Overdispersion with binomial distribution

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