How to interpret an average marginal effect in a multinomial logit model?

[Peter Brown]

Hi there!

I am currently analysing the average marginal effects of a multinomial logit model. Basically, you have three outcomes and four independent variables which are age (x1), IQ (x2), income (x3), and parent's education (x4). Lets assume outcome 2 would be travelling by plane instead of train (0) or bike (1).

How would I need to interpret theses outcome? Just x1 and x2 are statistically significant.

Is it correct to interpret or conclude that "if age increases by one unit - which is corresponding to one year - the average probability of choosing the outcome travelling by plane increases by 6.16, everything else being constant?

Is there anything else I can do in terms of interpreting these results? I thank you in advance.


Kind regards,

Peter

[Clyde Schechter]

This looks rather bizarre. While a logistic curve can have a very steep middle, steep enough that the marginal effect of a unit change in a predictor on probability could be as high as 6+, it is difficult to imagine a realistic situation where that arises. To achieve that, and not have age omitted from the model altogether because of perfect prediction, the range of values age takes on must be extremely narrow--not what one expects when age is denominated in years and the units of analysis are people (which I infer because these units of analysis have IQ, income, and parental education attributes.) This would be something you might expect to see if all the people in the subject were born the same week, so that the range of ages observed is a tiny fraction of a year. Possible, but seldom seen in the real world.

There is another way this can arise. If the effect of increasing age is to produce large decreases in the probabilities of the other three outcomes, then, since the total outcome probability must sum to 1, one could see a very steep gradient of probability of outcome 2 vs age like this. But again, the impacts of age on the other three outcomes would have to be almost as far-fetched as the scenario I just described.

So your interpretation is consistent with that output, in the way we typically abuse language when describing marginal effects. The definition of the marginal effect is dy/dx. Remember that in a non-linear model, there is no single marginal effect of a variable. The marginal effect changes with the value of the variable itself, as a consequence of the chain rule. So when we interpret a marginal effect, whether as here an average marginal effect, or a marginal effect at some specific level of the variable, bear in mind that when the predictor changes by 1 unit, the marginal effect changes with it--perhaps by a great deal. So saying that the marginal effect is the amount of change in the outcome associated with a unit change in the predictor is just a loose way of speaking. It's a bit like saying that we're going to travel 80 km in the next hour based on the fact that at this particular moment the speedometer reads 80km/h. This loose way of speaking is widely used, and allows those who had a painful experience taking calculus not to be reminded of that. So it persists. But it is inaccurate, and when using it you should always bear in mind that it is not literally correct.

All of that said, it is difficult for me to think of a realistic scenario that could produce the results you have gotten. So, while, taking into account the commonly accepted abuse of language, your interpretation of these results is correct, I'm suspicious that the results themselves are not.

[Jeff Wooldridge]

Could you show the command you used, and show the marginal effects for all three probabilities? I agree with Clyde that these seem very weird, unless this is the partial effect for outcome 2 relative to whatever has been chosen as the base.

[Peter Brown]

I thank you a lot for your responses. I have attached the marginal effects of all three probabilities including their commands. It also looked to me a bit weird since I was following the instructions of several papers on how to perform that analysis on Stata. However, the results looked quite high.

In terms of the interpretation, I am wondering if it should be more concrete or if the interpretation is widely accepted.

I also calculated marginal effects using the command mfx (please note that there were five variables earlier on). However, I am not sure what the exact difference is between the two ways. I suggest that the second one is at the means. Are 0.53, 0.40 and 0.07 the respective probabilities for outcome 0, 1 and 2? So this probability would increase by XX dy/dx when changing one unit in the independent variable?

[John Mullahy]

Peter: I agree with Clyde and Jeff that this is strange. Point estimates of the APEs/AMEs should be smaller than one in magnitude. One thought might be to compute by hand the marginal effects directly and see if they square with what margins is generating.

The attached gives what should be the relevant formulae (from a paper I wrote a few years ago; I hope the algebra is correct ). In this notation s(im) would translate as outcome y(m) for subject i.

[John Mullahy]

P.S. The conditional expectations expressed in the attachment should be interpreted in your case as conditional probabilities.