Do GEE and GLM estimate the same coefficients?
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
asked 4 hours ago
MarcelMarcel
403111
add a comment |
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
asked 4 hours ago
MarcelMarcel
403111
add a comment |
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
asked 4 hours ago
MarcelMarcel
403111
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
asked 4 hours ago
MarcelMarcel
403111
asked 4 hours ago
MarcelMarcel
403111
asked 4 hours ago
MarcelMarcel
403111
asked 4 hours ago
MarcelMarcel
403111
asked 4 hours ago
MarcelMarcel
403111
403111
add a comment |
add a comment |
1 Answer
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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$coefficients
[Estimate Std.err Wald Pr(>|W|)
(Intercept) 6.99850 0.2418413 8.374312e+02 0.0000000
week1 2.81525 0.2514376 1.253642e+02 0.0000000
week2 -0.15025 0.2051973 5.361492e-01 0.4640330
week3 0.00325 0.2075914 2.451027e-04 0.9875090
week4 -0.04700 0.2388983 3.870522e-02 0.8440338][1]
Note the first and second columns of the output in each model. They coefficients are identity, but standard errors differ.
answered 4 hours ago
StatsStudentStatsStudent
4,98332042
1
And does this remain the case when we choose a non-linear link function?
– Marcel
3 hours ago
The OP asked about a GLM, however, not a mixed GLM, correct? So it should beglm(fev ~ week)vs thegeeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/…
– Mark White
1 hour ago
@MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said.
– Marcel
1 hour ago
add a comment |
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Yes. GEE and GLM will indeed have the same coefficients, but different standard errors. To check, run an example in R. I've taken this example from Chapter 25 of Applied Regression Analysis and Other Multivariable Methods, 5th by Kleinbaum, et. al (just because it's on my desk and references GEE and GLM):
library(geepack)
library(lme4)
#get book data from
mydf<-read.table("http://www.hmwu.idv.tw/web/bigdata/rstudio-readData/tab/ch25q04.txt", header=TRUE)
mydf<-data.frame(subj=mydf$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.
answered 4 hours ago
StatsStudentStatsStudent
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 beglm(fev ~ week)vs thegeeglm. 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
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
add a comment |
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$coefficients
[Estimate Std.err Wald Pr(>|W|)
(Intercept) 6.99850 0.2418413 8.374312e+02 0.0000000
week1 2.81525 0.2514376 1.253642e+02 0.0000000
week2 -0.15025 0.2051973 5.361492e-01 0.4640330
week3 0.00325 0.2075914 2.451027e-04 0.9875090
week4 -0.04700 0.2388983 3.870522e-02 0.8440338][1]
Note the first and second columns of the output in each model. They coefficients are identity, but standard errors differ.
answered 4 hours ago
StatsStudentStatsStudent
4,98332042
1
And does this remain the case when we choose a non-linear link function?
– Marcel
3 hours ago
The OP asked about a GLM, however, not a mixed GLM, correct? So it should beglm(fev ~ week)vs thegeeglm. In that case, the GEE and GLM will agree, ut but differ in standard errors. This isn't necessarily the case for a multilevel model; see an example in a question I posted when the parameters of a multilevel model don't have anywhere near the same coefficients as a GEE: stats.stackexchange.com/questions/358231/…
– Mark White
1 hour ago
@MarkWhite nice catch, I didn't even notice he was fitting a mixed effects model. It is curious that they have the same coefficients, since I was under the impression that GLMM and GEE do not produce equivalent estimates, as you said.
– Marcel
1 hour ago
add a comment |
Yes. GEE and GLM will indeed have the same coefficients, but different standard errors. To check, run an example in R. I've taken this example from Chapter 25 of Applied Regression Analysis and Other Multivariable Methods, 5th by Kleinbaum, et. al (just because it's on my desk and references GEE and GLM):
library(geepack)
library(lme4)
#get book data from
mydf<-read.table("http://www.hmwu.idv.tw/web/bigdata/rstudio-readData/tab/ch25q04.txt", header=TRUE)
mydf<-data.frame(subj=mydf$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.
answered 4 hours ago
StatsStudentStatsStudent
4,98332042
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.
answered 4 hours ago
StatsStudentStatsStudent
4,98332042
answered 4 hours ago
StatsStudentStatsStudent
4,98332042
answered 4 hours ago
StatsStudentStatsStudent
4,98332042
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 beglm(fev ~ week)vs thegeeglm. 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
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
add a comment |
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 beglm(fev ~ week)vs thegeeglm. 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
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
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
1 hour 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
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
@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
add a comment |
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