Can studies where subjects have different follow-up times be analysed through binomial regression?

Penna (2011) asks whether subjects with different follow-up times can be analysed through binomial regression. The answer to this question is “yes”. If the rate for a particular person is λ and they have been observed for a period of time (t), then the probability (p) of having an event during that time is equal to 1-e-λt. The equation can be re-arranged to give log[log(1-p)] = log(λ)+log(t). This is a linear function of the log rate, enabling it to be modelled by regression, while the follow-up time values (t) can vary between individuals (Collett 1991). This approach is useful when the exact time of the event is unknown as, for example, in seroconversion studies like that of Gray et al. (2001). The regression technique is a generalised linear model with “complementary log-log link function”; the logarithm of time is used as an “offset”. As it uses the binomial distribution family, the heading “binomial regression” is appropriate. However, it differs from the model used by Mastrangelo et al. (2011) in that it yields rate ratios rather than risk differences.