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 analysis. 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).

Why the name marginal model?

In a typical multilevel model, there are level 1 & 2 residuals (R and U variables). The two variables form a joint distribution for the response variable ($Y_{ij}$). In a marginal model, we collapse over the level 1 & 2 residuals and thus marginalize (see also conditional probability) the joint distribution into an univariate normal distribution. In hierarchical linear modeling, we then fit the marginal model to data.

For example, for the following hierarchical model,

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

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

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.