Marginal model explained

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 ^{2} _{0}

Thus, the marginal model is,

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

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

References

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