Generalized Estimating Equations (GEEs Sample Clauses

Generalized Estimating Equations (GEEs. Consider the longitudinal observations (yij, xij) for times tij, j = 1, . . . , mi and sub- jects i = 1, . . . , n. Here, yij denotes the outcome variable and xij denotes a p × 1 vector of covariates. Let yi = (yi1, . . . , yimi )′ be an mi × 1 vector of outcomes for subject i and xi = (xi1, . . . , ximi )′. Further, let β be a p × 1 vector of regression parameters and ϕ denote a scale parameter. In this framework, the GEE approach models the mean and covariance matrix of Yi, i = 1, . . . , n as E (Yi) = µi (β) and V ar (Yi) = Vi (µ , α, ϕ) = ϕA1/2 (µ ) Ri (α) A1/2 (µ ) , ( ) where Ai is a diagonal matrix with var (Yij) = v µij as the jth diagonal element and Ri (α) is the ni × ni working correlation matrix for each Yi. It is assumed that Ri (α) is completely specified by an s × 1 vector of unknown parameters, α, which is the same for all subjects and can be estimated using the method of moments or another set of estimating equations. It follows that the GEE model is specified by n T ∂β i g (β; ϕ, µ ) = ∑ ∂µi V−1 (µ , α, ϕ) (Y i=1 i i − µ (β)) . The above model is sometimes referred to as a GEE1 model to distinguish it from the more recently developed GEE2 approach[47, 64, 88]. GEE1 models focus on esti- mation of the regression parameters and provide consistent, but not necessarily fully efficient, estimators regardless of whether the working correlation matrix is correctly specified [46]. Additionally, in contrast to XXX0 xxxxxx, XXX0 models assume or- thogonality of the estimating equations for the regression and association parameters [38].