Cross-sectional interaction in multi-level model

[Dudi Balson]

Hello,

I am trying to understand whether my code does what I think it does, and whether the analysis makes sense.

Originally, I am running a 5 level model:

mixed y || x1:|| x2: || x3: || x4:

Now, I would like to perform the following analysis - how does y changes with x4 interacted with some variable z. E.g. Say y is student grade, x1 is county, x2 city, x3 is school, x4 is principal. Z is city budget. I would like to interact school budget with principal, to understand whether principal affect student grade, and whether this effect changes with the size of the city's budget (higher budget can mean they can hire better principals, spend more funds on students, etc.) Then I guess I have 2 questions. 1) Can mixed models provide me with such analysis? 2) does the following model gives me the answer.

mixed y budget|| x1:|| x2: || x3: || x4: budget

Thank you in advance, the work you do here is such a blessing! Dudi

[Tom Scott]

Dudi, the principal coefficient would give you the "relationship" (not "effect" because I am assuming you don't have a research design that allows for causal inference) between the principal and student grades. A cross-level interaction between budget and principal would tell you if the relationship between principal and student grades depends on the city budget. I think you should read an article in the Journal of Applied Psychology: Understanding and estimating the power to detect cross-level interaction effects in multilevel modeling (Mathie et al., 2012)

[Dudi Balson]

Thank you Tom for your answer - I will check that article.
Does my second model describe a cross-sectional interaction between budget and principal in the right way in STATA?
Also, "if the relationship between principal and student grades depends on the city budget" - does it only show that there exist relationship (in the variance part of the table)/ or it also shows the direction of the effect (increases or decreases with the budget)? Since I am not sure the budget coefficient actually says that...

Thank you!