What is a simple description of how the GEE algorithm works? How exactly does the GEE process come up with the final estimates of the parameters?
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You assume:
- a link function $g(\mu_{ij})=\mathbf{x}_{ij}'\boldsymbol{\beta}$;
- the conditional variance of each $y_{ij}$, $\text{Var}(y_{ij}\mid\mathbf{x}_{ij})=\phi v(\mu_{ij})$;
- the pairwise within-subject association, $\mathbf{V}_i=\mathbf{A}^{1/2}_i\mathbf{C}_i\mathbf{A}^{1/2}_i$, where $\mathbf{A}_i=\text{diag}[\text{Var}(y_{ij}\mid\mathbf{x}_{ij})]$ and $\mathbf{C}_i$ is a correlation matrix depending on a set of parameters $\boldsymbol{\alpha}$.
GEE minimizes $\sum_i(y_i-\mu_{ij})'\mathbf{V}_i^{-1}(y_i-\mu_{ij})$, where $\mu_{ij}=g^{-1}(\mathbf{x}_{ij}'\boldsymbol{\beta})$, with respect to $\boldsymbol{\beta}$. To do so, it must solve the generalized estimating equations: $$\sum_{i=1}^N\mathbf{D}_i'\mathbf{V}^{-1}_i(\mathbf{y}_i-\boldsymbol\mu_i)=0$$ where $\mathbf{D}_i$ is the matrix of the derivatives of $\boldsymbol{\mu}_i$ with respect to $\boldsymbol\beta$ and is therefore a function of $\boldsymbol\beta$, while $\mathbf{V}_i$ is a function of $\boldsymbol\beta$, $\boldsymbol\alpha$, and $\phi$.
The estimation procedure estimates a first $\mathbf{V}_i$ matrix from the residuals of an OLS linear regression, then the two-stage iterative procedure is:
- given current estimate of $\mathbf{V}_i$, i.e. of $\boldsymbol\beta$ $\boldsymbol\alpha$ and $\phi$, compute an updated estimate of $\boldsymbol{\beta}$ as the solution to the generalized estimating equations;
- use this new estimate $\hat{\boldsymbol\beta}$ to update estimates of $\boldsymbol\alpha$ and $\phi$, and so of $\mathbf{V}_i$, from the standardized residuals.
