Is it plausible to have a positive coefficient with a negative marginal / impact effect after running multinomial logit model?
Solved – Positive coefficient but negative marginal effect in mlogit
marginal-distributionmlogitregression coefficients
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Both @Henry, and @JDav are pointing you in the right direction (+1 to each). However, I'm very visual and it helps me if I can see how this works. In that respect, here's a quick plot in which the first variable is confounded with group membership. If the groups are ignored, the correlation coefficient is positive (as can be seen in the figure), but in a multiple regression, $\beta_{x1}=-1$, albeit with different intercepts for the three groups.
As further food for thought, when all variables are categorical (instead of continuous as in this case) the phenomenon of reversing the apparent relationship upon inclusion of other variables is known as Simpson's paradox. Since it's ultimately quite similar, it may help to read about that as well. It is discussed on CV here.
First, your arm:age coefficient is not significant (p value 0.12 is greater than 10% and has "no stars" in the summary), so nothing can be said. However, assuming it were significant, you are right: Each year increases the ratio of the probabilities (odds ratio) Tatoo(arm)/tatoo(foot) by a factor of exp(0.04) approx 1.04 based on the previous year, so ten years more in age increase this ratio by a factor of 1.04^10 and so on. The same with lips: the ratio tatoo(lip)/tatoo(foot) decreases by a factor exp(-0.0778) per year. You calculate the odds ratio. Your last point is simple: change in odds ratio (probability ratio) tatoo(body part X)/tatoo(part Y) per year = exp(X.age-Y.age) with X.age,Y.age the corresponding coefficients.
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Best Answer
This is a version of a frequently asked question. The answer is that your variable is correlated with the other variables that are in the model. As a result, the estimates for the variable can be positive when the other variables are included even though the marginal association is negative. The basic story is discussed in many places on CV, but if you want a generic introduction, it may help you to read my answer here: Is there a difference between 'controlling for' and 'ignoring' other variables in multiple regression?