higher-order interaction terms in conditional logit: how to compute properly

[Eryk Krysowski]

Dear all,
I am having problems with using margins and inteff. Here is the story:

There is a famous paper by Norton et al. (2003) which describes why stata's wrong about interaction terms in logit regressions. They propose program called inteff that you can download for stata.
Maarten Buis also presents a way how to deal with it using odd ratios (http://www.maartenbuis.nl/publications/interactions.pdf).

The thing is that I am using 2nd order interaction terms. For conditional logit I have 11 possible decisions for each individual and only one is chosen. To check for the influence of personal characterics I need to regress on <choice_specific_variable>*sex*treatment, and not simply sex*treatment, because sex and treatment do not vary within each subject. However, when I type
"margins, over(<choice_specific_variable>#(treatment sex)) expression(exp(xb())) post" (look at Buis) stata says you cannot use interaction terms for margins. Do you have any idea how I proceed in such a case?

Using inteff (solution by Norton et al., 2003) is also problematic, because it doesn't work with conditional logit.

Thank you very much in advance for your reply and your time!
Best regards,
Eryk Krysowski

[Maarten Buis]

You can use the ratio of odds ratio interpretation without margins. In fact you can just use the main output of your clogit command with the or option.

[Eryk Krysowski]

You can use the ratio of odds ratio interpretation without margins. In fact you can just use the main output of your clogit command with the or option.

Dear Maarten,
Thanks a lot for your quick responce!
I do not misunderstand something about your tip then. The problem discussed by Norton et al. (2003) is - as you describe it:

The interaction effect should thus be the cross partial derivative of E(y) with respect to x₁ and x₂, that is, an approximation of how much the derivative of E(y) with respect to x₁ changes for a unit change in x₂. In non-linear models this is typically different from the first derivative of E(y) with respect to the multiplicative term x₁ × x₂.

And stata only shows the first derivative with respect to the multiplicative term x₁ × x₂, right? So the estimated coefficients may be - in the extreme case - even of opposite signs to those of true cross partial derivatives.

I thought I need margins to get true values. Presenting the outcomes of clogit in terms of odd ratios instead of coefficients does not change e.g. the p-values, so how can I know if the sign and significance level is correct?

Best regards,
Eryk

[daniel klein]

I am sure Maarten will get back to this, but in the meanwhile I would like to point out that the key to understanding the discussion lies in the following lines

The effect of x1, in the marginal effects metric, is the first derivative of the expected value of the dependent variable (E(y)) with respect to x1, [...]

and

However, the marginal effects and multiplicative effects answer subtly different questions, [...]

That is to say, the p-values indeed stay the same, but the interpretation changes. Maarten goes on to illustrate this point in the reminder of his article, so I suggest you read through the complete thing again, focusing on the different interpretations and trying to figure out which of the them best suits your research question. I admit, I also had to read it more than once, even though (or because?) Maarten gets pretty much straight to the point.

Best
Daniel

[Maarten Buis]

stata only shows the first derivative with respect to the multiplicative term x₁ × x₂, right?

No, clogit does not show you the derivative or marginal effect, it just shows you the ratio of odds ratios. The key step is to realize that there are different ways in which you can quantify an effect. A marginal effects after a clogit tries to approximate a difference in expected probabilities while odds ratios looks at ratios of odds (hence the name). Similarly, the interaction effect returned by clogit is a ratio of odds ratios. As long as you interpret the interation effects correctly as ratios of odds ratios, the sign, significance, and size of the interaction effects returned by clogit are exactly correct. Norton et al. only had a problem because they did not want to look at odds ratios but at marginal effects.

[Charlie Joyez]

Dear Statalisters,

I face a similar issue while interpreting interaction terms in a Nested Logit model (nlogit command), a discrete choice model on the same principle that the conditional logit, except that its hierarchical structure releases the independace of irrelative alternatives assumption.
However, unlike clogit , nlogit doesn't allow the or option to compute odds ratio.

How should we interpret the interaction term in a nlogit structure?
How to manage to get the coefficient represent a multiplicative effect, which wouldn't be subject to misunderstanding?

Ps : Sorry if this is not the proper place to ask for this question, but I found it made sens as topics are close enough.
Best regards
Charlie Joyez