statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, marginal models (Heagerty & Zeger, 2000) are a technique for obtaining regression estimates in

multilevel model Multilevel models (also known as hierarchical linear models, linear mixed-effect model, mixed models, nested data models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parame ...

ing, also called

hierarchical linear models Multilevel models (also known as hierarchical linear models, linear mixed-effect model, mixed models, nested data models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of param ...

. 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 In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...

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 Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...

for the response variable (

Y_

). In a marginal model, we collapse over the level 1 & 2 residuals and thus ''marginalize'' (see also

conditional probability In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occur ...

) the joint distribution into a univariate

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

. We then fit the marginal model to data. For example, for the following hierarchical model, :level 1:

Y_ = \beta_ + R_

, the residual is

R_

, and

\operatorname(R_) = \sigma^2

:level 2:

\beta_ = \gamma_ + U_

, the residual is

U_

, and

\operatorname(U_) = \tau_0^2

Thus, the marginal model is, :

Y_ \sim N(\gamma_,(\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. Regression models {{stats-stub