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Marginal model

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Marginal models l


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.

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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 i j ). 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 i j = β 0 j + R i j , the residual is R i j , and v a r ( R i j ) = σ 2 level 2: β 0 j = γ 00 + U 0 j , the residual is U 0 j , and v a r ( U 0 j ) = τ 0 2

Thus, the marginal model is,

Y i j ∼ N ( γ 00 , ( τ 0 2 + σ 2 ) )

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

References

Marginal model Wikipedia
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