lmer vs glmmPQL

Fabian Scheipl wrote:

On Tue, Jun 30, 2009 at 9:16 AM, Ken Beath<ken at kjbeath.com.au> wrote:

It appears that PQL with moderate random effect variance  
introduces a small
bias in a direction that reduces the MSE, at least in the  
simulations
chosen. For large variances the bias is probably excessive and the  
MSE will
increase using PQL.

 Hmmm.  How can bias, in any direction, reduce MSE?  (I can see that
there could be a tradeoff between bias and variance, but MSE
incorporates bias^2, right?  How about bias-corrected variants of
PQL (a la Raudenbush et al) -- mights those provide the best
of both worlds, or does the additional complexity inevitably
increase variance -> MSE?  (I don't know if those bias-corrected
variants are implemented anywhere other than MLWiN/HLM ... ?)

Results from simulations with sd(RandomIntercept)=3 instead of 1
(results attached) confirm your remark - with the possible exception
of very small data sets the performance (in rmse & bias) for Laplace
and AGQ is much much better than PQL.
I'm sorry for getting Ben Bolker and others all riled up with my  
earlier post.

 On the contrary, I think this is fascinating and worthwhile.
It amazes me that we still don't know these very basic things.

One more thing to consider though:
A random intercept variance of 1 in a logistic model means that the
medium  50% of subjects/groups are expected to have between about  
half
and about double the odds of a subject/group with random intercept=0,
which is already fairly large effect in my book.
##

qlnorm(c(.1, .25, .75, .9))

[1] 0.28 0.51 1.96 3.60
##

For a random intercept sd of 3, the multiplicative effect on the
baseline odds for the middle 50% is between  0.13 and  7.6,
##

qlnorm(c(.1, .25,  .75, .9), sdlog = 3)

[1]  0.021  0.132  7.565 46.743
##
which means really large inter-group/subject heterogeneity and might
not be encountered that frequently in real data (?) (or at least
suggest a mis-specified model that misses important
subject/group-level predictors...).

(Similar remarks concerning "effect size" of the random effect apply
to Poisson regression with log-link.)

So, what's the lesson --
Should we still prefer PQL if we expect to see small to intermediate
inter-group/subject heterogeneity?

Fabian

 good question.

-- 
Ben Bolker
Associate professor, Biology Dep't, Univ. of Florida
bolker at ufl.edu / www.zoology.ufl.edu/bolker
GPG key: www.zoology.ufl.edu/bolker/benbolker-publickey.asc