# Interpretation of random effects in mixed model with glmer()

I'm working on a final project and need to estimate a multilevel logistic model for analyzing dropout rates in higher education in a specific region.

The model has three levels, where intercepts vary across courses and institutions. I'm using the lme4 package and the glmer function. I understand the results for fixed effects, which are similar to traditional models, but I'm unsure about interpreting the random effects.

For institutions, the standard deviation was 0.45, and for courses within institutions, it was 0.34. What does this mean? Can I say that there is more variation in dropout chances among institutions than among courses within institutions?

It's not particularly common to want to interpret the random effects variances unless we are just fitting a variance components model (ie a model with random effects but no fixed effects). The random effects are usually thought of as being nuisance variables, that are estimated to handle the non-independence of observations in clustered data. Having said that, if you really want to say something about the random effects, then a standard deviation of 0.45 for institutions suggests that there is variability in the log odds of dropout rates across different institutions. A standard deviation of 0.34 for courses within institutions indicates variability in the log odds of dropout rates across different courses within the same institution.

And yes, you can say that there is more variation in dropout chances among institutions than among courses within institutions, suggesting that the factors influencing dropout rates might vary more between different institutions than they do among different courses within the same institution. You might want to go on and look at how the the whole unexplained variance is partitioned by calculating the variance partition coefficients:

VPC for courses:

$$VPC_{\text{course}} = \frac{\sigma^2_{\text{course}}}{\sigma^2_{\text{course}} + \sigma^2_{\text{institution}} + \frac{\pi^2}{3}}$$

and for institutions: $$VPC_{\text{institution}} = \frac{\sigma^2_{\text{institution}}}{\sigma^2_{\text{course}} + \sigma^2_{\text{institution}} + \frac{\pi^2}{3}}$$

• Thank you very much, @robert-long for the response, it was very helpful. Just to clarify further, my focus is on the fixed effects. However, as there is a question in the project regarding the evaluation of information from random effects, I wanted to make sure that I am correct in my interpretation and include this explanation based on the information reported by the glmer() function. So why isn't commom to interpret or report the random effects variances? Nov 16, 2023 at 17:22
• @gustavobrp you're welcome. The reason it's not common to interpret the random effects is simply because we are typically interested in the fixed effects and the random effects are nuisance variables. I gave an interpretation of the random effects in your study but I don't think it really adds much, since your interest is in the fixed effects. It's very much like the situation with experiments that use blocking, to get a more precise estimate of the fixed effects / reduce residual variation. We interpret the treatment (fixed effect) but we don't typically interpret the blocking factors..... Nov 16, 2023 at 17:26
• ...And if the blocking factor(s) have sufficient number of levels a mixed model with random effects for the blocking factors can be used, and again, we don't typically interpret the random effects. Basically, it all comes down to your research question. If it is about the fixed effects (what is the association between some outcome and some fixed effect?) then there is no need to interpret the random effects. Of course there could be situations where the research question does require the interpretation of the random effects Nov 16, 2023 at 17:30