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Randel
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This question is not easy to answer, and has been partially discussed at this site as

All terms are generally related to longitudinal/ panel data and repeated measures (in the format of advanced regression and ANOVA), but have multiple meanings in different context. I would like to answer it in formulas based on my knowledge.

Fixed-effects model

  • In biostatistics, fixed-effects usually comes together with random effects, denoted as $\boldsymbol\beta$ in the second formula. But the fixed-effects model is also defined to assume that the observations are independent, like cross-sectional setting, as in Longitudinal Data Analysis of Hedeker and Gibbons (2006).
  • In econometrics, the fixed-effects model is as $$ y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+u_i+\epsilon_{ij}$$ where $u_i$ is fixed (not random) intercept for each subject ($i$), we can also have a fixed-effect as $u_j$.
  • In meta-analysis, the fixed-effect model assumes underlying effect is the same across all studies (e.g. Mantel and Haenszel, 1959).

Random-effects model

  • In biostatistics, the random-effects model (Laird and Ware, 1982) is as $$ y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\boldsymbol z_{ij}^{'}\boldsymbol u_i+\epsilon_{ij}$$ where $\boldsymbol u_i$ are random variables.
  • In econometrics, the random-effects model may only refer to random intercept model as in biostatistics, i.e. $\boldsymbol z_{ij}^{'}=1$ and $\boldsymbol u_i$ is a scalar.
  • In meta-analysis, the random-effect model assumes heterogeneous effects across studies (DerSimonian and Laird, 1986).

Marginal model

Marginal model is generally compared to conditional model (random-effects model). It focuses on the population mean $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ but not the conditional mean $E(y_{ij}|\boldsymbol u_i)=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$. The interpretation and scale of the regression coefficients between marginal model and random-effects model is different for nonlinear models (e.g. logisitic regression). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

Randel
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