[R-sig-ME] Prediction of random effects in glmer()

[Ravi Varadhan]

Ben,
Thanks for explaining this.  It is quite obvious where the ||u||^2 comes from (after you pointed it out!).  I was looking at the paper by Booth and Hobert (JASA 1998) on computing the standard errors of predicted random effects.  Their Eqs. (6) and (8) are what I meant by conditional mean (E[u | y; \theta]) and conditional variance (Var[u | y; \theta]).
Best,
Ravi
________________________________
From: Ravi Varadhan <ravi.varadhan using jhu.edu>
Sent: Monday, February 15, 2021 3:24 PM
To: r-sig-mixed-models using r-project.org <r-sig-mixed-models using r-project.org>
Cc: bolker using mcmaster.ca <bolker using mcmaster.ca>
Subject: Re: Prediction of random effects in glmer()

Dear Ben,
Thanks for your response. I went back and looked at the draft JSS paper you sent me.  It does describe how the random effects are predicted as conditional modes, using a penalized, iteratively weighted least squares. However, I still have some questions.  Why is the penalty term ||u||^2 added? What does this mean?  Does glmer then provide standard errors for the predicted random effects (I don't think it does)?

One more question: it would be nice to also have an option for conditional mean and conditional variance of the random effect, although conditional variance would underestimate the true variance of the prediction.

Thank you,
Ravi
________________________________
From: Ravi Varadhan
Sent: Thursday, February 11, 2021 8:30 PM
To: r-sig-mixed-models using r-project.org <r-sig-mixed-models using r-project.org>
Subject: Prediction of random effects in glmer()

Hi,
I would like to know how the prediction of random effects is done in the GLMM modeling using the lme4::glmer function, i.e. how the BLUP-like predictions are made in the glmer() function?

Does it use frequentist prediction or empirical Bayes or full Bayes posterior?  Is there any documentation of the prediction methodology?

Thanks in advance.

Ravi

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