Generalized Linear Model – Understanding GEE, Quasi-Likelihood, and Their Generalizations

Question

Wikipedia formulates Generalized Estimating Equations (GEE) as

Given a mean model, $\mu_{ij}$, and variance structure, $V_{i}$, the estimating equation is formed via:
$$
U(\beta) = \sum_{i=1}^N \frac{\partial \mu_{ij}}{\partial \beta_k} V_i^{-1} \{ Y_i – \mu_i(\beta)\} \,\!
$$
The parameter estimates solve U(β)=0 and are typically obtained via the Newton-Raphson algorithm.

1. Does GEE belong to maximum quasi-likelihood method (is the maximum quasi-likelihood method same as quasi-likelihood estimation?) If yes, what is
   its quasi-likeilhood function, or does GEE maximizes some quasi-likelihood
   function?

2. What does GEE "generalize"? Is it estimating equation method for
   estimation?

   In what sense is GEE "generalized"? Is it similar to
   the way in which the generalized linear model generalizes the linear model?

Answer 1

1. Your link actually leads to "quasi-maximum likelihood" or more formally "composite likelihood". You can find a good review about composite likelihood here. Composite likelihood sometimes was called quasi-likelihood, such as Hjort and Omre (1994), Glasbey (2001) and Hjort and Varin (2008). However, composite likelihood, which can be applied in space-time models and longitudinal data, was proposed by Besag (1974, 1975), and quasi-likelihood was introduced by Wedderburn (1974) and mainly used in generalized linear models. As I discussed here, GEE only uses the mean ($\mu$) and variance ($V$) of the outcome and reaches the quasi-likelihood, $$Q(\mu,y)=\int^{\mu}_y(y-t)^TV^{-1}dt,$$ and the quasi-likelihood estimating equations (quasi-score function) for the estimation is $$\sum_i\frac{\partial{\mu_i^{'}}}{\partial{\beta}}V_i^{-1}(y_i-\mu_i)=0.$$
2. GEE is an extension of generalized linear models to the analysis of longitudinal data. I prefer to think the word "generalized" came from GLM, but you can also regard it as a generalization to longitudinal data.