[R-sig-ME] Results from vcov() in lme4.

[Rolf Turner]

I am rather puzzled by the results of applying vcov() to (binomial) 
models fitted by glmer().  The linear predictor in the models is of the form

      alpha_i + beta_i * x + <random effects>

where "i" corresponds to "treatment group" and x is numeric.  There are 
six treatment groups so the covariance matrix returned by vcov() is
12 x 12.

When I fit the model with nAGQ = 0 in the call to glmer() the resulting 
matrix is quite sparse --- there is non-zero covariance only between 
parameter estimates corresponding to the same value of "i" (i.e. to the 
same treatment group).  In other words, under the appropriate ordering 
of the estimated parameters, the covariance matrix is block diagonal, 
with six 2 x 2 blocks.

When I fit the model with nAGQ set equal to 1, the resulting covariance 
matrix is non-sparse; all entries are non-zero.  (The entries outside of 
the "block diagonal structure" are, I suppose, "relatively" small --- 
they range from -0.0145 to 0.0010 --- but are well away from being 
"approximately zero".)

I would like a better understanding of the reason for the difference.  I 
was under the impression that setting nAGQ = 0 gives a somewhat 
quick-and-dirty fit of the model; less accurate and reliable than with
nAGQ = 1.  Why does setting nAGQ = 0 (apparently) cause the covariance 
between parameter estimates corresponding to different treatment groups 
to be *exactly* zero?

If I remember my childhood teaching correctly, in a *linear* model, such 
covariances would indeed be exactly zero, so one might expect 
"approximately" zero in the generalised linear model setting.  But why 
should setting nAGQ = 0 result in a covariance matrix which is "just 
like" one from the linear model setting?

I guess it doesn't really matter a damn, but I'd like to understand what 
is going on, at least "in rough intuitive terms".

Can anyone enlighten me?

Thanks.

cheers,

Rolf Turner

-- 
Honorary Research Fellow
Department of Statistics
University of Auckland
Phone: +64-9-373-7599 ext. 88276