Can I specify a random intercept for trial in mixed logistic regression with binomial distribution?

Question

I have disaggregated/long-format data representing a binary outcome (success/failure). Cases are described by a single covariate called "covariate1". In each case, there are multiple participants, so each row represents an individual's response in that particular case (a trial). I'm interested in the effect that case characteristics have on the probability of observing successes.

I want to model this as mixed logistic regression with a binomial response because participants can appear in more than one case, so trials across cases can be clustered by their ID. The code for my models is shown below:

#If data were aggregated, information about the dependency across trials would be lost:
library(lme4)
glmer(cbind(resp,trials) ~ cov1
        family=binomial, agg.data)

#Disaggregated data
glmer(resp ~ cov1 + (1|ID),
        family=binomial, disagg.data)

Is it correct to specify a random intercept for participant ID across trials? What would be the interpretation of the random intercept estimates?

Answer 1

Your specification of the model is correct as described: your model attempts to predict the binary response of the trials using a single fixed effect (the lone covariate in your syntax) and random intercepts for subjects.

The random intercepts in this case essentially tell you how much on average each subject deviates from the mean logit of the response. So if you have one subject who has a random intercept of -2, this means that on average this person has far less successes than the others on trials. As an example, if the response was a reading test, this would suggest that this particular subject is a poorer reader than the others.

The random intercept variance from the model also tells you just how much subjects fluctuate. If the variance is low, then the clusters don't really tell us much more about the model. But if the variance is high, then a substantial part of our normal error can be attributed to by-subject differences, and adjusts our fixed effects to be more accurate (versus what would be biased coefficients if this random effect variance was unmodeled).