Meaning of perfect correlation between by-intercept and by-slope adjustments
On Fri, May 13, 2011 at 3:32 PM, Petar Milin <pmilin at ff.uns.ac.rs> wrote:
On 13/05/11 22:00, Douglas Bates wrote:
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))
I am always hesitant to go for scalar version. As far as I understand, this implies homoscedasticity across levels of F1, but correct me if I am wrong. In my model, I am not sure if that would be correct.
You are correct. However, the model with vector-valued random effects is not supported by the data in the sense that it converges to a singular variance-covariance matrix. When you have 5 random effects associated with each level of participant and you allow the 5 by 5 positive semi-definite variance-covariance matrix you are attempting to estimate 15 variance parameters for that one term. You need a lot of data to be able to do that.