Marginal model

In statistics, marginal models (Heagerty & Zeger, 2000) are a technique for obtaining regression estimates in multilevel modeling, also called hierarchical linear models. People often want to know the effect of a predictor/explanatory variable X, on a response variable Y. One way to get an estimate for such effects is through regression analysis.

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 a univariate normal distribution. 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 $$\operatorname{var}(R_{ij}) = \sigma^2$$


 * level 2: $$\beta_{0j} = \gamma_{00} + U_{0j}$$, the residual is $$U_{0j}$$, and $$\operatorname{var}(U_{0j}) = \tau_0^2$$

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.