*2.2. Generalized Estimating Equations*

The joint likelihood function for longitudinally clustered response *Yij* is generally difficult to specify. Liang and Zeger [20] developed the generalized estimating equations (GEE) method to account for the intra-cluster correlation. It models the marginal instead of the conditional distribution given the previous observations and only requires a working correlation structure for *Yij* to be specified.

The first two marginal moments of *Yij* are denoted by E(*Yij*) = *μij* = *Z<sup>T</sup> ij β* and Var(*Yij*) = *υ*(*μij*), respectively, where *υ* is a known variance function. Then, the estimating equation for *β* is defined as:

$$\sum\_{i=1}^{n} \frac{\partial \mu\_i(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} V\_i^{-1} (Y\_i - \mu\_i(\boldsymbol{\beta})) = 0,\tag{2}$$

where *μi*(*β*)=(*μi*1(*β*), ..., *μik*(*β*)), *Yi* = (*Yi*1, ...,*Yik*) and *Vi* is the covariance matrix of *Yi*. The first term in (2), *∂μi*(*β*) *∂β* , reduces to *Zi* = (*Zi*1, ..., *Zik*), which corresponds to the *k* × (1 + *q* + *p* + *pq*) matrix of the main and interaction effects.

*Vi* is often unknown in practice and difficult to estimate especially when the number of variance components is large. In GEE, the covariance matrix *Vi* is specified through a simplified working correlation matrix *R*(*η*) as *Vi* = *A* 1 2 *<sup>i</sup> R*(*η*)*A* 1 2 *<sup>i</sup>* , with the diagonal marginal variance matrix *Ai* = diag{Var(*Yi*1), ..., Var(*Yik*)}. *R*(*η*) is characterized by a finite-dimensional nuisance parameter *η*. Commonly adopted correlation structures for *R*(*η*) can be independent, AR(1), and exchangeable, among others. Liang and Zeger [20] showed that if *η* can be consistently estimated, the GEE estimator of the regression coefficient is consistent. Furthermore, the consistency holds even when the working correlation structure is misspecified.
