# Centering and assumptions of 3-level multilevel model

I am struggling with formulating a multilevel model for my analysis and would really appreciate if you could give me some insight. This is a 3-level model where the response (Y), exposure (X), and other covariates I will control for, are all numerical. Y is an individual (level-1) variable, I have no level-1 variable to control for, X is a level-2 variable and the rest are level-3 variables. I am only interested in seeing the fixed and random effects of X, for all the other covariates I will only include the fixed effects. My questions are:

1. Do I need to center X and other covariates? I am reading this book and some other bunch of literature and all emphasize centering level-1 variables for correct interpretability of the intercept term. Since none of my variables are level-1 and I won’t consider any interaction term, am I really required to center my variables? I have read centering at the grand mean is a good practice, though.
2. How do I test the assumptions? ‘This book’ explains the assumptions to be tested for a level-2 model. Does the same logic apply for the level-3 model? How much can I be affected if my assumptions are violated?
3. Finally, I am having a hard time to write the syntax for my analysis. I will be using SAS for the analysis. I have written my code as:

proc mixed covtest noclprint;
class level-2 level-3;
model Y= X A B C/solution ddfm=bw;
random intercept X/sub=level-3;
random intercept X/sub=level-2 (level-3);
run;


Specifically, I am not sure how does these two random effects of X interpret.