Generalized estimating equation

In statistics, a generalized estimating equation (GEE) is used to estimate the parameters of a generalized linear model with a possible unknown correlation between outcomes.[1][2]

Parameter estimates from the GEE are consistent even when the covariance structure is misspecified, under mild regularity conditions. The focus of the GEE is on estimating the average response over the population ("population-averaged" effects) rather than the regression parameters that would enable prediction of the effect of changing one or more covariates on a given individual. GEEs are usually used in conjunction with Huber–White standard error estimates, also known as "robust standard error" or "sandwich variance" estimates. In the case of a linear model with a working independence variance structure, these are known as "heteroscedasticity consistent standard error" estimators. Indeed, the GEE unified several independent formulations of these standard error estimators in a general framework.

GEEs belong to a class of regression techniques that are referred to as semiparametric because they rely on specification of only the first two moments. They are a popular alternative to the likelihood–based generalized linear mixed model which is more sensitive to variance structure specification.[3] They are commonly used in large epidemiological studies, especially multi-site cohort studies, because they can handle many types of unmeasured dependence between outcomes.

Formulation

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]

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

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 Quasi-AIC (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:

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

Computation

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.

See also

• Generalized method of moments
• Repeated measures design

References

1. Kung-Yee Liang and Scott Zeger (1986). "Longitudinal data analysis using generalized linear models". Biometrika. 73 (1): 13–22. doi:10.1093/biomet/73.1.13.CS1 maint: uses authors parameter (link)
2. Hardin, James; Hilbe, Joseph (2003). Generalized Estimating Equations. London: Chapman and Hall/CRC. ISBN 978-1-58488-307-4.
3. Fong, Y; Rue, H; Wakefield, J (2010). "Bayesian inference for generalized linear mixed models". Biostatistics. 11 (3): 397–412. doi:10.1093/biostatistics/kxp053. PMC 2883299. PMID 19966070.
4. Diggle, Peter J.; Patrick Heagerty; Kung-Yee Liang; Scott L. Zeger (2002). Analysis of Longitudinal Data. Oxford Statistical Science Series. ISBN 978-0-19-852484-7.
5. Pan, W. (2001), "Akaike's information criterion in generalized estimating equations", Biometrics, 57 (1): 120–125, doi:10.1111/j.0006-341X.2001.00120.x, PMID 11252586.
6. Breitung, Jörg; Chaganty, N. Rao; Daniel, Rhian M.; Kenward, Michael G.; Lechner, Michael; Martus, Peter; Sabo, Roy T.; Wang, You-Gan; Zorn, Christopher (2010). "Discussion of 'Generalized Estimating Equations: Notes on the Choice of the Working Correlation Matrix'". Methods of Information in Medicine. 49 (5): 426–432. doi:10.1055/s-0038-1625133. S2CID 3213776.
7. Sarah J. Ratcliffe and Justine Shults (2008). "GEEQBOX: A MATLAB Toolbox for Generalized Estimating Equations and Quasi-Least Squares". Journal of Statistical Software. 25 (14): 1–14.CS1 maint: uses authors parameter (link)
8. "The GENMOD Procedure". The SAS Institute.
9. "IBM SPSS Advanced Statistics". IBM SPSS website.
10. "Stata's implementation of GEE" (PDF). Stata website.
11. "gee: Generalized Estimation Equation solver". CRAN.
12. geepack: Generalized Estimating Equation Package, CRAN
13. multgee: GEE solver for correlated nominal or ordinal multinomial responses using a local odds ratios parameterization, CRAN
14. https://www.statsmodels.org/devel/gee.html
15. Andreas Ziegler and Ulrike Grömping (1998). "The generalised estimating equations: a comparison of procedures available in commercial statistical software packages". Biometrical Journal. 40 (3): 245–260. doi:10.1002/(sici)1521-4036(199807)40:3<245::aid-bimj245>3.0.co;2-n.CS1 maint: uses authors parameter (link)
16. Nicholas J. HORTON and Stuart R. LIPSITZ (1999). "Review of software to fit generalized estimating equation regression models". The American Statistician. 53 (2): 160–169. CiteSeerX 10.1.1.22.9325. doi:10.1080/00031305.1999.10474451.CS1 maint: uses authors parameter (link)
17. Nazanin Nooraee, Geert Molenberghs, and Edwin R. van den Heuvel (2014). "GEE for longitudinal ordinal data: Comparing R-geepack, R-multgee, R-repolr, SAS-GENMOD, SPSS-GENLIN" (PDF). Computational Statistics & Data Analysis. 77: 70–83. doi:10.1016/j.csda.2014.03.009.CS1 maint: uses authors parameter (link)

Further reading

• 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.