Hierarchical mixed-effects models notation I have a hierarchical mixed-effects model where observations i, from group S_2 (representing municipalities), are nested inside group S_1 (representing states), which in turn are nested inside N (representing countries). Here is the notation I used to describe this model:

I based this notation on some references I found but I am not sure if I am representing the hierarchical structure (in red) the right way.
I am also calling δ_(NS_1 S_2 ) and θ_(NS_1 S_2 ) as "design matrices" but is this the right term to use?
Any references or feedback are appreciated. Thank you in advance!

Admittedly going two or more levels down, can lead to an indexing spaghetti but I think your notation is fine. The only obvious issue for me is that you $i$ at the front where it should be something like $N,S1,S2,i$. I would suggests looking at some papers on "Four-level hierarchical models" to get some further ideas on how to properly notate things. (e.g. Latent Variable Regression Four-Level Hierarchical Model Using Multisite Multiple-Cohort
Longitudinal Data (2018) by Choi and Kim) Finally a standard way to refer to the matrices denoting the grouping hierarchy is "random-effects design matrix" - usually denoted with $Z$ (to distinguish it from the "fixed-effects design matrix" - usually denoted with $X$).
