# Models for Generalized Estimating Equation?

From Wikipedia, Generalized Estimating Equation (GEE) is a method to estimate the parameters of a generalized linear model (with an exponential family distribution for the response).

By reading other references online, I am confused whether GEE is an estimation method, or a statistical model like the generalized linear model, but I am inclined to think of GEE as an estimation method, as an alternative to MLE. Am I right?

But it only relies on the mean model (as a function of the parameter), and the variance function (as a function of the mean). If I am correct, it is a maximum quasi-likelihood method (although I also want to ask what a maximum quasi-likelihood method is?).

So GEE doesn't use the entire likelihood offered by the generalized linear model, but only part of it. Is it correct that GEE should be able to apply to bigger models than the generalized linear model?

Thanks and regards!

• What do you mean by "bigger models"? In terms of broadness, I think regression can be categorized as GLM, outcome transformed linear regression and nonlinear regression. – Randel Jun 29 '13 at 20:34
• By a model, I mean things like a generalized linear model (such as Gaussian linear model, a logistic model, ...), a generalized mixed linear model. By a model which is bigger, I mean that GEE can also be applied to models which are not generalized linear models. – Tim Jun 29 '13 at 21:18
• Some texts and papers also call "GEE models", e.g. Hedeker, D., & Gibbons, R. D. (2006). Longitudinal data analysis. Wiley-Interscience. I guess it is to separate it from random effects models, since GEE is mainly regarded as fixed effects models (or marginal models). In Page 126 of Diggle et al. (2002), they discussed two extensions of GLMs for longitudinal data which can be estimated by GEE, i.e. marginal models and transition (Markov) models. – Randel Jun 30 '13 at 16:30

## 1 Answer

I prefer to call GEE an estimation method compared to ML or REML, since it combines quasi-likelihood estimation with robust variance estimation to estimate generalized linear marginal models for longitudinal data. Some texts and papers also call "GEE models", e.g. Hedeker, D., & Gibbons, R. D. (2006). Longitudinal data analysis. Wiley-Interscience. I guess it is to separate it from subject-specific (fixed and random effects) models, since GEE is mainly regarded as or marginal (population average) models.

We have no idea about the distribution function of the outcome, but we know its mean ($\mu$) and variance ($V$). So we cannot do ML but we can turn to the quasi-likelihood,

$$Q(\mu,y)=\int^{\mu}_y(y-t)^TV^{-1}dt,$$

and the quasi-likelihood estimating equations (quasi-score function) is

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

Thus the estimating equations are derived without specifying the joint distribution of a outcomes but they reduce to the score equations (marginal distributions). The approach based on maximum likelihood (ML) estimation specifies the joint multivariate normal distribution of outcome variables, while the approach of GEE based on the quasi-likelihood specifies only the marginal distributions.

I have seen GEE was applied in statistical genetics, but I am afraid it is also under the framework of generalized linear models.

• Thanks! "the approach of GEE based on the quasi-likelihood specifies only the marginal distributions", So is GEE based on maximizing a quasi likelihood function? How is the quasi likelihood function constructed, and is it related to the marginal distributions? – Tim Jun 30 '13 at 12:18
• I think GEE is based on maximizing a quasi likelihood function. You can find details in Liang and Zeger (1986), Zeger and Liang (1986) and Prentice (1988). As explained in Fitzmaurice et al. (2004), “the term marginal in this context indicates that the model for the mean response depends only on the covariates of interest, and not on any random effects or previous responses.” – Randel Jun 30 '13 at 16:43