogous interpretation for the output for z. The calculation of these marginal effects reflects the inclusion of the interaction term in the model, which permits the odds ratio associated with x to vary depending on the level of z (and vice versa). There is, of course, no such thing as a marginal effect of the interaction term itself.

[Meng Song]

Thank you very much for your reply Clyde. Does this mean I shall interpret the marginal effects as normal, ie. the effect of a unit change in x on y, when the interaction effect of x and z is controlled for, ceteris paribus?

[Clyde Schechter]

Does this mean I shall interpret the marginal effects as normal, ie. the effect of a unit change in x on y, when the interaction effect of x and z is controlled for, ceteris paribus?

An unfortunate fact of language is that it is difficult to express this properly and it is very easy to say things that sound correct but don't actually make any sense.

There is no such thing as controlling for the interaction effect of x and z. (And this is not about "control" vs "adjust," there is also no such thing as adjusting for the interaction effect of x and z.) Indeed, the term interaction effect is, itself, a misnomer, because it does not represent an effect: it represents the difference between effects. I prefer to call it the interaction "term," to avoid confusion.

An accurate way to describe the output of -margins, dydx(x)- is: the average difference in ("effect on" if you wish to use language with causal connotations) the expected value of the outcome y associated with a 1-unit change in x, averaged over the distribution of z in the data. The calculation of it recognizes and accounts for the fact that the marginal effect of x on the expected outcome is a non-constant function of z (i.e. there is an x#z interaction).

[Richard Williams]

I've always liked the way Vince Wiggins explained this:

https://www.stata.com/statalist/arch.../msg00293.html

[Meng Song]

Thank you very much Clyde. Your suggested interpretation makes good sense. Basically what I plan to do is to present the marginal effects from "margins, dydx(x)", as well as the graphs showing interaction effects from marginsplot. I gather I shall provide some interpretation for the marginal effects, thus asking for an appropriate way to do this. Many thanks indeed.

[Meng Song]

Thanks a lot Richard for your reply. I appreciate Vince's advice of visualising the interaction effects over a range of values. As the estimated coefficients are not directly interpretable (please correct me if I'm mistaken here), marginal effects are going to be presented together with graphs showing interaction effects. That's why I wonder about the correct way to interpret these marginal effects.

[Meng Song]

The -margins, dydx()- output for x, represents the average marginal return of x, adjusted to the distributions of both x and z in the estimation sample. There is an analogous interpretation for the output for z. The calculation of these marginal effects reflects the inclusion of the interaction term in the model, which permits the odds ratio associated with x to vary depending on the level of z (and vice versa). There is, of course, no such thing as a marginal effect of the interaction term itself.

Dear Clyde,
May I ask a following question, please? Would it be correct to tell the significance of the interaction term from the direct output of "xtlogit y c.x##c.z, re"? The graph from "marginsplot" shows different effects of the interaction term, but I'm not sure if I can say this is significant. Many thanks.

[Clyde Schechter]

Would it be correct to tell the significance of the interaction term from the direct output of "xtlogit y c.x##c.z, re"?

Yes.

The graph from "marginsplot" shows different effects of the interaction term,

No, it does not. Nor does it show unicorns, because neither unicorns nor "effects of interaction terms" exist. When dealing with interaction models it is very important to be careful how you say things. It is far too easy to blurt out phrases that sound nice but are meaningless: "effect of the interaction term" is one of those. The interaction term, in your model, shows the extent to which the effect of z depends on c, and vice versa. It does not, itself, represent an effect. I know I'm being very pedantic here, but it is hard enough for people to understand interaction models when everything is said carefully. When loose language is thrown into the mix, it becomes impossible.

You don't show the -margins- command you ran before the -marginsplot-, so I can't comment on what the graph shows, nor advise you how to interpret it.