Equation for model with only level 2 covariate (studying ICC)

Question

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 = μ + a𝑖 + b𝑖𝑗

is my unrestricted model where 𝑦𝑖𝑗 is is the outcome for group member 𝑗 from group 𝑖, 𝜇 is the population mean, 𝑎𝑖 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?

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,

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

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