Marginal model

What is a marginal model?

Marginal model is one that is used to provide regression estimates, especially in multilevel models or hierarchical linear models (Heagerty & Zeger, 2000).

How does a marginal model work?

In a marginal model, we collapse over the level 1 & 2 random residuals (R and U variables) and thus marginalize the joint distribution of the response variable ($Y_{ij}$) into an univariate distribution. In hierarchical linear modeling, we fit the marginal model to data.

For example, for the following hierarchical model,

$Y_{ij}=\beta _{0j}+R_{ij}=\gamma _{00}+U_{0j}+R_{ij}$

The level 1 random residual: $R_{ij}$

The level 2 random residual: $U_{0j}$

The marginal model is,

$Y_{ij}\sim N(\gamma _{00},(\tau _{0}^{2}+\sigma ^{2}))$

Reference

Heagerty, P. J., & Zeger, S. L. (2000). Marginalized multilevel models and likelihood inference. Statistical Science, 15(1), 1-26.