Marginal model

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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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