[R-sig-ME] lmer vs glmmPQL

[Ben Bolker]

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