[R-sig-ME] LRT between GLMM and GLM to test a Single Random Intercept

[Juho Kristian Ruohonen]

My readers are likely to want to see a p-value on the only random effect
(an intercept) in my logistic GLMM.

Supposedly, if I fit the model using Laplace approximation, then the
likelihood is comparable with that of the fixed-effects model, so the
p-value from a LRT (divided by two) can be used. But I don't trust the
Laplace approximation much. I'd rather use at least 10 quadrature points
for improved accuracy. This also results in a more flattering (smaller)
random-effect variance and hence a lower reported intraclass correlation.
But if I use any more than 1 quadrature point, I can no longer report a
p-value on the random effect because *anova()* refuses to compare the
models, citing incomparable likelihood functions. I thought of calculating
the log-likelihood of the GLMM manually using *dbinom()*, the data and the
fitted values, but this thread
<https://stats.stackexchange.com/questions/381085/calculating-log-likelihood-of-logistic-adaptive-quadrature-glmm-for-comparison-w>
says I can't use the binomial PMF for that.


Is there a way I can have my cake (many quadrature points) and eat it too
(get a p-value for the random effect)? That parametric bootstrap procedure
sounds neat, but I'd still be running into the same problem: the LRT
calculated at each iteration compares a fixed and a mixed model, hence the
likelihoods cannot be compared.

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