There are problems with the MASS library's
glmmPQL. It does not return a log-likelihood, so model selection will be difficult.
As for what to look at: check for
warnings(), look at fixed effect estimates and significance, look at random effects variance estimates (and confidence intervals for those if you can), and look at the overdispersion parameter estimate. I would say to also look at the log-likelihood so you can compare nested models, but... I've already said that could be an issue.
There are, however, a few other options for you.
Older versions of
lme4 had a
method parameter for
glmer which allowed you to specify PQL. While the error reporting was sometimes scant (you needed to check warnings about separability to see if your variance estimates were nonsense). However, apart from that issue the code was better in that it returned a log-likelihood which you could use.
You could also try
glmer's Laplace or quadrature (say, for 2-3 points) methods. Those are typically slower than PQL, but they are similar in that they approximate the integrand and may be more useful than an old version of
glmer or not getting a log-likelihood back from