Options when linear mixed model residuals are non-normally distributed

[Dougie Jones]

Hey everyone,

In a project I've been working on I've run into non-normally distributed residuals in a linear mixed model, with some fixed effects and just random intercepts (no random slopes). Non-normality of the residuals, of course, violates the assumption that the error term is normally distributed. I have uploaded a qqplot and histogram of the level 1 residuals.

The model has dependent variable time (0, 1, 3, 4, 5, 9, 14, ..., 432) and several binary and categorical variables, as well as one continuous variable.

predict lev1, rstandard
qnorm lev1
histogram lev1, rstandard

Ultimately I want to visualize the random intercept, and I am afraid that a log-transformation may eliminate that option. Moreover, the fact that interpretation of linear mixed models is relatively simple is something I like.

What approaches that allow visualizing the random intercepts would you recommend for me? Are perhaps GLMMS a good choice? And with what family and link function?

[Clyde Schechter]

If you use the -vce(robust)- option, then the assumption of normality of residuals no longer applies.

[Dougie Jones]

Thank you, Clyde.

However, using vce(robust) makes it impossible to predict the standard errors of the random effects predictions.

mixed x y || id:, vce(robust)
predict random_effects, reffects reses(newvar)

Is there any way around this issue? Or an alternative to show the uncertainty surround the random effects predictions?

[Joseph Coveney]

You mentioned generalized linear multilevel / mixed models. Have you considered something like the following?

meglm . . . || . . . : , family(gaussian) link(log)

[Bruce Weaver]

If you use the -vce(robust)- option, then the assumption of normality of residuals no longer applies.

Hi Clyde, and others. There are lots of books and articles one can cite to support use of Huber-White robust standard errors to deal with heteroscedasticity. I wonder if anyone can point to any reputable textbooks or articles to support Clyde's point about non-normality. Thus far, all I've come up with is the following excerpt from this UCLA web-book on regression (emphasis added):

The Stata regress command includes a robust option for estimating the standard errors using the Huber-White sandwich estimators. Such robust standard errors can deal with a collection of minor concerns about failure to meet assumptions, such as minor problems about normality, heteroscedasticity, or some observations that exhibit large residuals, leverage or influence. For such minor problems, the robust option may effectively deal with these concerns.

With the robust option, the point estimates of the coefficients are exactly the same as in ordinary OLS, but the standard errors take into account issues concerning heterogeneity and lack of normality.

It's not that I doubt what you and the web-book author are saying, Clyde. I'm just imagining that my boss or other collaborators may need something in print in a reputable book or journal to be convinced.

Cheers,
Bruce