mean centering in no-constant model

[Will Hauser]

Apologies for starting a new thread but I have a fundamentally different question, hence the new topic.

So I am estimating a logit model of the following form:

logit c.xvar1 i.xvar2 ibn.xvar3 if sample, or nocons vce(cluster xvar3)

My interest is in xvar3 which is a categorical variable identifying the group to which the observations belong; there are 20 groups. Initially, I grand mean centered all the predictors except xvar3 because this was to be a mixed effects model and mean centering would have been appropriate in that context. However, with the fixed effects model I assumed that the coefficients would not be sensitive to mean centering since the intercept is omitted. This proves only partially true. The dummies for the groups (var3) change if the other predictors are mean centered (or not) while all other variables in the model (including other dummy variables) are not sensitive to the mean centering. What is the proper interpretation of the xvar3 coefficients (the reference category-less dummies) under a mean centered model?

Ideally, I would like to get the odds (or probability) of the outcome occurring for an overall 'average' case in each of the 20 groups identified by xvar3. My sense is that this is what I am getting by mean centering all variable except xvar3 and running the model specifying ibn.xvar3 and using the "noconstant" option as described above.... but I am unsure.

Obviously, I could fit the usual model with the constant and then use margins to get the average marginal effect at the mean but I think that's an unnecessary level of complexity if I can just make sense of the no constant model.

[Clyde Schechter]

Your code is confusing. Stata ignores the c. you prefixed to xvar1. But if xvar1 is truly a continuous variable, it isn't really suitable as an outcome variable in a logistic regression. Mind, it's not a violation of syntax that Stata will complain about, but it's odd at best.

Putting that aside and just focusing on the interpretation of coefficients in a no-constant model, using ibn.xvar3 and -nocons-, you end up with as many indicator variables for xvar3 as it has levels in the estimation sample: you do not get the usual omission of one reference category. In this set up, the coefficient of one of those levels is your model's estimate of the log odds of a positive outcome when xvar3 takes on that value and all the other predictors are zero. So if you mean-center all the other variables, then we're talking about those underlying variables taking on their mean values. If you don't mean-center the other variables and leave them in their native form, then we're talking about their actually being zero.

This explains why the coefficients of the xvar3 indicators differ when you center and when you don't: they're predicting probabilities for entirely different conditions.

Obviously, I could fit the usual model with the constant and then use margins to get the average marginal effect at the mean but I think that's an unnecessary level of complexity if I can just make sense of the no constant model.

I would say that if you have to post a question on the Forum to get an explanation of your findings, when you could have used -margins- and gotten straightforward answers with minimal effort and little thought, your approach adds, rather than avoids, an extra level of complexity, no?

[Will Hauser]

Thanks Clyde for another insightful answer. And yes, I accidentally omitted the outcome (yvar) from the hypothetical code, sorry for the confusion; the c. prefix was simply to show that the model contained both dummy and continuous covariates. It sounds like I had the interpretation correct, I was simply unsure. I expected the other coefficients to be unaffected by centering but did not expect centering the other coefficients to affect the uncentered and reference-less set of dummy variables. I see now the mistake in my math.

As to margins - it's a great command to be sure and I use it a lot. If I hadn't received such a helpful answer here I would have resorted to experimenting with it to make sense of what I was seeing in the no constant model. I want to thank you again for the helpful and quick response.