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