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 = β_0j + R_ij, the residual is R_ij, and var(R_ij) = σ²

level 2: β_0j = γ₀₀ + 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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