Multicollinearity test after estimating a multilevel model

[Jessica Rayse]

Dear colleagues,

I would like to perform a multicollinearity test between variables after estimating a multilevel model. Could someone recommend me a test for this?

Thanks,

Jéssica

[Clyde Schechter]

Multicolinearity is a property of the data, not the regression model. For whatever reason, -estat vif- only runs after-regress-. But you can just run -regress- on the same variables as are in your multilevel model (leaving out the random effects) and then run -estat vif- to get colinearity statistics. If somebody is demanding a multicolinearity test, that's what you can do.

That said, it is hard to find a bigger waste of time and energy in statistics than testing for multicolinearity. Multicolinearity does not bias parameter estimates. What it can do is degrade precision, resulting in larger standard errors and wider confidence intervals (and smaller test statistics, with larger p-values). The only variables that are affected by this are the ones that are participating in the multicolinear relationship. In most situations, we can classify our predictor variables into two groups: the ones we are really interested in, and the ones that are included only to adjust for their nuisance contribution to outcome variance (often called "control variables.") If only the latter are involved in a multicolinearity relationship, then it makes no difference: we don't care about estimating them with precision--we just want to account for their contribution to the outcome variance, and the multicolinearity doesn't interfere with that. If one or more of the variables you are actually interested in is involved in the multicolinearity, then you may have a problem. Or perhaps not.

The way to see if you have a problem (as opposed to just having multicolinearity that is not a problem) is to just look at the standard errors or confidence intervals of the coefficients of the predictors of interest. If they are small enough that your results are precise enough to answer your research questions, then there is no problem. If they are so large that your study is inconclusive in regard to one of its goals, then you have a problem. Now, here's the bad news. If you do have a problem, there is nothing you can do about it with the existing data. The only solutions are to get a larger data sample, typically a much larger sample, or to scrap your data and start over with a different design that breaks the multicolinear relationship, for example by some kind of matching or restrictions on inclusion.

Notice, therefore, that you don't need any kind of test for multicolinearity. You can identify whether you have a multicolinearity problem just by looking at your standard errors (or confidence intervals) for the key predictor variables of interest.

For a much more entertaining explanation of this, see Arthur Goldberger's A Course in Econometrics, which has a chapter devoted to this and introducing a better term for the phenomenon: hyponumerosity.

[Bruce Weaver]

Well said, Clyde Schechter! And for anyone who is not familiar with Goldberger's comments on micronumerosity, see this nice blog post by Dave Giles:

• https://davegiles.blogspot.com/2011/...umerosity.html