Marginal model

What is a marginal model?

People often want to know the effect of X, the predictor/explanatory variable, on Y, the response variable. One way to get an estimate for such effects in through regression. And Marginal model is one technique that is employed to obtain the regression estimates in the field of multilevel modeling, a.k.a. 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,

level 1: $Y_{ij}=\beta _{0j}+R_{ij},var(R_{ij})=\sigma ^{2}$

level 2: $\beta _{0j}=\gamma _{00}+U_{0j},var(U_{0j})=\tau _{0}^{2}$

Thus, the marginal model is,

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

This model is what is used to fit to data in order to get regression estimates.

Reference

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