I posted this question in response to an earlier question asked (Is ICC in random-intercept models restricted to the null-model?). I then realized that it is not recommend to write follow-up questions.
I am trying to write the equation of a model with only level 2 covariate. How would you write it? I am looking at the ICC in an unrestricted model and then want to see how the ICC decrease when I control for covariates that all members of a group share.
𝑦𝑖𝑗 = μ + a𝑖 + b𝑖𝑗
is my unrestricted model where 𝑦𝑖𝑗 is is the outcome for group member 𝑗 from group 𝑖, 𝜇 is the population mean, 𝑎𝑖 is a group-specific factor shared by all members from group 𝑖, and 𝑏𝑖𝑗 is an individual-specific factor unique to individual 𝑗 from group 𝑖.
If I want to write an equation that shows that I added a level 2 variable (X) in the model (which has the same value for all member of a group), what should it look like?