11.5 Overdispersion
As has been alluded to in previous sections, each of the error distributions makes an assumption about the relationship between the mean and variance (Fig.1.9^{1}). In the general linear model with normal residuals, the assumption was that the variance remained constant (i.e., we assumed homoscedasticity  see Fig.1.9a). However, in the Binomial and Poisson distributions, this is not necessarily the case.
Figure 1.9: Theoretical expectations of the relationship between the variance and the mean in the Binomial and Poisson distributions. Axis labels also include the theoretical mean and variance, represented in terms of distribution parameters.
For the Binomial, we expect a humpshaped relationship. In the Poisson, where the variance is equal to the mean, there is a 1:1 linear relationship. In fact, this is one of the reasons we would expect to see a pattern such as that observed in the first diagnostic plot in Fig.1.7. As the expected value increases, variability increases.
However, in some cases, the observed variability in data may exceed these expectations. For example, in the case of loglinear regression, it is not uncommon for the variance to increase much greater than the 1:1 relationship that is expected under the Poisson distribution. In fact, it is common for biological data to exhibit this phenomenon. In these cases, we say that the data are “overdispersed”. Unfortunately, an implication of such overdispersion is that the residual deviance tends to be inflated, thus masking the potential effects of predictor variables.
Well, how do we identify overdispersion? One approach is to compare the
residual deviance to the residual df. They should theoretically be
equal. However, as discussed in [ and [, this must be done with caution.
Overdispersion may be an indicator that important predictor variables
have not been considered, and variability in these factors may result in
our data being biased. “Overdispersion happens for real, scientifically
important reasons, and these reasons may throw doubt upon our ability to
interpret the experiment in an unbiased way” [7]. As an
example, let’s reconsider the bromeliad data. Fortunately, in this case
(see the summary for model m2
in this section), the residual deviance,
175.06, is not too different than the residual degrees of freedom, 188.
However, one thing that commonly leads to overdispersion in biological
data is an abundance of zero counts. To illustrate, let’s randomly
replace 30 observations of S.d
with zeros. This can easily be done
using the following code.
> m3 = glm(S.d ~ log(debris.wt), family = poisson,
+ d)
> summary(m3)
Call:
glm(formula = S.d ~ log(debris.wt), family = poisson, data = d)
Deviance Residuals:
Min 1Q Median 3Q Max
5.1754 0.7284 0.2409 1.0149 2.9512
Coefficients:
Estimate Std. Error z value Pr( > z)
(Intercept) 1.25268 0.07538 16.62 < 2e16 ***
log(debris.wt) 0.28483 0.02377 11.98 < 2e16 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 789.1 on 189 degrees of freedom
Residual deviance: 634.0 on 188 degrees of freedom
AIC: 1273.7
Number of Fisher Scoring iterations: 5
Here, we see that the residual deviance is several times larger than the residual df. What to do?
Well, at this point, it is worth asking, “is there any potentially
important drivers that have been left out of this analysis that could be
confounding these results?” If, after careful consideration, you wish to
proceed, there are options for attempting to deal with the
overdispersion. R implements two distributions within glm
that are
useful in such situations. These are the “quasibinomial” and the
“quasipoisson”. In these distributions, the dispersion parameter,
$\tau $, instead of being fixed at 1, as is the case of the Binomial
and Poisson, is estimated as if it were an unknown parameter. As was
mentioned in section 1.2.2, the variance of the error
distribution for the quasibinomial is N$\rho (1\rho )\tau $. For
the quasipossson, it is $\lambda \tau $. The dispersion parameter
estimate,$\hat\tau $, then is used in the calculation of standard
errors.
+ d)
> summary(m4)
Call:
glm(formula = S.d ~ log(debris.wt), family = quasipoisson, data = d)
Deviance Residuals:
Min 1Q Median 3Q Max
5.1754 0.7284 0.2409 1.0149 2.9512
Coefficients:
Estimate Std. Error t value Pr( > t)
(Intercept) 1.25268 0.11460 10.93 < 2e16 ***
log(debris.wt) 0.28483 0.03615 7.88 2.56e13 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for quasipoisson family taken to be 2.311609)
Null deviance: 789.1 on 189 degrees of freedom
Residual deviance: 634.0 on 188 degrees of freedom
AIC: NA
Number of Fisher Scoring iterations: 5
Notice that much of the output is exactly the same as we saw for the loglinear model. The only exceptions are:
a value for the dispersion parameter is reported,
the standard error estimates have changed and are now tested using ttests, and
the AIC is not reported.
Parameter estimates and values for deviances are unchanged. However, instead of relying on maximum likelihood estimation, we are really maximizing the “quasilikelihood.” As a result, the ANODEV table can no longer rely on the likelihood ratio $\chi ^{2}$ test (and the AIC is undefined). Instead, an approximate Ftest that uses the dispersion parameter can be requested.
Analysis of Deviance Table
Model: quasipoisson, link: log
Response: S.d
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev F
NULL 189 789.1
log(debris.wt) 1 155.1 188 634.0 67.096
Pr( > F)
NULL
log(debris.wt) 3.879e14 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
The results of the Ftest are consistent with the results of the ttest
found in the model summary. There is a significant relationship between
S.d
and log(debris.wt)
.

This figure is based, in part, on p. 511 in [ ↩