[R-sig-ME] lmer vs glmmPQL

[Ken Beath]

On 27/06/2009, at 4:21 AM, Ben Bolker wrote:

>  That's really interesting and kind of scary.
>  Do you have any thoughts on why this should be so?
>  I know of a few simulation studies (Browne and Draper, Breslow) that
> test PQL and generally find reasonably "significant" bias for binary
> data with large random variance components.  I guess I had simply
> assumed that Laplace/AG(H)Q would be better.  (There are also some
> theoretical demonstrations (Jiang?) that PQL is asymptotically
> inconsistent, I think ...)
>

MLE for nonlinear models are biased, so it is not unexpected that  
Laplace etc will be biased.

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.

Ken




>  * Are you working in a different regime from previous studies
> (smaller data sets, or some other point)?
>  * Does considering RMSE rather than bias give a qualitatively
> different conclusion (i.e., PQL is biased but has lower variance)?
>  * ?
>
>  Since in a recent paper I recommended Laplace/AGHQ out of principle,
> and Wald tests out of pragmatism, and thought the former  
> recommendation
> was reliable but the latter was not, it's interesting to be having
> my world turned upside down ...
>
>  Would welcome opinions & pointers to other studies ...
>
>  Ben Bolker
>
>
> @article{browne_comparison_2006,
> 	title = {A comparison of Bayesian and likelihood-based methods for
> fitting multilevel models},
> 	volume = {1},
> 	url = {http://ba.stat.cmu.edu/journal/2006/vol01/issue03/ 
> draper2.pdf},
> 	number = {3},
> 	journal = {Bayesian Analysis},
> 	author = {William J. Browne and David Draper},
> 	year = {2006},
> 	pages = {473--514}
> }
>
> @incollection{breslow_whither_2004,
> 	title = {Whither {PQL?}},
> 	isbn = {0387208623},
> 	booktitle = {Proceedings of the second Seattle symposium in
> biostatistics: Analysis of correlated data},
> 	publisher = {Springer},
> 	author = {N. E. Breslow},
> 	editor = {Danyu Y. Lin and P. J. Heagerty},
> 	year = {2004},
> 	pages = {1–22}
> }
> Fabian Scheipl wrote:
>> Ben Bolker said:
>>> My take would be to pick lmer over glmmPQL every time, provided
>>> it can handle your problem -- in general it should be more accurate.
>>
>> That's what I wanted to demonstrate to my students last week, so I  
>> did
>> a small simulation study with a logit-model with random intercepts:
>>
>> logit(P(y_ij=1)) =  x_ij + b_i;
>> b_i ~N(0,1);
>> x_ij ~U[-1,1];
>> i=1,..,m;
>> j=1,...,n_i
>>
>> The pdfs with the results are attached (m subjects, ni obs/subject,
>> RPQL is PQL with iterated REML fits on the working observations
>> instead of ML, nAGQ=11 for AGQ).
>> The results surprised me :
>> - For the estimated standard deviation of the random intercepts, PQL
>> actually has (much) lower rmse for small and medium-sized data sets
>> and bias is about the same for LA, AGQ and PQL for small datasets.
>> - There were no relevant differences in rmse or bias for the  
>> estimates
>> of the fixed effects.
>>
>> Differences for poisson data should be even smaller, since their
>> likelihood is more normal-ish.
>> glmer may still be preferrable since its much faster and more stable
>> than glmmPQL, but accuracy for smaller datasets may be better for  
>> PQL.
>>
>> Best,
>> Fabian
>>
>
>
> -- 
> 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
>
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