Generalized estimating equation

Given a mean model ${\displaystyle \mu _{ij}}$ for subject ${\displaystyle i}$ and time ${\displaystyle j}$ that depends upon regression parameters ${\displaystyle \beta _{k}}$, and variance structure, ${\displaystyle V_{i}}$, the estimating equation is formed via:[4]

${\displaystyle U(\beta )=\sum _{i=1}^{N}{\frac {\partial \mu _{i}}{\partial \beta }}V_{i}^{-1}\{Y_{i}-\mu _{i}(\beta )\}\,\!}$

The parameters ${\displaystyle \beta _{k}}$ are estimated by solving ${\displaystyle U(\beta )=0}$ and are typically obtained via the Newton–Raphson algorithm. The variance structure is chosen to improve the efficiency of the parameter estimates. The Hessian of the solution to the GEEs in the parameter space can be used to calculate robust standard error estimates. The term "variance structure" refers to the algebraic form of the covariance matrix between outcomes, Y, in the sample. Examples of variance structure specifications include independence, exchangeable, autoregressive, stationary m-dependent, and unstructured. The most popular form of inference on GEE regression parameters is the Wald test using naive or robust standard errors, though the Score test is also valid and preferable when it is difficult to obtain estimates of information under the alternative hypothesis. The likelihood ratio test is not valid in this setting because the estimating equations are not necessarily likelihood equations. Model selection can be performed with the GEE equivalent of the Akaike Information Criterion (AIC), the Quasilikelihood under the Independence model Criterion (QIC).[5]

Relationship with Generalized Method of Moments

The generalized estimating equation is a special case of the generalized method of moments (GMM).[6] This relationship is immediately obvious from the requirement that the score function satisfy the equation:

$${\displaystyle \mathbb {E} [U(\beta )]={1 \over {N}}\sum _{i=1}^{N}{\frac {\partial \mu _{i}}{\partial \beta }}V_{i}^{-1}\{Y_{i}-\mu _{i}(\beta )\}\,\!=0}$$

Software for solving generalized estimating equations is available in MATLAB,[7] SAS (proc genmod[8]), SPSS (the gee procedure[9]), Stata (the xtgee command[10]), R (packages gee,[11] geepack[12] and multgee[13]), and Python (package statsmodels[14]).

Comparisons among software packages for the analysis of binary correlated data [15][16] and ordinal correlated data[17] via GEE are available.

• Generalized method of moments
• Repeated measures design

• Hardin, James; Hilbe, Joseph (2003). Generalized Estimating Equations. London: Chapman and Hall/CRC. ISBN 978-1-58488-307-4.
• Ziegler, A. (2011). Generalized Estimating Equations. Springer. ISBN 978-1-4614-0498-9.