# Sample size for comparing negative binomial rates in noninferiority and equivalence trials with unequal follow-up times

## TL;DR

This paper develops sample size formulas for comparing negative binomial rates in noninferiority and equivalence trials with unequal follow-up times, accounting for varying dispersion parameters and follow-up durations.

## Contribution

It introduces new sample size calculation methods that incorporate treatment-specific dispersion and unequal follow-up times, improving accuracy over existing approaches.

## Key findings

- Ignoring follow-up time variation underestimates sample size requirements.
- The proposed formulas are validated through simulations showing improved accuracy.
- Methods for estimating dispersion parameters from published data are provided.

## Abstract

We derive the sample size formulae for comparing two negative binomial rates based on both the relative and absolute rate difference metrics in noninferiority and equivalence trials with unequal follow-up times, and establish an approximate relationship between the sample sizes required for the treatment comparison based on the two treatment effect metrics. The proposed method allows the dispersion parameter to vary by treatment groups. The accuracy of these methods is assessed by simulations. It is demonstrated that ignoring the between-subject variation in the follow-up time by setting the follow-up time for all individuals to be the mean follow-up time may greatly underestimate the required size, resulting in underpowered studies. Methods are provided for back-calculating the dispersion parameter based on the published summary results.