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From |
Richard Williams <richardwilliams.ndu@gmail.com> |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: Marginal Effects for Logistic Mulitlevel Model withInteractionTerms |

Date |
Sun, 29 Aug 2010 20:22:26 -0500 |

A belated followup to Maarten's lengthy post on this:

sysuse auto logit foreign price mpg c.mpg#c.price, nolog margins, dydx(*)

At 02:29 AM 8/20/2010, Maarten buis wrote:

--- On Thu, 19/8/10, Iris.C.Goensch@zeu.uni-giessen.de wrote: > -mfx does not recognize interaction terms and thus does not > calculate the marginal effects for interaction terms in the > correct way (the problems resulting are explained in Ai/ > Norton and Greene (2010). The problem is that marginal effects try to force a from of effect onto a model that is not native to that model. The consequence is that it will always give some problems, like the difficulty you get with getting interaction effects. It is like fitting a square peg into a round hole, you might force it in with a hammer (possible add a bit of duct tape), but the result will never be nice. The obvious solution is to use the form of effect that is native to the model, unless you for substantive reasons really need that other type of effect. In case of (multilevel) logistic regression that means that you need to look at odds ratios, and the interaction effects will give you ratios of odds ratios. This is discussed in Buis (2010). So, based on my 2010 article I recommend that you just report odds ratios, and interpret the interaction terms of ratios of odds ratios. This article gives you a concrete example of what such a discussion of results could look like, it is actually much easier than many people think. This article has led to quite a bit of private communication with people who are so used to the idea that interactions are very hard in non-linear models that they are surprised that the solution can be so easy. So below I will give a bit longer explanation that liberally borrows from these private conversations. Remember that an effect is nothing other than a comparison of the expected outcome across real or counter-factual groups. We observe incomes for a set of males, we find a comparable set of females who work (that can be hard, which is why we have models like -heckman-) and a comparison of the average wages of the males and females is the effect of gender. We can compare average wages by looking at the difference or at the ratio, i.e. women earn x euros/pounds/yen/... less then men, or women earn y% less then men. Both are completely valid quantifications of the effect of gender on income. We usually start our statistical (econometric/psychometric/ ...) modelling education with a linear model. Linear models are naturally designed for a comparison of groups in terms of differences. If we add a continuous variable x than its parameter says if we get one more unit of x we can expect b more units of y, regardless of how many units of y you had to start with. So we took all possible comparisons that where one unit apart and constrained the differences in expected y to be equal. We can relax such assumptions, but it shows that effects in terms of differences is the native way of thinking about effects in linear models. The native way of thinking about effects is in terms of ratios in non-linear models that include a log in their link function (e.g. (multi-level) logistic regression that models the log(odds), Poisson that models the log(count), survival models that model the log(hazard) or log(time), etc.). If we add a continuous variable in one of those models we say that expected value of y increases by a factor of exp(b) when you get an additional unit of x, regardless of how many units of y you had to start with. Again this assumption can be relaxed, but it shows that effects in terms of ratios are the native way of thinking in this type of non-linear models. Now you can use effects in terms of ratios in linear models (e.g. elasticities) and effects in terms of differences in non-linear models (marginal effects), but since they are not the native way of thinking in those models and thus there will always be some friction. You cannot represent an effect that is constant in terms of ratios with one effect in terms of differences, and vice versa (except for the trivial case where there is no effect). This type of problems multiply when looking at interaction effects. This is why Ai & Norton (2004) had to go through all the effort and than present their interaction effects in terms of graphs, while I could just exponentiate my coefficient and had my interaction effect as one number. The mathematical proof is straightforward, and can be directly derived from the properties of the logarithm. A version of that proof can actually be found in the Stata Journal article of Edward Norton, Hua Wang, and Chunrong Ai (2004), where they discuss the implementation of their technique in Stata. In section 2.6 of (Norton et al. 2004) they showed that the exponentiated interaction effect is the ratio of odds ratios. They claim that nobody can understand that, in my article (Buis 2010) I show that that is actually not that hard. In short Norton et al. and I don't disagree on the math. The choice between which method to choose is really one that needs to be based on substantive and pragmatic reasoning. If your theory gives you a clue whether or not you want to control for differences in the baseline odds, than the choice is easy: with level effects you don't control for such differences, with ratio effects you do. When your theory is not that precise, then you need to use pragmatic reasoning. With odds ratios you need to put a bit of effort into explaining your results as a lot of people find them hard. With marginal effects you have the awkwardness of trying to force a linear model on top of a non-linear one. Sometimes it is nice to present both, in the way that I have done in my article. Hope this helps, Maarten M.L. Buis (2010) "Stata tip 87: Interpretation of interactions in non-linear models", The Stata Journal, 10(2), pp. 305-308. Edward Norton, Hua Wang, and Chunrong Ai (2004) "Computing interaction effects and standard errors in logit and probit models" The Stata Journal 4(2):154--167. <http://www.stata-journal.com/article.html?article=st0063> -------------------------- Maarten L. Buis Institut fuer Soziologie Universitaet Tuebingen Wilhelmstrasse 36 72074 Tuebingen Germany http://www.maartenbuis.nl -------------------------- * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

------------------------------------------- Richard Williams, Notre Dame Dept of Sociology OFFICE: (574)631-6668, (574)631-6463 HOME: (574)289-5227 EMAIL: Richard.A.Williams.5@ND.Edu WWW: http://www.nd.edu/~rwilliam * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**Follow-Ups**:**Re: st: Marginal Effects for Logistic Mulitlevel Model withInteractionTerms***From:*Maarten buis <maartenbuis@yahoo.co.uk>

**References**:**Re: st: Marginal Effects for Logistic Mulitlevel Model withInteractionTerms***From:*Iris.C.Goensch@zeu.uni-giessen.de

**Re: st: Marginal Effects for Logistic Mulitlevel Model withInteractionTerms***From:*Maarten buis <maartenbuis@yahoo.co.uk>

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