Do GEE and GLM estimate the same coefficients?












3














In a GLM, the likelihood equations depend on the assumed distribution only through the mean and the variance. The likelihood equations are



$$sum_i^n (frac{partial mu_i}{partial eta_i}) frac{y_i - mu_i}{Var(Y_i)}x_{ij} = 0, quad (j = 1, ..., p)$$



and in the quasi-likelihood case, we just let $Var(Y_i) = v(mu_i)$ be some function of the mean. For GEE, the response is extended to be multivariate, with an assumed correlation structure, with the quasi-likelihood equations.



Does this imply that GEE and GLM will have the same parameter (say $beta$) estimates (population averaged) with the only difference being correct standard errors in the GEE case (Assuming clustered data?)



If the estimated coefficients are not the same, then what is the difference?






regression clustering generalized-linear-model estimation gee







share|cite|improve this question



























    3














    In a GLM, the likelihood equations depend on the assumed distribution only through the mean and the variance. The likelihood equations are



    $$sum_i^n (frac{partial mu_i}{partial eta_i}) frac{y_i - mu_i}{Var(Y_i)}x_{ij} = 0, quad (j = 1, ..., p)$$



    and in the quasi-likelihood case, we just let $Var(Y_i) = v(mu_i)$ be some function of the mean. For GEE, the response is extended to be multivariate, with an assumed correlation structure, with the quasi-likelihood equations.



    Does this imply that GEE and GLM will have the same parameter (say $beta$) estimates (population averaged) with the only difference being correct standard errors in the GEE case (Assuming clustered data?)



    If the estimated coefficients are not the same, then what is the difference?






    regression clustering generalized-linear-model estimation gee







    share|cite|improve this question

























      3












      3








      3







      In a GLM, the likelihood equations depend on the assumed distribution only through the mean and the variance. The likelihood equations are



      $$sum_i^n (frac{partial mu_i}{partial eta_i}) frac{y_i - mu_i}{Var(Y_i)}x_{ij} = 0, quad (j = 1, ..., p)$$



      and in the quasi-likelihood case, we just let $Var(Y_i) = v(mu_i)$ be some function of the mean. For GEE, the response is extended to be multivariate, with an assumed correlation structure, with the quasi-likelihood equations.



      Does this imply that GEE and GLM will have the same parameter (say $beta$) estimates (population averaged) with the only difference being correct standard errors in the GEE case (Assuming clustered data?)



      If the estimated coefficients are not the same, then what is the difference?






      regression clustering generalized-linear-model estimation gee







      share|cite|improve this question













      In a GLM, the likelihood equations depend on the assumed distribution only through the mean and the variance. The likelihood equations are



      $$sum_i^n (frac{partial mu_i}{partial eta_i}) frac{y_i - mu_i}{Var(Y_i)}x_{ij} = 0, quad (j = 1, ..., p)$$



      and in the quasi-likelihood case, we just let $Var(Y_i) = v(mu_i)$ be some function of the mean. For GEE, the response is extended to be multivariate, with an assumed correlation structure, with the quasi-likelihood equations.



      Does this imply that GEE and GLM will have the same parameter (say $beta$) estimates (population averaged) with the only difference being correct standard errors in the GEE case (Assuming clustered data?)



      If the estimated coefficients are not the same, then what is the difference?






      regression clustering generalized-linear-model estimation gee




      regression clustering generalized-linear-model estimation gee






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 4 hours ago









      MarcelMarcel

      403111




      403111






















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          Yes. GEE and GLM will indeed have the same coefficients, but different standard errors. To check, run an example in R. I've taken this example from Chapter 25 of Applied Regression Analysis and Other Multivariable Methods, 5th by Kleinbaum, et. al (just because it's on my desk and references GEE and GLM):



          library(geepack)
          
library(lme4)
          

          
#get book data from 
          
mydf<-read.table("http://www.hmwu.idv.tw/web/bigdata/rstudio-readData/tab/ch25q04.txt", header=TRUE)
          
mydf<-data.frame(subj=mydf$subj, week=as.factor(mydf$week), fev=mydf$fev)
          
#Make 5th level the reference level to match book results
          
mydf$          week<-relevel(mydf$week, ref="5")
          

          
#Fit GLM Mixed Model
          
mixed.model<-summary(lme4::lmer(fev~week+(1|subj),data=mydf))
          
mixed.model$coefficients
          

          
                Estimate Std. Error     t value
          
(Intercept)  6.99850  0.2590243 27.01870247
          
week1        2.81525  0.2439374 11.54087244
          
week2       -0.15025  0.2439374 -0.61593680
          
week3        0.00325  0.2439374  0.01332309
          
week4       -0.04700  0.2439374 -0.19267241
          

          
#Fit a gee model with any correlation structure.  In this case AR1
          
gee.model<-summary(geeglm(fev~week, id=subj, waves=week, corstr="ar1", data=mydf))
          
gee.model$coefficients
          

          
            [Estimate   Std.err         Wald  Pr(>|W|)
          
(Intercept)  6.99850 0.2418413 8.374312e+02 0.0000000
          
week1        2.81525 0.2514376 1.253642e+02 0.0000000
          
week2       -0.15025 0.2051973 5.361492e-01 0.4640330
          
week3        0.00325 0.2075914 2.451027e-04 0.9875090
          
week4       -0.04700 0.2388983 3.870522e-02 0.8440338][1]


          Note the first and second columns of the output in each model. They coefficients are identity, but standard errors differ.






          share|cite|improve this answer

















          • 1




            And does this remain the case when we choose a non-linear link function?
            – Marcel
            3 hours ago










          • The OP asked about a GLM, however, not a mixed GLM, correct? So it should be glm(fev ~ week) vs the geeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/…
            – Mark White
            2 hours ago












          • @MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said.
            – Marcel
            1 hour ago











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          Yes. GEE and GLM will indeed have the same coefficients, but different standard errors. To check, run an example in R. I've taken this example from Chapter 25 of Applied Regression Analysis and Other Multivariable Methods, 5th by Kleinbaum, et. al (just because it's on my desk and references GEE and GLM):



          library(geepack)
          
library(lme4)
          

          
#get book data from 
          
mydf<-read.table("http://www.hmwu.idv.tw/web/bigdata/rstudio-readData/tab/ch25q04.txt", header=TRUE)
          
mydf<-data.frame(subj=mydf$subj, week=as.factor(mydf$week), fev=mydf$fev)
          
#Make 5th level the reference level to match book results
          
mydf$          week<-relevel(mydf$week, ref="5")
          

          
#Fit GLM Mixed Model
          
mixed.model<-summary(lme4::lmer(fev~week+(1|subj),data=mydf))
          
mixed.model$coefficients
          

          
                Estimate Std. Error     t value
          
(Intercept)  6.99850  0.2590243 27.01870247
          
week1        2.81525  0.2439374 11.54087244
          
week2       -0.15025  0.2439374 -0.61593680
          
week3        0.00325  0.2439374  0.01332309
          
week4       -0.04700  0.2439374 -0.19267241
          

          
#Fit a gee model with any correlation structure.  In this case AR1
          
gee.model<-summary(geeglm(fev~week, id=subj, waves=week, corstr="ar1", data=mydf))
          
gee.model$coefficients
          

          
            [Estimate   Std.err         Wald  Pr(>|W|)
          
(Intercept)  6.99850 0.2418413 8.374312e+02 0.0000000
          
week1        2.81525 0.2514376 1.253642e+02 0.0000000
          
week2       -0.15025 0.2051973 5.361492e-01 0.4640330
          
week3        0.00325 0.2075914 2.451027e-04 0.9875090
          
week4       -0.04700 0.2388983 3.870522e-02 0.8440338][1]


          Note the first and second columns of the output in each model. They coefficients are identity, but standard errors differ.






          share|cite|improve this answer

















          • 1




            And does this remain the case when we choose a non-linear link function?
            – Marcel
            3 hours ago










          • The OP asked about a GLM, however, not a mixed GLM, correct? So it should be glm(fev ~ week) vs the geeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/…
            – Mark White
            2 hours ago












          • @MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said.
            – Marcel
            1 hour ago
















          2














          Yes. GEE and GLM will indeed have the same coefficients, but different standard errors. To check, run an example in R. I've taken this example from Chapter 25 of Applied Regression Analysis and Other Multivariable Methods, 5th by Kleinbaum, et. al (just because it's on my desk and references GEE and GLM):



          library(geepack)
          
library(lme4)
          

          
#get book data from 
          
mydf<-read.table("http://www.hmwu.idv.tw/web/bigdata/rstudio-readData/tab/ch25q04.txt", header=TRUE)
          
mydf<-data.frame(subj=mydf$subj, week=as.factor(mydf$week), fev=mydf$fev)
          
#Make 5th level the reference level to match book results
          
mydf$          week<-relevel(mydf$week, ref="5")
          

          
#Fit GLM Mixed Model
          
mixed.model<-summary(lme4::lmer(fev~week+(1|subj),data=mydf))
          
mixed.model$coefficients
          

          
                Estimate Std. Error     t value
          
(Intercept)  6.99850  0.2590243 27.01870247
          
week1        2.81525  0.2439374 11.54087244
          
week2       -0.15025  0.2439374 -0.61593680
          
week3        0.00325  0.2439374  0.01332309
          
week4       -0.04700  0.2439374 -0.19267241
          

          
#Fit a gee model with any correlation structure.  In this case AR1
          
gee.model<-summary(geeglm(fev~week, id=subj, waves=week, corstr="ar1", data=mydf))
          
gee.model$coefficients
          

          
            [Estimate   Std.err         Wald  Pr(>|W|)
          
(Intercept)  6.99850 0.2418413 8.374312e+02 0.0000000
          
week1        2.81525 0.2514376 1.253642e+02 0.0000000
          
week2       -0.15025 0.2051973 5.361492e-01 0.4640330
          
week3        0.00325 0.2075914 2.451027e-04 0.9875090
          
week4       -0.04700 0.2388983 3.870522e-02 0.8440338][1]


          Note the first and second columns of the output in each model. They coefficients are identity, but standard errors differ.






          share|cite|improve this answer

















          • 1




            And does this remain the case when we choose a non-linear link function?
            – Marcel
            3 hours ago










          • The OP asked about a GLM, however, not a mixed GLM, correct? So it should be glm(fev ~ week) vs the geeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/…
            – Mark White
            2 hours ago












          • @MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said.
            – Marcel
            1 hour ago














          2












          2








          2






          Yes. GEE and GLM will indeed have the same coefficients, but different standard errors. To check, run an example in R. I've taken this example from Chapter 25 of Applied Regression Analysis and Other Multivariable Methods, 5th by Kleinbaum, et. al (just because it's on my desk and references GEE and GLM):



          library(geepack)
          
library(lme4)
          

          
#get book data from 
          
mydf<-read.table("http://www.hmwu.idv.tw/web/bigdata/rstudio-readData/tab/ch25q04.txt", header=TRUE)
          
mydf<-data.frame(subj=mydf$subj, week=as.factor(mydf$week), fev=mydf$fev)
          
#Make 5th level the reference level to match book results
          
mydf$          week<-relevel(mydf$week, ref="5")
          

          
#Fit GLM Mixed Model
          
mixed.model<-summary(lme4::lmer(fev~week+(1|subj),data=mydf))
          
mixed.model$coefficients
          

          
                Estimate Std. Error     t value
          
(Intercept)  6.99850  0.2590243 27.01870247
          
week1        2.81525  0.2439374 11.54087244
          
week2       -0.15025  0.2439374 -0.61593680
          
week3        0.00325  0.2439374  0.01332309
          
week4       -0.04700  0.2439374 -0.19267241
          

          
#Fit a gee model with any correlation structure.  In this case AR1
          
gee.model<-summary(geeglm(fev~week, id=subj, waves=week, corstr="ar1", data=mydf))
          
gee.model$coefficients
          

          
            [Estimate   Std.err         Wald  Pr(>|W|)
          
(Intercept)  6.99850 0.2418413 8.374312e+02 0.0000000
          
week1        2.81525 0.2514376 1.253642e+02 0.0000000
          
week2       -0.15025 0.2051973 5.361492e-01 0.4640330
          
week3        0.00325 0.2075914 2.451027e-04 0.9875090
          
week4       -0.04700 0.2388983 3.870522e-02 0.8440338][1]


          Note the first and second columns of the output in each model. They coefficients are identity, but standard errors differ.






          share|cite|improve this answer












          Yes. GEE and GLM will indeed have the same coefficients, but different standard errors. To check, run an example in R. I've taken this example from Chapter 25 of Applied Regression Analysis and Other Multivariable Methods, 5th by Kleinbaum, et. al (just because it's on my desk and references GEE and GLM):



          library(geepack)
          
library(lme4)
          

          
#get book data from 
          
mydf<-read.table("http://www.hmwu.idv.tw/web/bigdata/rstudio-readData/tab/ch25q04.txt", header=TRUE)
          
mydf<-data.frame(subj=mydf$subj, week=as.factor(mydf$week), fev=mydf$fev)
          
#Make 5th level the reference level to match book results
          
mydf$          week<-relevel(mydf$week, ref="5")
          

          
#Fit GLM Mixed Model
          
mixed.model<-summary(lme4::lmer(fev~week+(1|subj),data=mydf))
          
mixed.model$coefficients
          

          
                Estimate Std. Error     t value
          
(Intercept)  6.99850  0.2590243 27.01870247
          
week1        2.81525  0.2439374 11.54087244
          
week2       -0.15025  0.2439374 -0.61593680
          
week3        0.00325  0.2439374  0.01332309
          
week4       -0.04700  0.2439374 -0.19267241
          

          
#Fit a gee model with any correlation structure.  In this case AR1
          
gee.model<-summary(geeglm(fev~week, id=subj, waves=week, corstr="ar1", data=mydf))
          
gee.model$coefficients
          

          
            [Estimate   Std.err         Wald  Pr(>|W|)
          
(Intercept)  6.99850 0.2418413 8.374312e+02 0.0000000
          
week1        2.81525 0.2514376 1.253642e+02 0.0000000
          
week2       -0.15025 0.2051973 5.361492e-01 0.4640330
          
week3        0.00325 0.2075914 2.451027e-04 0.9875090
          
week4       -0.04700 0.2388983 3.870522e-02 0.8440338][1]


          Note the first and second columns of the output in each model. They coefficients are identity, but standard errors differ.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 4 hours ago









          StatsStudentStatsStudent

          4,98332042




          4,98332042








          • 1




            And does this remain the case when we choose a non-linear link function?
            – Marcel
            3 hours ago










          • The OP asked about a GLM, however, not a mixed GLM, correct? So it should be glm(fev ~ week) vs the geeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/…
            – Mark White
            2 hours ago












          • @MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said.
            – Marcel
            1 hour ago














          • 1




            And does this remain the case when we choose a non-linear link function?
            – Marcel
            3 hours ago










          • The OP asked about a GLM, however, not a mixed GLM, correct? So it should be glm(fev ~ week) vs the geeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/…
            – Mark White
            2 hours ago












          • @MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said.
            – Marcel
            1 hour ago








          1




          1




          And does this remain the case when we choose a non-linear link function?
          – Marcel
          3 hours ago




          And does this remain the case when we choose a non-linear link function?
          – Marcel
          3 hours ago












          The OP asked about a GLM, however, not a mixed GLM, correct? So it should be glm(fev ~ week) vs the geeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/…
          – Mark White
          2 hours ago






          The OP asked about a GLM, however, not a mixed GLM, correct? So it should be glm(fev ~ week) vs the geeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/…
          – Mark White
          2 hours ago














          @MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said.
          – Marcel
          1 hour ago




          @MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said.
          – Marcel
          1 hour ago


















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          Eduardo VII do Reino Unido

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          Eduardo VII Rei do Reino Unido da Grã-Bretanha e Irlanda e dos Domínios Britânicos de Além-mar Imperador da Índia Rei do Reino Unido e Imperador da Índia Reinado 22 de janeiro de 1901 a 6 de maio de 1910 Coroação 9 de agosto de 1902 Delhi Durbar 1 de janeiro de 1903 Predecessora Vitória Sucessor Jorge V   Esposa Alexandra da Dinamarca Descendência Alberto Vitor, Duque de Clarence e Avondale Jorge V do Reino Unido Luísa, Princesa Real Vitória de Gales Maud de Gales Alexandre João de Gales Casa Saxe-Coburgo-Gota Nome completo Alberto Eduardo Nascimento 9 de novembro de 1841   Palácio de Buckingham, Londres, Reino Unido Morte 6 de maio de 1910 (68 anos)   Palácio de Buckingham, Londres, Reino Unido Enterro Capela de São Jorge, Windsor, Berkshire, Reino Unido 20 de maio de 1910 Pai Alberto de Saxe-Coburgo-Gota Mãe Vitória do Reino Unido Assinatura ...
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