6 replaced http://stats.stackexchange.com/ with https://stats.stackexchange.com/

This question has been partially discussed at this site as below, and opinions seem mixed.

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

## Fixed-effects model

• In biostatistics, fixed-effects, denoted as $$\color{red}{\boldsymbol\beta}$$ in Equation (*) below, usually comes together with random effects. 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 can be written as $$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\color{red}{u_i}+\epsilon_{ij}$$ where $$\color{red}{u_i}$$ is fixed (not random) intercept for each subject ($$i$$), or we can also have a fixed-effect as $$u_j$$ for each repeated measurement ($$j$$); $$\boldsymbol x_{ij}$$ denotes covariates.
• 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) can be written as $$\tag{*} y_{ij}=\boldsymbol x_{ij}^{'}\color{red}{\boldsymbol\beta}+\boldsymbol z_{ij}^{'}\color{blue}{\boldsymbol u_i}+e_{ij}$$ where $$\color{blue}{\boldsymbol u_i}$$ is assumed to follow a distribution. $$\boldsymbol x_{ij}$$ denotes covariates for fixed effects, and $$\boldsymbol z_{ij}$$ denotes covariates for random effects.
• 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), and the former focuses on the population mean (take linear model for an example) $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ while the latter deals with 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 would be different for nonlinear models (e.g. logistic regression). Let $$h(E(y_{ij}|\boldsymbol u_i))=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$$, then $$E(y_{ij})=E(E(y_{ij}|\boldsymbol u_i))=E(h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i))\neq h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta),$$ unless trivially the link function $$h$$ is the identity link (linear model), or $$u_i=0$$ (no random-effects). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

This question has been partially discussed at this site as below, and opinions seem mixed.

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

## Fixed-effects model

• In biostatistics, fixed-effects, denoted as $$\color{red}{\boldsymbol\beta}$$ in Equation (*) below, usually comes together with random effects. 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 can be written as $$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\color{red}{u_i}+\epsilon_{ij}$$ where $$\color{red}{u_i}$$ is fixed (not random) intercept for each subject ($$i$$), or we can also have a fixed-effect as $$u_j$$ for each repeated measurement ($$j$$); $$\boldsymbol x_{ij}$$ denotes covariates.
• 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) can be written as $$\tag{*} y_{ij}=\boldsymbol x_{ij}^{'}\color{red}{\boldsymbol\beta}+\boldsymbol z_{ij}^{'}\color{blue}{\boldsymbol u_i}+e_{ij}$$ where $$\color{blue}{\boldsymbol u_i}$$ is assumed to follow a distribution. $$\boldsymbol x_{ij}$$ denotes covariates for fixed effects, and $$\boldsymbol z_{ij}$$ denotes covariates for random effects.
• 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), and the former focuses on the population mean (take linear model for an example) $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ while the latter deals with 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 would be different for nonlinear models (e.g. logistic regression). Let $$h(E(y_{ij}|\boldsymbol u_i))=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$$, then $$E(y_{ij})=E(E(y_{ij}|\boldsymbol u_i))=E(h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i))\neq h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta),$$ unless trivially the link function $$h$$ is the identity link (linear model), or $$u_i=0$$ (no random-effects). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

This question has been partially discussed at this site as below, and opinions seem mixed.

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

## Fixed-effects model

• In biostatistics, fixed-effects, denoted as $$\color{red}{\boldsymbol\beta}$$ in Equation (*) below, usually comes together with random effects. 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 can be written as $$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\color{red}{u_i}+\epsilon_{ij}$$ where $$\color{red}{u_i}$$ is fixed (not random) intercept for each subject ($$i$$), or we can also have a fixed-effect as $$u_j$$ for each repeated measurement ($$j$$); $$\boldsymbol x_{ij}$$ denotes covariates.
• 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) can be written as $$\tag{*} y_{ij}=\boldsymbol x_{ij}^{'}\color{red}{\boldsymbol\beta}+\boldsymbol z_{ij}^{'}\color{blue}{\boldsymbol u_i}+e_{ij}$$ where $$\color{blue}{\boldsymbol u_i}$$ is assumed to follow a distribution. $$\boldsymbol x_{ij}$$ denotes covariates for fixed effects, and $$\boldsymbol z_{ij}$$ denotes covariates for random effects.
• 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), and the former focuses on the population mean (take linear model for an example) $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ while the latter deals with 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 would be different for nonlinear models (e.g. logistic regression). Let $$h(E(y_{ij}|\boldsymbol u_i))=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$$, then $$E(y_{ij})=E(E(y_{ij}|\boldsymbol u_i))=E(h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i))\neq h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta),$$ unless trivially the link function $$h$$ is the identity link (linear model), or $$u_i=0$$ (no random-effects). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

This question has been partially discussed at this site as below, and opinions seem mixed.

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

## Fixed-effects model

• In biostatistics, fixed-effects, denoted as $$\color{red}{\boldsymbol\beta}$$ in Equation (*) below, usually comes together with random effects. 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 can be written as $$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\color{red}{u_i}+\epsilon_{ij}$$ where $$\color{red}{u_i}$$ is fixed (not random) intercept for each subject ($$i$$), or we can also have a fixed-effect as $$u_j$$ for each repeated measurement ($$j$$); $$\boldsymbol x_{ij}$$ denotes covariates.
• 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) can be written as $$\tag{*} y_{ij}=\boldsymbol x_{ij}^{'}\color{red}{\boldsymbol\beta}+\boldsymbol z_{ij}^{'}\color{blue}{\boldsymbol u_i}+e_{ij}$$ where $$\color{blue}{\boldsymbol u_i}$$ is assumed to follow a distribution. $$\boldsymbol x_{ij}$$ denotes covariates for fixed effects, and $$\boldsymbol z_{ij}$$ denotes covariates for random effects.
• 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), and the former focuses on the population mean (take linear model for an example) $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ while the latter deals with 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 would be different for nonlinear models (e.g. logistic regression). Let $$h(E(y_{ij}|\boldsymbol u_i))=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$$, then $$E(y_{ij})=E(E(y_{ij}|\boldsymbol u_i))=E(h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i))\neq h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta),$$ unless trivially the link function $$h$$ is the identity link (linear model), or $$u_i=0$$ (no random-effects). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

This question has been partially discussed at this site as below, and opinions seem mixed.

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

## Fixed-effects model

• In biostatistics, fixed-effects, denoted as $$\color{red}{\boldsymbol\beta}$$ in Equation (*) below, usually comes together with random effects. 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 can be written as $$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\color{red}{u_i}+\epsilon_{ij}$$ where $$\color{red}{u_i}$$ is fixed (not random) intercept for each subject ($$i$$), or we can also have a fixed-effect as $$u_j$$ for each repeated measurement ($$j$$).
• 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) can be written as $$\tag{*} y_{ij}=\boldsymbol x_{ij}^{'}\color{red}{\boldsymbol\beta}+\boldsymbol z_{ij}^{'}\color{blue}{\boldsymbol u_i}+e_{ij}$$ where $$\color{blue}{\boldsymbol u_i}$$ is assumed to follow a distribution.
• 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), and the former focuses on the population mean (take linear model for an example) $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ while the latter deals with 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 would be different for nonlinear models (e.g. logistic regression). Let $$h(E(y_{ij}|\boldsymbol u_i))=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$$, then $$E(y_{ij})=E(E(y_{ij}|\boldsymbol u_i))=E(h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i))\neq h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta),$$ unless trivially the link function $$h$$ is the identity link (linear model), or $$u_i=0$$ (no random-effects). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

This question has been partially discussed at this site as below, and opinions seem mixed.

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

## Fixed-effects model

• In biostatistics, fixed-effects, denoted as $$\color{red}{\boldsymbol\beta}$$ in Equation (*) below, usually comes together with random effects. 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 can be written as $$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\color{red}{u_i}+\epsilon_{ij}$$ where $$\color{red}{u_i}$$ is fixed (not random) intercept for each subject ($$i$$), or we can also have a fixed-effect as $$u_j$$ for each repeated measurement ($$j$$); $$\boldsymbol x_{ij}$$ denotes covariates.
• 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) can be written as $$\tag{*} y_{ij}=\boldsymbol x_{ij}^{'}\color{red}{\boldsymbol\beta}+\boldsymbol z_{ij}^{'}\color{blue}{\boldsymbol u_i}+e_{ij}$$ where $$\color{blue}{\boldsymbol u_i}$$ is assumed to follow a distribution. $$\boldsymbol x_{ij}$$ denotes covariates for fixed effects, and $$\boldsymbol z_{ij}$$ denotes covariates for random effects.
• 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), and the former focuses on the population mean (take linear model for an example) $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ while the latter deals with 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 would be different for nonlinear models (e.g. logistic regression). Let $$h(E(y_{ij}|\boldsymbol u_i))=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$$, then $$E(y_{ij})=E(E(y_{ij}|\boldsymbol u_i))=E(h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i))\neq h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta),$$ unless trivially the link function $$h$$ is the identity link (linear model), or $$u_i=0$$ (no random-effects). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

4 modified according to @amoeba's suggestions

This question is not easy to answer, and has been partially discussed at this site as below, and opinions seem mixed.

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

## Fixed-effects model

• In biostatistics, fixed-effects, denoted as $$\color{red}{\boldsymbol\beta}$$ in Equation (*) below, 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 iscan be written as $$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+u_i+\epsilon_{ij}$$$$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\color{red}{u_i}+\epsilon_{ij}$$ where $$u_i$$$$\color{red}{u_i}$$ is fixed (not random) intercept for each subject ($$i$$), or we can also have a fixed-effect as $$u_j$$ for each repeated measurement ($$j$$).
• 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) iscan be written as $$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\boldsymbol z_{ij}^{'}\boldsymbol u_i+\epsilon_{ij}$$$$\tag{*} y_{ij}=\boldsymbol x_{ij}^{'}\color{red}{\boldsymbol\beta}+\boldsymbol z_{ij}^{'}\color{blue}{\boldsymbol u_i}+e_{ij}$$ where $$\boldsymbol u_i$$ are$$\color{blue}{\boldsymbol u_i}$$ is assumed to follow a distribution.
• 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), and the former focuses on the population mean (take linear model for an example) $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ while the latter deals with 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 would be different for nonlinear models (e.g. logistic regression). Let $$h(E(y_{ij}|\boldsymbol u_i))=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$$, then $$E(y_{ij})=E(E(y_{ij}|\boldsymbol u_i))=E(h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i))\neq h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta),$$ unless trivially the link function $$h$$ is the identity link (linear model), or $$u_i=0$$ (no random-effects). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

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 the question 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 assumed to follow a distribution.
• 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), and the former focuses on the population mean $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ while the latter deals with 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 would be different for nonlinear models (e.g. logistic regression). Let $$h(E(y_{ij}|\boldsymbol u_i))=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$$, then $$E(y_{ij})=E(E(y_{ij}|\boldsymbol u_i))=E(h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i))\neq h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta),$$ unless trivially the link function $$h$$ is the identity link (linear model), or $$u_i=0$$ (no random-effects). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

This question has been partially discussed at this site as below, and opinions seem mixed.

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

## Fixed-effects model

• In biostatistics, fixed-effects, denoted as $$\color{red}{\boldsymbol\beta}$$ in Equation (*) below, usually comes together with random effects. 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 can be written as $$y_{ij}=\boldsymbol x_{ij}^{'}\boldsymbol\beta+\color{red}{u_i}+\epsilon_{ij}$$ where $$\color{red}{u_i}$$ is fixed (not random) intercept for each subject ($$i$$), or we can also have a fixed-effect as $$u_j$$ for each repeated measurement ($$j$$).
• 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) can be written as $$\tag{*} y_{ij}=\boldsymbol x_{ij}^{'}\color{red}{\boldsymbol\beta}+\boldsymbol z_{ij}^{'}\color{blue}{\boldsymbol u_i}+e_{ij}$$ where $$\color{blue}{\boldsymbol u_i}$$ is assumed to follow a distribution.
• 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), and the former focuses on the population mean (take linear model for an example) $$E(y_{ij})=\boldsymbol x_{ij}^{'}\boldsymbol\beta,$$ while the latter deals with 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 would be different for nonlinear models (e.g. logistic regression). Let $$h(E(y_{ij}|\boldsymbol u_i))=\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i$$, then $$E(y_{ij})=E(E(y_{ij}|\boldsymbol u_i))=E(h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta + \boldsymbol z_{ij}^{'}\boldsymbol u_i))\neq h^{-1}(\boldsymbol x_{ij}^{'}\boldsymbol\beta),$$ unless trivially the link function $$h$$ is the identity link (linear model), or $$u_i=0$$ (no random-effects). Good examples include generalized estimating equations (GEE; Zeger, Liang and Albert, 1988) and marginalized multilevel models (Heagerty and Zeger, 2000).

Bounty Ended with 50 reputation awarded by gung
3 fixed typo
2 expand the marginal model
1