# uncorrelated random effects with lmer and glmer

I am modelling the activity of animals (subjects). Each subject is measured for activity repeatedly (which requires a random intercept associated with subject). Subjects have also been measured in two environments (a and b). I specifically want to see if the activity level of a given subject in environment a correlates with its activity level in environment b.

In line with this really useful answer (How to specify uncorrelated random slopes in lmer() syntax?), I tried doing a log-likelyhood ratio test on these models:

full.model <-
lmer(activity ~ (1+environment|subject), family='binomial')
no.correlation.model <-
lmer(activity ~ (1|subject) + (environment-1|subject), family='binomial')


It appears that this code works nicely for gaussian data, but that it does not for binomial data. In this case, no.correlation.model has two random intercepts, a 'slope' associated with environment and a correlation.

Is there a way to test whether the correlation estimated in full.model is significant?

Here is a reproducible dataset.

activity <-rbinom(n=1000, size=1, prob=0.5)
subject <- rep(1:100,each=10)
environment <- rep(c('a', 'b'), 500)


Many thanks in advance!