What are some good examples of working through a multilevel model by hand? I've been learning about multilevel models lately, and I understand the concept of shrinkage and partial pooling (I think), but I'm still confused to some extent on how partial pooling actually happens.  What I've found about my own learning is that I tend to really only 'get' something if I've worked through the mechanics of it.  In light of that, my question is:
What are some good examples of working through a simple multilevel model (with partial pooling) by hand?  I'm thinking of web pages, but books and papers would also be great.
What I'm thinking of is an example like: 2 units, each with 2 subunits, or: 3 units, each with 3 subunits.  Those are my guess regarding what would make good examples, but I could be mistaken, so please let me know if there's something better.
I'm looking for examples with numbers rather than variables, both because it helps me to understand things better, and because I'd like to be able to check my work by running the model in lmer() and comparing numerical results.
The things I'd like to learn are:


*

*How do the subunits pool?

*How do the units pool?

*How do the subunits interact with the units in the pooling?


I'd like to be able to write down all the relevant equations and work through the algebra.  For the actual adding/subtracting/multiplying/dividing I'll use a calculator or Excel or R, but I'd like to work through the equations by hand.
I'll take as many suggestions as possible!
 A: This link will at least get you started with some model equations for linear mixed effects models as well as linear models which assume no pooling or complete pooling: http://www.utstat.toronto.edu/~guerzhoy/303/lec/lec9/multi.pdf. 
This link is a bit more explicit in terms of how partial pooling works in a linear mixed effects model: http://www.win-vector.com/blog/2017/09/partial-pooling-for-lower-variance-variable-encoding/#more-5245. 
This link will give you visual insights into partial pooling compared to no pooling and complete pooling: https://www.r-bloggers.com/plotting-partial-pooling-in-mixed-effects-models/. 
These notes (section 20.7) also talk about pooling in a linear mixed effects model context and explain how pooling depends on the size of the group: https://jrnold.github.io/bayesian_notes/multilevel-models.html. 
These slides talk about the degree of pooling and how to quantify it: https://web.as.uky.edu/statistics/users/pbreheny/701/S13/notes/4-18.pdf. 
For pooling in a generalized linear mixed effects model, check out this link: https://cran.r-project.org/web/packages/rstanarm/vignettes/pooling.html.
