Negative Binomial Regression: Offset Variable and Dispersion Parameter My case is that previously it's assumed that the counts of events follows a negative binomial distribution, and the annualized exacerbation rate is 1 with a dispersion parameter of 1.5, and all 500 patients will be followed up for 1 year. But due to some operation reasons, the follow-up period have to be shorten from 1 year to around 0.5 year for each patient. My understanding is that the annualized exacerbation rate is still 1, just including 0.5 year as a offset variable when doing the NB regression (correct me if I am wrong). My question is will the dispersion parameter be impacted due to the fact that the followup period become shorter? (I have to do some simulation using R, i need to know how should the dispersion parameter as well as the mean of the NB distribution be setup).
Thank you.
 A: Using an log(duration of follow-up) offset (i.e. log(0.5) rather than 0.5) is indeed one way of doing this. Note that, if you have drop-outs this targets a hypothetical estimand (i.e. implicitly imputes data, as if all patients had continued as before the drop-out), which many would consider sensible, but may not be what all stakeholders may desire as the estimand in a regulatory clinical trial.
The disperion parameter indicates how much the true annualized event rate differs between patients. That suggest that it could be affected by the duration e.g. because of


*

*seasonal variability: if the outcome varies by season, then a 1-year trial approximately balances this out, while in a 0.5 year trial patients will differ more depending on when they are included (i.e. higher between patient variability in half-year trial).

*divering patient characteristics: Patients being more similar at the start of the trial, but developing differently over time (i.e. lower between patient variability in half-year trial).


Particularly point (1) can be addressed by simulating data separately for each month to see the imapct. For that you can exploit that the negative binomial is a gamma-Poisson mixture. I.e. you can create a random patient effect $u_i$ from a gamma distribution and then simulation each month as $Y_{ij} \sim \text{Poisson}(u_i \times \mu_j / 12)$, if the assumed annualized rate in month $j$ is $\mu_j$.
