[R-sig-ME] Correlation of Fixed Effects

[Douglas Bates]

On Mon, Feb 9, 2009 at 3:52 PM, Gorjanc Gregor
<Gregor.Gorjanc at bfro.uni-lj.si> wrote:

> The default print method outputs also Correlation of Fixed Effects.
> How is this computed and what does it actually represent?

It is an approximate correlation of the estimator of the fixed
effects.  (I include the word "approximate" because I should but in
this case the approximation is very good.)  I'm not sure how to
explain it better than that.  Suppose that you took an MCMC sample
from the parameters in the model, then you would expect the sample of
the fixed-effects parameters to display a correlation structure like
this matrix.

As for how it is calculated, look in the vignettes for the definition
of a p by p upper triangular matrix called R_X.  It is returned as the
RX slot in the fitted model.  This matrix is part of the Cholesky
factor in the combined model matrices for the penalized least squares
problem that determines the conditional modes of the random effects
and the conditional estimates of the fixed effects.  If we didn't have
any random effects this would be the R matrix from the QR
decomposition of X.  The same calculation that creates the correlation
of the coefficients in a fixed-effects model from R creates the
correlation of the fixed-effects coefficients from RX here.  See

?chol2inv

> I have two models
> that essentially give me the same message, but in one model the correlations
> between covariates are really high 0.9 and higher, while in other model use of
> poly(), reduced correlations a lot! Should I care?
>
> Thanks!
>
> Lep pozdrav / With regards,
>    Gregor Gorjanc
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