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 = ΞΌ + a𝑖 + b𝑖𝑗

is my unrestricted model where 𝑦𝑖𝑗 is is the outcome for group member 𝑗 from group 𝑖, πœ‡ is the population mean, π‘Žπ‘– 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?


1 Answer 1


I am not familiar with the notation you used (and I am not sure it is common or correct, besides usually subscript $i$ refers to individual and $j$ to the group). But I think it won't matter here. You said you already estimated unconstrained model. So we can write the system of equations for this model

$$\begin{align} \text{Level 1: } Y_{ij} & = \beta_{0j} + r_{ij} \\\ \text{Level 2: } \beta_{0j} & = \gamma_{00} + u_{0j} \end{align}$$

If we substitute level-2 equation into level-1:

$$Y_{ij} = \gamma_{00} + u_{0j} + r_{ij}$$

Moving from this unconstrained model to a model with only a level-2 predictor is straightforward:

$$\begin{align} \text{Level 1: } Y_{ij} & = \beta_{0j} + r_{ij} \\\ \text{Level 2: } \beta_{0j} & = \gamma_{00} + \gamma_{01}W_j + u_{0j} \end{align}$$

So, level-1 equation is same, and we add only $W_j$ as a level-2 predictor:

$$Y_{ij} = \gamma_{00} + \gamma_{01}W_j + u_{0j} + r_{ij}$$

This is a variant of random-intercept model where the intercept ($\beta_{0j}$ in level-1 equation) is a function of level-2 predictor and variability. In your case, there is an additional level-1 predictor ($X_{ij}$). Assuming that its effect is constrained to be invariant across level-2 units:

$$\begin{align} \text{Level 1: } Y_{ij} & = \beta_{0j} + \beta_{1j}X_{ij} + r_{ij} \\\ \text{Level 2: } \beta_{0j} & = \gamma_{00} + \gamma_{01}W_j + u_{0j} \\\ \beta_{1j} & = \gamma_{10} \end{align}$$

The model is,

$$Y_{ij} = \gamma_{00} + \gamma_{01}W_j + \gamma_{10}X_{ij} + u_{0j} + r_{ij} $$

Luke, D. A. (2004). Multilevel Modeling. London: Sage Publications.


Your Answer

By clicking β€œPost Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.