# Overdispersed poisson-distribution and offset --> standard errors?

I am modeling count data using R and doing a fixed-effects/random-effects model and thus limited on functions and therefore cannot use a quasipoisson model or negative binomial distribution, but poisson.

The outcome in my model is number of sickness absence spells a person has during the exposure time. This is done for a number of periods for the same group of people and thus fixed/random-effects model seems appropriate.

The count data is rather overdispersed (mean = 4.66, var = 28.81 --> var/mean ~6), but the exposure time during which the sickness absence spells are counted also varies a lot from person to person (between 1 and 1095 days). Therefore the actual variable to be explained is rate data. The common method to "compensate" for this variance in exposure is to use it as an offset in the model [in R:

sickness_absence ~ explaining_variable1+
explaining_variable2+offset(log(exposure))


Because of the log-link of the Poisson-distribution, it seems to me that the offset is actually used to divide the explained variable (sickness_absence in this case). If I calculate the var(sickness_absence / exposure)/mean(sickness_absence / exposure) it will be 0.5.

Does this mean that using a Poisson distribution with offset would be justified and overdispersion wouldn't need to be addressed when calculating the standard errors? Or can the effect be considered a minor one and can be mitigated by correcting the standard errors by robust or sandwich estimator?

## 1 Answer

tl;dr the offset might ameliorate your apparent overdispersion, but it's probably better to evaluate overdispersion based on the conditional distribution (e.g., compute (residual deviance/residual df) or (sum(Pearson resids^2)/residual df))) rather than trying to guess from the marginal distribution of your data. You're presumably fitting a model with some covariates, so the variance/mean ratio of your response variable alone doesn't tell you very much at all.

That said, I'm not sure your options for handling overdispersion are as limited as you think they are:

• quasi-likelihood is possible, either in MASS::glmmPQL or rolling it yourself;
• lme4 has a glmer.nb function; glmmTMB has two different negative binomial families
• you can use observation-level random effects to account for overdispersion

See the relevant section of the GLMM FAQ for more info on all of these options.