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Overdispersion with binomial distribution
4 messages · Jessica L Hite/hitejl/O/VCU, Brian Ripley, Ben Bolker
Jessica L Hite/hitejl/O/VCU <hitejl <at> vcu.edu> writes:
I am attempting to run a glm with a binomial model to analyze proportion data. I have been following Crawley's book closely and am wondering if there is an accepted standard for how much is too much overdispersion? (e.g. change in AIC has an accepted standard of 2).
In principle, in the null case (i.e. data are really binomial) the deviance is chi-squared distributed with the df equal to the residual df. For example: example(glm) deviance(glm.D93) ## 5.13 summary(glm.D93)$dispersion ## 1 (by definition) dfr <- df.residual(glm.D93) deviance(glm.D93)/dfr ## 1.28 d2 <- sum(residuals(glm.D93,"pearson")^2) ## 5.17 (disp2 <- d2/dfr) ## 1.293 gg2 <- update(glm.D93,family=quasipoisson) summary(gg2)$dispersion ## 1.293, same as above pchisq(d2,df=dfr,lower.tail=FALSE) all.equal(coef(glm.D93),coef(gg2)) ## TRUE se1 <- coef(summary(glm.D93))[,"Std. Error"] se2 <- coef(summary(gg2))[,"Std. Error"] se2/se1 # (Intercept) outcome2 outcome3 treatment2 treatment3 # 1.137234 1.137234 1.137234 1.137234 1.137234 sqrt(disp2) # [1] 1.137234
My code and output are below, given the example in the book, these data are WAY overdispersed .....do I mention this and go on or does this signal the need to try a different model? If so, any suggestions on the type of distribution (gamma or negative binomial ?)?
Way overdispersed may indicate model lack of fit. Have you examined residuals/data for outliers etc.? quasibinomial should be fine, or you can try beta-binomial (see the aod package) ...
attach(Clutch2)
y<-cbind(Total,Size-Total)
glm1<-glm(y~Pred,"binomial")
summary(glm1)
Call:
glm(formula = y ~ Pred, family = "binomial")
Deviance Residuals:
Min 1Q Median 3Q Max
-9.1022 -2.7899 -0.4781 2.6058 8.4852
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.35095 0.06612 20.433 < 2e-16 ***
PredF -0.34811 0.11719 -2.970 0.00297 **
PredSN -3.29156 0.10691 -30.788 < 2e-16 ***
PredW -1.46451 0.12787 -11.453 < 2e-16 ***
PredWF -0.56412 0.13178 -4.281 1.86e-05 ***
---
#### the output for residual deviance and df does not change even when I
use quasibinomial, is this ok? #####
That's as expected.
library(MASS)
you don't really need MASS for quasibinomial.
glm2<-glm(y~Pred,"quasibinomial") summary(glm2)
Call:
glm(formula = y ~ Pred, family = "quasibinomial")
Deviance Residuals:
Min 1Q Median 3Q Max
-9.1022 -2.7899 -0.4781 2.6058 8.4852
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.3510 0.2398 5.633 1.52e-07 ***
PredF -0.3481 0.4251 -0.819 0.41471
PredSN -3.2916 0.3878 -8.488 1.56e-13 ***
PredW -1.4645 0.4638 -3.157 0.00208 **
PredWF -0.5641 0.4780 -1.180 0.24063
---
Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
(Dispersion parameter for quasibinomial family taken to be 13.15786)
Null deviance: 2815.5 on 108 degrees of freedom
Residual deviance: 1323.5 on 104 degrees of freedom
(3 observations deleted due to missingness)
AIC: NA
Number of Fisher Scoring iterations: 5
anova(glm2,test="F")
Analysis of Deviance Table
Model: quasibinomial, link: logit
Response: y
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev F Pr(>F)
NULL 108 2815.5
Pred 4 1492.0 104 1323.5 28.349 6.28e-16 ***
---
Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
model1<-update(glm2,~.-Pred) anova(glm2,model1,test="F")
Analysis of Deviance Table Model 1: y ~ Pred Model 2: y ~ 1 Resid. Df Resid. Dev Df Deviance F Pr(>F) 1 104 1323.5 2 108 2815.5 -4 -1492.0 28.349 6.28e-16 *** --- Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
coef(glm2)
(Intercept) PredF PredSN PredW PredWF 1.3509550 -0.3481096 -3.2915601 -1.4645097 -0.5641223 Thanks Jessica hitejl <at> vcu.edu
On Tue, 17 Feb 2009, Ben Bolker wrote:
Jessica L Hite/hitejl/O/VCU <hitejl <at> vcu.edu> writes:
I am attempting to run a glm with a binomial model to analyze proportion data. I have been following Crawley's book closely and am wondering if there is an accepted standard for how much is too much overdispersion? (e.g. change in AIC has an accepted standard of 2).
In principle, in the null case (i.e. data are really binomial) the deviance is chi-squared distributed with the df equal to the residual df.
*Approximately*, provided the expected counts are not near or below one. See MASS ?7.5 for an analysis of the size of the approximation errors (which can be large and in both directions). Given that I once had a consulting job where the over-dispersion was causing something close ot panic and was entirely illusory, the lack of the 'approximately' can have quite serious consequences.
For example: example(glm) deviance(glm.D93) ## 5.13 summary(glm.D93)$dispersion ## 1 (by definition) dfr <- df.residual(glm.D93) deviance(glm.D93)/dfr ## 1.28 d2 <- sum(residuals(glm.D93,"pearson")^2) ## 5.17 (disp2 <- d2/dfr) ## 1.293 gg2 <- update(glm.D93,family=quasipoisson) summary(gg2)$dispersion ## 1.293, same as above pchisq(d2,df=dfr,lower.tail=FALSE) all.equal(coef(glm.D93),coef(gg2)) ## TRUE se1 <- coef(summary(glm.D93))[,"Std. Error"] se2 <- coef(summary(gg2))[,"Std. Error"] se2/se1 # (Intercept) outcome2 outcome3 treatment2 treatment3 # 1.137234 1.137234 1.137234 1.137234 1.137234 sqrt(disp2) # [1] 1.137234
My code and output are below, given the example in the book, these data are WAY overdispersed .....do I mention this and go on or does this signal the need to try a different model? If so, any suggestions on the type of distribution (gamma or negative binomial ?)?
Way overdispersed may indicate model lack of fit. Have you examined residuals/data for outliers etc.? quasibinomial should be fine, or you can try beta-binomial (see the aod package) ...
attach(Clutch2)
y<-cbind(Total,Size-Total)
glm1<-glm(y~Pred,"binomial")
summary(glm1)
Call:
glm(formula = y ~ Pred, family = "binomial")
Deviance Residuals:
Min 1Q Median 3Q Max
-9.1022 -2.7899 -0.4781 2.6058 8.4852
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.35095 0.06612 20.433 < 2e-16 ***
PredF -0.34811 0.11719 -2.970 0.00297 **
PredSN -3.29156 0.10691 -30.788 < 2e-16 ***
PredW -1.46451 0.12787 -11.453 < 2e-16 ***
PredWF -0.56412 0.13178 -4.281 1.86e-05 ***
---
#### the output for residual deviance and df does not change even when I
use quasibinomial, is this ok? #####
That's as expected.
library(MASS)
you don't really need MASS for quasibinomial.
glm2<-glm(y~Pred,"quasibinomial") summary(glm2)
Call:
glm(formula = y ~ Pred, family = "quasibinomial")
Deviance Residuals:
Min 1Q Median 3Q Max
-9.1022 -2.7899 -0.4781 2.6058 8.4852
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.3510 0.2398 5.633 1.52e-07 ***
PredF -0.3481 0.4251 -0.819 0.41471
PredSN -3.2916 0.3878 -8.488 1.56e-13 ***
PredW -1.4645 0.4638 -3.157 0.00208 **
PredWF -0.5641 0.4780 -1.180 0.24063
---
Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
(Dispersion parameter for quasibinomial family taken to be 13.15786)
Null deviance: 2815.5 on 108 degrees of freedom
Residual deviance: 1323.5 on 104 degrees of freedom
(3 observations deleted due to missingness)
AIC: NA
Number of Fisher Scoring iterations: 5
anova(glm2,test="F")
Analysis of Deviance Table
Model: quasibinomial, link: logit
Response: y
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev F Pr(>F)
NULL 108 2815.5
Pred 4 1492.0 104 1323.5 28.349 6.28e-16 ***
---
Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
model1<-update(glm2,~.-Pred) anova(glm2,model1,test="F")
Analysis of Deviance Table Model 1: y ~ Pred Model 2: y ~ 1 Resid. Df Resid. Dev Df Deviance F Pr(>F) 1 104 1323.5 2 108 2815.5 -4 -1492.0 28.349 6.28e-16 *** --- Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
coef(glm2)
(Intercept) PredF PredSN PredW PredWF 1.3509550 -0.3481096 -3.2915601 -1.4645097 -0.5641223 Thanks Jessica hitejl <at> vcu.edu
Brian D. Ripley, ripley at stats.ox.ac.uk Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/ University of Oxford, Tel: +44 1865 272861 (self) 1 South Parks Road, +44 1865 272866 (PA) Oxford OX1 3TG, UK Fax: +44 1865 272595
Thanks for the clarification. I actually had MASS open to that page while I was composing my reply but forgot to mention it (trying to do too many things at once) ... Ben Bolker
Prof Brian Ripley wrote:
On Tue, 17 Feb 2009, Ben Bolker wrote:
Jessica L Hite/hitejl/O/VCU <hitejl <at> vcu.edu> writes:
I am attempting to run a glm with a binomial model to analyze proportion data. I have been following Crawley's book closely and am wondering if there is an accepted standard for how much is too much overdispersion? (e.g. change in AIC has an accepted standard of 2).
In principle, in the null case (i.e. data are really binomial) the deviance is chi-squared distributed with the df equal to the residual df.
*Approximately*, provided the expected counts are not near or below one. See MASS ?7.5 for an analysis of the size of the approximation errors (which can be large and in both directions). Given that I once had a consulting job where the over-dispersion was causing something close ot panic and was entirely illusory, the lack of the 'approximately' can have quite serious consequences.
For example: example(glm) deviance(glm.D93) ## 5.13 summary(glm.D93)$dispersion ## 1 (by definition) dfr <- df.residual(glm.D93) deviance(glm.D93)/dfr ## 1.28 d2 <- sum(residuals(glm.D93,"pearson")^2) ## 5.17 (disp2 <- d2/dfr) ## 1.293 gg2 <- update(glm.D93,family=quasipoisson) summary(gg2)$dispersion ## 1.293, same as above pchisq(d2,df=dfr,lower.tail=FALSE) all.equal(coef(glm.D93),coef(gg2)) ## TRUE se1 <- coef(summary(glm.D93))[,"Std. Error"] se2 <- coef(summary(gg2))[,"Std. Error"] se2/se1 # (Intercept) outcome2 outcome3 treatment2 treatment3 # 1.137234 1.137234 1.137234 1.137234 1.137234 sqrt(disp2) # [1] 1.137234
My code and output are below, given the example in the book, these data are WAY overdispersed .....do I mention this and go on or does this signal the need to try a different model? If so, any suggestions on the type of distribution (gamma or negative binomial ?)?
Way overdispersed may indicate model lack of fit. Have you examined residuals/data for outliers etc.? quasibinomial should be fine, or you can try beta-binomial (see the aod package) ...
attach(Clutch2)
y<-cbind(Total,Size-Total)
glm1<-glm(y~Pred,"binomial")
summary(glm1)
Call:
glm(formula = y ~ Pred, family = "binomial")
Deviance Residuals:
Min 1Q Median 3Q Max
-9.1022 -2.7899 -0.4781 2.6058 8.4852
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.35095 0.06612 20.433 < 2e-16 ***
PredF -0.34811 0.11719 -2.970 0.00297 **
PredSN -3.29156 0.10691 -30.788 < 2e-16 ***
PredW -1.46451 0.12787 -11.453 < 2e-16 ***
PredWF -0.56412 0.13178 -4.281 1.86e-05 ***
---
#### the output for residual deviance and df does not change even when I
use quasibinomial, is this ok? #####
That's as expected.
library(MASS)
you don't really need MASS for quasibinomial.
glm2<-glm(y~Pred,"quasibinomial") summary(glm2)
Call:
glm(formula = y ~ Pred, family = "quasibinomial")
Deviance Residuals:
Min 1Q Median 3Q Max
-9.1022 -2.7899 -0.4781 2.6058 8.4852
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.3510 0.2398 5.633 1.52e-07 ***
PredF -0.3481 0.4251 -0.819 0.41471
PredSN -3.2916 0.3878 -8.488 1.56e-13 ***
PredW -1.4645 0.4638 -3.157 0.00208 **
PredWF -0.5641 0.4780 -1.180 0.24063
---
Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
(Dispersion parameter for quasibinomial family taken to be 13.15786)
Null deviance: 2815.5 on 108 degrees of freedom
Residual deviance: 1323.5 on 104 degrees of freedom
(3 observations deleted due to missingness)
AIC: NA
Number of Fisher Scoring iterations: 5
anova(glm2,test="F")
Analysis of Deviance Table
Model: quasibinomial, link: logit
Response: y
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev F Pr(>F)
NULL 108 2815.5
Pred 4 1492.0 104 1323.5 28.349 6.28e-16 ***
---
Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
model1<-update(glm2,~.-Pred) anova(glm2,model1,test="F")
Analysis of Deviance Table Model 1: y ~ Pred Model 2: y ~ 1 Resid. Df Resid. Dev Df Deviance F Pr(>F) 1 104 1323.5 2 108 2815.5 -4 -1492.0 28.349 6.28e-16 *** --- Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
coef(glm2)
(Intercept) PredF PredSN PredW PredWF 1.3509550 -0.3481096 -3.2915601 -1.4645097 -0.5641223 Thanks Jessica hitejl <at> vcu.edu
Ben Bolker Associate professor, Biology Dep't, Univ. of Florida bolker at ufl.edu / www.zoology.ufl.edu/bolker GPG key: www.zoology.ufl.edu/bolker/benbolker-publickey.asc