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 (${\displaystyle Y_{ij}}$). In a marginal model, we collapse over the level 1 & 2 residuals and thus marginalize (see also conditional probability) the joint distribution into a univariate normal distribution. We then fit the marginal model to data.

For example, for the following hierarchical model,

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

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

Thus, the marginal model is,

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

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

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