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  • Logit with interaction

    I want to estimate a logit model with a binary dependent variable and two categorical independent variables, Concern and Motivation, each with five levels.

    In my specification, I interact these variables as i.Concern##i.Motivation. When I examine the output of the model, most of the interaction term coefficients are statistically insignificant.

    However, I understand that in nonlinear models like logit, the coefficients on interaction terms are not directly interpretable in the way they are in linear models. So instead, I used Stata’s margins command:
    margins Motivation, dydx(Concern),
    to calculate the marginal effects of each level of Concern at each level of Intrinsic Motivation.

    This gives me a set of marginal effects that are all statistically significant, in contrast to the insignificant interaction coefficients from the logit model itself. I wanted to confirm whether this is the correct approach for interpreting the interaction — are these marginal effects the appropriate quantities to focus on in this case?

  • #2
    In light of findings of Ai and Norton (2003), the coefficient on the interaction term that you find is not the true interaction effect. There are many user-written commands on Stata which address this issue, such as inteff (journals.sagepub.com/doi/pdf/10.1177/1536867X0400400206).

    Taking the marginal effects of one variable at various levels of the other, is a correct approach.

    In sum, do not interpret the coefficient on an interaction term in a non-linear model as an interaction effect.

    Shang et al. (2018) do highlight one exception in Poisson though: when the interaction is between a binary and continuous variable, its coefficient is directly interpretable as a difference in semi-elasticity.

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    • #3
      Well, that is not a universally held opinion. The key issue is that Ai and Norton claim that economists are not smart enough to understand the concept of a ratio of ratios. Personally, I have met enough smart economists to doubt that statement. However, I am sure they know more economists than I do, so if they think the average economist can't do multiplication or division then I will defer to their judgement.

      An interaction is just a comparison of comparisons. You can compare groups by computing differences or by computing ratios. So in a additive (linear) model an interaction would be a difference in difference, while in a multiplicative (like a logit or a Poisson) model an interaction is a ratio of ratios.
      ---------------------------------
      Maarten L. Buis
      University of Konstanz
      Department of history and sociology
      box 40
      78457 Konstanz
      Germany
      http://www.maartenbuis.nl
      ---------------------------------

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      • #4
        I don't want to get into the debate over the proper way to represent interactions in non-linear models. On the one hand, I agree with Maarten Buis that differences in differences and ratios of ratios are both legitimate ways of doing that. On the other hand, I think that in practical terms, the audience that can properly understand differences in differences (or think they can) is appreciably larger than the audience that can understand ratios of ratios.

        Be that as it may, I'm concerned that O.P. is misunderstanding the outputs of -logistic- and -margins-:

        This gives me a set of marginal effects that are all statistically significant, in contrast to the insignificant interaction coefficients from the logit model itself.
        The output of -margins Motivation, dydx(Concern)- does not directly provide assessments of interactions (in the sense of differences in differences). It provides the estimates of marginal effects of Concern conditional on each level of motivation. The question of interaction is not answered directly by those results: the extent of interaction is answered by the differences among those conditional marginal effects. So the fact that all of those -margins- outputs are "significant" tells you only that regardless of the level of Motivation, the marginal effect of Concern is "statistically significant." It does not tell you whether the differences between marginal effects of Concern associated with different levels of Motivation are "significantly different." For that you need to calculate those differences, the simplest way being -margins Motivation, dydx(Concern) pwcompare-.

        By contrast, the logistic regression coefficients of the interaction terms are actual measures of the extent of interaction, albeit in the ratio of ratios metric rather than difference in differences. And, also important to recognize, the interaction terms in the logistic regression do not directly give you the conditional marginal effects: those require further calculation, which is what -margins, dydx()- provides.

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