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

[Ben Bolker]

   This follows an earlier private conversation that didn't quite get 
resolved.

   I'm interpreting "how the BLUP-like predictions are made" as "how do 
you estimate the conditional modes"?  "Conditional modes" is what we 
call the predicted deviations of each group's effects (intercept, slope, 
whatever) from the population mean (i.e. the fixed-effect estimate for 
that thing).  For classic LMMs, conditional modes==BLUPs, but not 
otherwise: see Doug Bates's comments here

https://stat.ethz.ch/pipermail/r-sig-mixed-models/2011q2/016047.html

   There is a long-neglected GLMM manuscript that builds on the 
Bates/Maechler/Bolker/Walker JSS paper which I should clean up at least 
enough to be able to post it publicly.  In the meantime, what it says is 
that the conditional modes are determined using a penalized iteratively 
reweighted least squares algorithm.

1. Given parameter values, β and θ, and starting estimates, u0 , 
evaluate the linear predictor, η, the corresponding conditional mean,
μ_{Y|U} =u , and the conditional variance. Establish the weights as the 
inverse of the variance. We write these weights in the form
of a diagonal weight matrix, W , although they are stored and 
manipulated as a vector.


2. Solve the penalized, weighted, nonlinear least squares problem

    (arg min of [L2 norm of weighted residual vector] + [L2 norm of 
conditional modes])

3. Update the weights, W , and check for convergence. If not converged, 
go to step 2.

  Gauss-Newton, blah blah blah blah ...

   I'm happy to send the draft to anyone who asks for it.

  HOWEVER, when I sent Ravi the draft it seemed as though it didn't 
answer the question. So Ravi, maybe you could clarify?

   I would classify this as "empirical Bayes" since the θ parameters 
(the vector of elements of the Cholesky factors that define the 
covariance matrices of the random effects) are determined from data 
without an explicit prior.


On 2/12/21 5:43 PM, Juho Kristian Ruohonen wrote:
> Following. I'd like to know this as well.
> 
> J
> 
> pe 12. helmik. 2021 klo 3.37 Ravi Varadhan (ravi.varadhan using jhu.edu)
> kirjoitti:
> 
>> 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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