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| In [[statistics]], '''marginal models''' (Heagerty & Zeger, 2000) are a technique for obtaining regression estimates in [[multilevel model]]ing, also called [[hierarchical linear models]].
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| 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?==
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| 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 (<math>Y_{ij}</math>). 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.
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| For example, for the following hierarchical model,
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| :level 1: <math>Y_{ij} = \beta_{0j} + R_{ij}</math>, the residual is <math>R_{ij}</math>, and <math>var(R_{ij}) = \sigma^2</math>
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| :level 2: <math>\beta_{0j} = \gamma_{00} + U_{0j}</math>, the residual is <math>U_{0j}</math>, and <math>var(U_{0j}) = \tau_0^2</math>
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| Thus, the marginal model is,
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| :<math>Y_{ij} \sim N(\gamma_{00},(\tau_0^2+\sigma^2))</math> | |
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| This model is what is used to fit to data in order to get regression estimates.
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| ==References==
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| Heagerty, P. J., & Zeger, S. L. (2000). Marginalized multilevel models and likelihood inference. ''Statistical Science, 15(1)'', 1-26.
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| [[Category:Statistical models]] | |
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| {{stats-stub}}
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