[R-sig-ME] Meaning of perfect correlation between by-intercept and by-slope adjustments

[Douglas Bates]

On Fri, May 13, 2011 at 12:35 PM, Petar Milin <pmilin at ff.uns.ac.rs> wrote:
> Hello!

> Simplified model that I have is:
> lmer(Y ~ F1 + F2 + C1 + (1+F1|participants) + (1|items))
> F1 and F2 are categorical predictors (factors) and C1 is a covariable
> (continuous predictor). F1 has five levels.
> By-participant adjustments for F1 are justified (likelihood ratio test is
> highly significant). However, what puzzles me is perfect correlation between
> two levels of F1. Others are quite high, but not perfect. I wonder what this
> means, exactly? Is there some "lack of information" which leads to problems
> in estimating variances?

I think of the estimation criterion for mixed models (the REML
criterion or the deviance) as being like a smoothing criterion that
seeks to balance complexity of the model versus fidelity to the data.
It happens that models in which the variance covariance matrix of the
random effects is singular or nearly singular are considered to have
low complexity so the criterion will push the optimization to that
extreme when doing so does not introduce substantially worse fits.

One way around this is to avoid fitting models with vector-valued
random effects and, instead, use two terms with simple scalar random
effects, as in

lmer(Y ~ F1 + F2 + C1 + (1|participants) + (1|F1:participants) + (1|items))