What do "marginal" and "conditional" mean in "marginal models" and "conditional models"?

Question

What do "marginal" and "conditional" mean in "marginal models" and "conditional models"?

Are they related to marginal distributions and conditional distributions?

Thanks!

Answer 1

In longitudinal data analysis, marginal models refers to population average models, e.g., generalized estimating equations (GEE) models; conditional models refers to subject specific models, e.g., mixed-effects models. The two models address different questions.

• Let's start from a linear mixed-effects model, $$ y_{ij}=\mathbf{x}_{ij}^{'}\boldsymbol{\beta}+\mathbf {z}_{ij}^{'}\mathbf{u}_i+\epsilon_{ij}.$$

  The mean of outcome conditional on the random effects $\mathbf{u}_i$ is $$\mu_{ij}^c=E(y_{ij}|\mathbf u_i)=\mathbf x_{ij}^{'}\boldsymbol\beta + \mathbf z_{ij}^{'}\mathbf u_i,$$ and the marginal mean of outcome (average over the distribution of random effects) is $$\mu_{ij}^m=E(y_{ij})=E(E(y_{ij}|\mathbf u_i))=\mathbf x_{ij}^{'}\boldsymbol\beta,$$ since we assume $\mathbf u_i$ has mean 0. The $\boldsymbol\beta$ coincides in marginal and conditional models.

• However, for nonlinear models, $\boldsymbol\beta$ in the two models would differ in both interpretation (population average vs. subject specific) and scale of coefficients, $$E(\mu_{ij}^c)=E(h^{-1}(\mathbf x_{ij}^{'}\boldsymbol\beta^c + \mathbf z_{ij}^{'}\mathbf u_i))\neq h^{-1}(\mathbf x_{ij}^{'}\boldsymbol\beta^m)=\mu_{ij}^m,$$ where $h$ is the link function, e.g., logit or probit link for binary data, log link for count data.