[R-sig-ME] How does blme carry out estimation and inference.

[Kotecha, Gopal]

Dear all,

I have two questions about blme.  Apologies if any of this has been 
asked before.

1. How does estimation occur?
2. How does inference occur for fixed effects parameters? Specifically 
when you use the “summary” method on a bglmer object, how are standard 
errors estimated?

I have written up some more details below, and also pasted a LaTeX image 
here for easier reading:  https://imgur.com/a/syL1LWx

I am using the default settings and the following logistic model:

P(Y_{ij}=1|\beta, X_{ij},T_{ij},\theta_j) = F(X^T_{ij}  \beta + \theta_j)
individuals i, groups j, where scalar \theta_j is distributed 
Normal(0,\sigma_\theta), outcome Y_{ij} is binary, X_{ij} is a vector of 
binary covariates with corresponding fixed effect parameter vector \beta.
F(t) =(1+\exp(-t))^{-1}.

My understanding is the following:

  - BLME uses a penalized log likelihood approach, letting us place a 
prior on $\sigma_\theta$ to stabilize the inference and prevent zero 
values.  We can also place a prior on $\beta$

- Inference is first carried out on the marginal log posterior density
l(\sigma_\theta|Y_{ij}) where \beta and \theta_j are both integrated out 
to find the MAP estimate \hat\sigma_\theta.  This is then substituted 
into the original likelihood l(\hat\sigma_\theta,\theta|Y_{ij}) which is 
maximized to get \hat\beta. This is as far as I can tell from Chung 
et.a. 2013.  Is this correct?

- How does blme estimate the standard errors of fixed effect estimates 
\hat\beta (i.e. what does summary on a model object produce)?  I can't 
find anything relating to this in the documentation.  Am I correct in 
thinking the marginal likelihood with the MAP estimate of 
\hat\sigma_\theta plugged in is profiled (i.e. a profile likelihood 
approach on l(\theta,\hat\sigma_\theta|Y_{ij}))?

Any pointers, tips or references I have missed would be super useful!

Thanks in advance!
Gopal

Refs:

Chung, Yeojin, et al. "A nondegenerate penalized likelihood estimator 
for variance parameters in multilevel models." Psychometrika 78 (2013): 
685-709.
Bates, D., Mächler, M., Bolker, B., $\&$ Walker, S. (2015). Fitting 
Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 
67(1), 1–48. https://doi.org/10.18637/jss.v067.i01