Solved – What do “marginal” and “conditional” mean in “marginal models” and “conditional models”

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

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