# GEE, quasi-likelihood and what it generalizes

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?

• For your first question, you can find the Wiki page of "Quasi-likelihood" from the "See also" column of Wiki page of "Quasi-maximum_likelihood". I think all the three Wiki pages need to be edited and expanded. As discussed in Page 139 of Diggle et al. (2002), "in the absence of a convenient likelihood function to work with, it is sensible to estimate $\beta$ by solving a multivariate analogue of the quasi-score function (Wedderburn, 1974)." – Randel Jun 30 '13 at 16:42

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 $$\sum_i\frac{\partial{\mu_i^{'}}}{\partial{\beta}}V_i^{-1}(y_i-\mu_i)=0.$$