Generalized estimating equation

In statistics, a generalized estimating equation (GEE) is used to estimate the parameters of a generalized linear model with a possible unmeasured correlation between observations from different timepoints.{{cite journal | journal=Biometrika | volume=73 | issue=1 | pages=13–22 | title=Longitudinal data analysis using generalized linear models | author1 = Kung-Yee Liang |author2-link=Scott Zeger | author2 = Scott Zeger | year=1986 | doi=10.1093/biomet/73.1.13| doi-access=free }}{{cite book | last = Hardin| first = James |author2=Hilbe, Joseph |authorlink2=Joseph Hilbe | title = Generalized Estimating Equations | url = https://archive.org/details/generalizedestim0000hard| url-access = registration| publisher = London: Chapman and Hall/CRC | year = 2003 |isbn=978-1-58488-307-4 }}

Regression beta coefficient estimates from the Liang-Zeger GEE are consistent, unbiased, and asymptotically normal even when the working correlation is misspecified, under mild regularity conditions. GEE is higher in efficiency than generalized linear models (GLMs) in the presence of high autocorrelation. When the true working correlation is known, consistency does not require the assumption that missing data is missing completely at random. Huber-White standard errors improve the efficiency of Liang-Zeger GEE in the absence of serial autocorrelation but may remove the marginal interpretation. GEE estimates the average response over the population ("population-averaged" effects) with Liang-Zeger standard errors, and in individuals using Huber-White standard errors, also known as "robust standard error" or "sandwich variance" estimates.{{cite journal

| last1 = Abadie | first1 = Alberto

| last2 = Athey | first2 = Susan

| last3 = Imbens | first3 = Guido W

| last4 = Wooldridge | first4 = Jeffrey M

| arxiv = 1710.02926

| date = October 2022

| doi = 10.1093/qje/qjac038

| issue = 1

| journal = The Quarterly Journal of Economics

| pages = 1–35

| title = When Should You Adjust Standard Errors for Clustering?

| url = https://www.nber.org/papers/w24003

| volume = 138}} Huber-White GEE was used since 1997, and Liang-Zeger GEE dates to the 1980s based on a limited literature review.{{Cite journal |last1=Wolfe |first1=Frederick |last2=Anderson |first2=Janice |last3=Harkness |first3=Deborah |last4=Bennett |first4=Robert M. |last5=Caro |first5=Xavier J. |last6=Goldenberg |first6=Don L. |last7=Russell |first7=I. Jon |last8=Yunus |first8=Muhammad B. |date=1997 |title=A prospective, longitudinal, multicenter study of service utilization and costs in fibromyalgia |url=https://onlinelibrary.wiley.com/doi/10.1002/art.1780400904 |journal=Arthritis & Rheumatism |language=en |volume=40 |issue=9 |pages=1560–1570 |doi=10.1002/art.1780400904|pmid=9324009 }} Several independent formulations of these standard error estimators contribute to GEE theory. Placing the independent standard error estimators under the umbrella term "GEE" may exemplify abuse of terminology.

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 at risk for consistency loss at variance structure specification.{{cite journal |pmc=2883299|year=2010|last1=Fong|first1=Y|title=Bayesian inference for generalized linear mixed models|journal=Biostatistics|volume=11|issue=3|pages=397–412|last2=Rue|first2=H|last3=Wakefield|first3=J|doi=10.1093/biostatistics/kxp053|pmid=19966070}} The trade-off of variance-structure misspecification and consistent regression coefficient estimates is loss of efficiency, yielding inflated Wald test p-values as a result of higher variance of standard errors than that of the most optimal.{{Cite journal |last1=O'Brien |first1=Liam M. |last2=Fitzmaurice |first2=Garrett M. |last3=Horton |first3=Nicholas J. |date=October 2006 |title=Maximum Likelihood Estimation of Marginal Pairwise Associations with Multiple Source Predictors |journal=Biometrical Journal |language=en |volume=48 |issue=5 |pages=860–875 |doi=10.1002/bimj.200510227 |issn=0323-3847 |pmc=1764610 |pmid=17094349}} 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 $\mu_{ij}$ for subject $i$ and time $j$ that depends upon regression parameters $\beta_k$ , and variance structure, $V_{i}$ , the estimating equation is formed via:{{cite book | last = Diggle| first = Peter J. |author2=Patrick Heagerty |author3=Kung-Yee Liang |author4=Scott L. Zeger| title = Analysis of Longitudinal Data | publisher = Oxford Statistical Science Series | year = 2002 |isbn=978-0-19-852484-7}}

: $U(\beta) = \sum_{i=1}^N \frac{\partial \mu_{i}}{\partial \beta} V_i^{-1} \{ Y_i - \mu_i(\beta)\} \,\!$

The parameters $\beta_k$ are estimated by solving $U(\beta)=0$ 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-likelihood under the independence model criterion (QIC).{{Citation | last= Pan | first= W. | title= Akaike's information criterion in generalized estimating equations | journal= Biometrics | year= 2001 | volume= 57 | issue= 1 | pages= 120–125 | doi= 10.1111/j.0006-341X.2001.00120.x| pmid= 11252586 | s2cid= 7862441 }}.

= Relationship with Generalized Method of Moments =

The generalized estimating equation is a special case of the generalized method of moments (GMM).{{Cite journal|last1=Breitung|first1=Jörg|last2=Chaganty|first2=N. Rao|last3=Daniel|first3=Rhian M.|last4=Kenward|first4=Michael G.|last5=Lechner|first5=Michael|last6=Martus|first6=Peter|last7=Sabo|first7=Roy T.|last8=Wang|first8=You-Gan|last9=Zorn|first9=Christopher|s2cid=3213776|date=2010|title=Discussion of 'Generalized Estimating Equations: Notes on the Choice of the Working Correlation Matrix'|journal=Methods of Information in Medicine|volume=49|issue=5|pages=426–432|doi=10.1055/s-0038-1625133}} This relationship is immediately obvious from the requirement that the score function satisfy the equation: $\mathbb{E}[U(\beta)] = {1\over{N}}\sum_{i=1}^N \frac{\partial \mu_{i}}{\partial \beta} V_i^{-1} \{ Y_i - \mu_i(\beta)\} \,\! = 0$

Computation

Software for solving generalized estimating equations is available in MATLAB,{{cite journal | journal=Journal of Statistical Software | volume=25 | issue=14 | pages=1–14 | title=GEEQBOX: A MATLAB Toolbox for Generalized Estimating Equations and Quasi-Least Squares | author1 = Sarah J. Ratcliffe | author2 = Justine Shults | year=2008 |url = http://www.jstatsoft.org/v25/i14}} SAS (proc genmod{{cite web|url=http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/genmod_toc.htm |title=The GENMOD Procedure |location=The SAS Institute}}), SPSS (the gee procedure{{cite web|url=http://www.spss.com/software/statistics/advanced-statistics/ |title=IBM SPSS Advanced Statistics|date=5 April 2024 |location=IBM SPSS website}}), Stata (the xtgee command{{cite web|url=https://www.stata.com/manuals15/xtxtgee.pdf |title=Stata's implementation of GEE |location=Stata website}}), R (packages glmtoolbox,{{cite web|url=https://cran.r-project.org/web/packages/glmtoolbox/index.html |location=CRAN |title=glmtoolbox: Set of Tools to Data Analysis using Generalized Linear Models|date=10 October 2023 }} gee,{{cite web|url=https://cran.r-project.org/web/packages/gee/index.html |location=CRAN |title=gee: Generalized Estimation Equation solver|date=7 November 2019 }} geepack{{citation |url=https://cran.r-project.org/web/packages/geepack/index.html |location=CRAN |title=geepack: Generalized Estimating Equation Package|date=18 December 2020 }} and multgee{{citation |url=https://cran.r-project.org/web/packages/multgee/index.html |location=CRAN |title=multgee: GEE solver for correlated nominal or ordinal multinomial responses using a local odds ratios parameterization|date=13 May 2021 }}), Julia (package GEE.jl{{cite web |last1=Shedden |first1=Kerby |title=Generalized Estimating Equations in Julia |url=https://github.com/kshedden/GEE.jl |website=GitHub |access-date=24 June 2022 |date=23 June 2022}}) and Python (package statsmodels{{Cite web|url=https://www.statsmodels.org/devel/gee.html|title=Generalized Estimating Equations — statsmodels}}).

Comparisons among software packages for the analysis of binary correlated data {{cite journal | journal=Biometrical Journal | volume=40 | issue=3 | pages=245–260 | title=The generalised estimating equations: a comparison of procedures available in commercial statistical software packages| author1 =Andreas Ziegler | author2 = Ulrike Grömping | year=1998 | doi=10.1002/(sici)1521-4036(199807)40:3<245::aid-bimj245>3.0.co;2-n}}{{cite journal | journal=The American Statistician | volume=53 | issue=2 | title=Review of software to fit generalized estimating equation regression models | author1 = Nicholas J. HORTON | author2 = Stuart R. LIPSITZ | year=1999 | doi = 10.1080/00031305.1999.10474451 | pages=160–169| citeseerx=10.1.1.22.9325 }} and ordinal correlated data{{cite journal | journal=Computational Statistics & Data Analysis | title=GEE for longitudinal ordinal data: Comparing R-geepack, R-multgee, R-repolr, SAS-GENMOD, SPSS-GENLIN| author1 = Nazanin Nooraee | author2 = Geert Molenberghs | author3 = Edwin R. van den Heuvel | year=2014 | doi=10.1016/j.csda.2014.03.009 | volume=77 | pages=70–83| s2cid=15063953| url=https://pure.rug.nl/ws/files/17588929/Title_and_contents_.pdf}} via GEE are available.

References

External links

[https://online.stat.psu.edu/stat504/lesson/12 Advanced Topics I - Generalized Estimating Equations (GEE)]

Category:Regression analysis

Category:Estimation methods

Category:M-estimators

Category:Semi-parametric models