interpreting the interaction effects of odds ratios in femlogit

[Alan Muller]

I am using a fixed effects multinomial logit model (femlogit) with interaction terms. There are three possible outcomes, with outcome(1) being the baseline/default. I have an interaction between two continuous variables, and I am unsure how to interpret the odds ratios, especially since the odds ratios associated with the interacted variables' main effects really jump around. I understand that being a ratio means such fluctuations are possible, but I am:

1) concerned that the fluctuations somehow mean something is "wrong", and
2) unsure how to make sense of the two main effects plus the interaction.

femlogit f.y x1-x10 x1*x2, robust or

A few things: femlogit doesn't allow operators like "#", so interaction terms need to be calculated by hand (hence "x1*x2" as the interaction here). Next, x1 is log transformed.

I'm not reproducing the entire output, but here's what the odds ratios of interest look like in the case of outcome 3 (relative to outcome 1):

x1	1.04*	1.04*
	-1.99	-2.04
x2	0.24	1.04e+06*
	(-0.70)	-2.34
x3	11.52	7.53
	-0.82	-0.67
x4	0.42*	0.44*
	(-2.56)	(-2.45)
x5	0.94	0.98
	(-0.29)	(-0.10)
x6	0.92	0.92
	(-0.93)	(-0.93)
x7	0.92	1.02
	(-0.01)	0
x8	0.02	0.03
	(-1.25)	(-1.15)
x9	1	0.99
	(-0.06)	(-0.10)
x10	1.72**	1.74**
	-10.48	-10.6
x1*x2		0.16**
		(-2.80)

As said, the odds ratio for x2 has me concerned. As for the interaction term, I read a post by Maarten Buis in a related thread in which he said the way to interpret the interaction effect is to calculate it across a meaningful range of x1 or x2 (I guess either). If I were interested in -1SD to +1 SD for x1, that range would be 5 (like I said, this is a log-transformed currency based value). That would mean like di (exp(_b[c.x1#c.x2]*5)-1)*100 = 25%. That is certainly interesting, if true. But what about the ORs for x1 and x2 in isolation? Are they not to be included in this calculation somehow?

Sorry if this is really basic. If anyone has a simple formula for how to explain the relationships of interest in a straightforward yet meaningful way, I'm all ears.

[Clyde Schechter]

Well, the fact that the ORs associated with x1 and x2 differ with and without the interaction term is nothing to be concerned about. However, the fact that the OR for x2 is 1.04 x 10⁶ in the presence of the interaction term is strongly suggestive that something is wrong. Odds ratios like that (which are, for practical purposes infinite) almost always reflect some kind of error. In the case of logistic models (including multivariate logistic models) the error is often in the data: it sounds like outcome1 is never, or only extremely rarely, 0 when x1 is zero. (The "x2 odds ratio," in the presence of the x1*x2 interaction is, in fact, not the x2 odds ratio; it's the odds ratio for x2 conditional on x1 being 0.) If that's true, then your data really doesn't lend itself to this kind of model.

That said, let's review the interpretation of odds ratios when there are interactions. The "odds ratios" for x1 and x2 are not what they appear to be. The "odds ratio for x1" is, in fact, the odds ratio for x1 conditional on x2 = 0, and vice versa. In fact, in an interaction model, there is no such thing as "the odds ratio for x1." There is a different x1 odds ratio for every value of x2; no one of them can claim to be "the x1 odds ratio"; and the output shown is simply the one for x2 = 0. Then there is the "odds ratio for x1*x2." That's even worse, because it isn't an odds ratio at all. It's a ratio of odds ratios. Given values of x1 = a, and x2 = b or c, it means that the odds ratio of x2 given x1 = a and x2 = b is equal to (0.16)^b-c times the odds ratio of x2 given x1 = a and x2 = c.

If that seems difficult to intuit, that's because it is. Interactions between continuous variables are hard enough to wrap your mind around in linear models. They are an order of magnitude harder in logistic models, and add another one or two orders of magnitude of difficulty in multivariate logistic models.

Finally, I'll just add that the fact that one of the variables is the logarithm of something else doesn't change any of this.

[Alan Muller]

Thank you Clyde, this is definitely food for thought. I understand the point about the one outcome basically never occurring when x1 is zero, because this had me questioning the wisdom of the fixed effects approach. When I considered this before I wondered if a remlogit using qsem would be a smarter way to go?

[Clyde Schechter]

You can try it, but I doubt that will get you any better results.

A more fruitful approach might be to re-center the variable x1 around some value that is fairly far from 0 (but not so far that outcome1 = 0 almost always happens at x1 = that value) and using the re-centered variable in the model instead of x1. That should produce more reasonable odds ratios.

[Alan Muller]

That worked great, and based on your first reply makes a lot of sense. As a first shot, I mean-centered x1 (x1's original mean was 8.2 with a SD of 2.2). Rerunning the model, the OR and significance level for the interaction x1*x2 stayed essentially the same, while the OR for x2 in the interaction specification went from 1.04 x 10⁶ to a much more respectable 0.31. It is now also non-significant (p-value = 0.61), however. Does that make sense, given the mean centering, in light of what you said before (different ORs for x2 depending on the level of x1)?

[Clyde Schechter]

Yes, it makes sense. You should ignore the p-values of the X1 (original or re-centered) and X2 odds ratios in any case--in an interaction model they are almost always useless. The p-values of the interaction term, and the joint p-value of X1 and X1*X2, (or X2 and X1*X2) may have some usefulness, but not those of X1 or X2 alone.