Negative Binomial Regression: Offset Variable and Dispersion Parameter

Question

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.

Answer 1

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

1. 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).
2. 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 $\mu_j$.