Smooth coarea formula

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 $var(R_{ij})=\sigma ^{2}$

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.

References

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

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