Polynomial regression with multilevel data

Question

I want to estimate a polynomial regression to test the effect of self and follower-perception of leadership behavior on some outcome (e.g., followers' job satisfaction). Hence, I have a multilevel data structure with followers' assessments of their supervisors' leadership ($X_1$) as well as followers' job satisfaction ($Y$) on level 1 and supervisors' self-assessment of their leadership behavior ($X_2$) on level 2.

Ignoring the multilevel structure of the data, I would fit the following model:

$$ Job Satisfaction = b_0 + b_1SelfRating + b_2FollowerRating + b_3SelfRaring^2 + b_4SelfRating \times FollowerRating + b_5FollowerRating + e $$

My question: Can I fit a similar model as a hierarchical linear model? And if yes, are there any things, like mandatory random effects, I have to consider?

In my mind the final model would look (something) like this:

$$ Z_{ij} = \gamma_{00_j} + \gamma_{01_j}Y_j + \gamma_{02_j}Y_j^2 + \gamma_{10_j} X_{ij} + \gamma_{11_j}X_{ij}Y_j + \gamma_{20_j}X_{ij}^2 + u_{0_j}+ u_{1_j} + u_{2_j} + e_{ij} $$

Thanks for your help!

Answer 1

I found an article by Nestler, Humberg and Schönbrodt forthcoming in Psychological Methods titled "Response surface analysis with multilevel data - Illustration for the case of congruence hypotheses". it is not yet published but you can find the preprint here.

For a case with two level-1 independent variables, they argue for including fixed effects for group-mean centered first-order terms, group-mean centered higher-order terms, and the group means. They also include all possible random effects.

For a case with one level-1 and one level-2 independet variable they advise "to group-mean center the Level 1 congruence variable and including the group-mean-centered Level 1 congruence variable, the Level 2 congruence variable, and the resulting group means in the model to disentangle the Level 1 and Level 2 effects in the response surface analysis."