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

[Douglas Bates]

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.