[R-sig-ME] Likelihood ratio statistics when one model is probably overfitting

[Petar Milin]

Hello ALL,
Sorry for being stubborn, but I would really like to clarify one issue 
that remained open after my latest post.

In brief, I started with a model that had vector-valued random effect, 
like (1+A|item). Unfortunately, it got perfect correlations between some 
levels of A
(the variance covariance matrix of the random effects is singular or 
nearly singular). Then, I got advice to go for scalar random effect, 
like (1|A+item). Finally, I got another advice to compare two models -- 
vector-valued and scalar -- by means of likelihood ratio.

My question is: whether one should compare model that has (nearly) 
singular variance/covariance matrix of random effects at all? Using 
fit-statistics of a model that has "wrong fit" (overfit) seems 
unnecessary, if not odd. (Moreover, I secretly did that, and the 
vector-valued won!)

Please, can anyone more experienced clarify this?

Thanks,
PM