# Using random intercepts in a multilevel model as dependent variables in a linear model

I have a mixed model with 3 levels: individual, city, and state, and so I get random intercepts for both cities and states. I understand that since cities are nested in their state, their intercepts are comparable only within that state. Now I want to estimate the city-level factors such as GDP per capita and racial profile against these city intercepts as dependent variables. There are two ways I can think of doing:

First, dependent variables are city intercepts, and independent variables are city-level factors subtracted by the state means. This model would estimate the city-level impacts with state-level impacts being controlled.

Second, dependent variables are city intercepts plus state intercepts, and independent variables are city-level factors. This model would estimate the city-level impacts combined with state-level impacts.

Are these models reasonable? If so, how can I weight these models? This is particularly interesting for the second model. I understand that within a state, city intercepts have 0 covariance. But what if I add the city intercepts with state intercepts? What would be the var/covar matrix like after this addition?

UPDATE: I also understand I could've thrown the city intercepts right into the multilevel models, but since I have 4 city-level factors, these models never converged by using clmm() function in the ordinal package in R. BTW, since all of my individual-level fixed effects are ordinal/nominal, using the lme4 would sacrifice a lot of information. That's why I chose to have a 2-step estimation.

• Are you using the same fixed effects that are included to estimate the random intercepts in the mixed model as regressors in a model for the outcome? – AdamO May 6 at 17:18
• @AdamO No. The second model has a different set of regressors. For example, the mixed model has individual's income and gender from a survey as fixed effects, while the second model would have GDP per capita and percent of non-white from census as regressors. – blueprint May 6 at 18:15