[R-sig-ME] AIC and BIC with ML and REML

[Dimitris Rizopoulos]

Gorjanc Gregor wrote:
>> In a way this question relates to the earlier discussion on whether to
>> regard the random effects as parameters or as unobserved random
>> variables.  The difference between -2 * l + 2 * p and -2 * l + 2 * (p
>> + q) can be considered to be a question of how many parameters there
>> are in the model.  In particular, do the random effects count as
>> parameters?  I had an interesting discussion with Georges Monette
>> about this a few days ago and both of us feel the saying the random
>> effects don't affect the parameter count is underestimating the
>> complexity of the model but saying they should add q to the number of
>> parameters is overestimating the complexity.
> 
> Thanks for this comment. I will take this as a yes to: "Are there are several
> definitions of AIC and BIC lurking around?".
> 
> DIC "solves" the issue of effective number of parameters to some extent
> though it also has its own problems
> 
> http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/dicpage.shtml

there is also the conditional AIC of Vaida and Blanchard (Biometrika, 
2005, 351--370) that calculates the effective degrees of freedom for the 
random effects. Both these authors and the DIC authors distinguish 
between the focus of inference (or better the focus of prediction) in 
mixed models, i.e., either the marginal log-likelihood or the 
conditional on the random effects log-likelihood.

Best,
Dimitris


>> I don't know a good answer to the question of how to "count" the
>> number of parameters in a mixed model (but I also don't feel that AIC
>> or BIC should be taken too seriously - these quantities are, at best,
>> a guide for model comparisons).
> 
> Sure.
> 
> gg
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-- 
Dimitris Rizopoulos
Assistant Professor
Department of Biostatistics
Erasmus University Medical Center

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