# Estimating parameters of the Poisson distribution using multiple datasets

I'm reading a paper Henzi et al. 2010. Infanticide and reproductive restraint in a polygynous social mammal. PNAS 107: 2130-2135 where they test the fit of a data to a Poisson distribution. However, I'm struggling to understand their fitting procedure and there is no mention of it in the Methods section.

The basic problem is that they have several different time intervals of varying lengths (in this case alpha male tenure lengths i.e. the length of time an alpha male spends as the alpha male), and a number of events (in this case subordinate male conceptions) within each time interval. They expect the baseline distribution of events against time interval to follow a Poisson distribution. Essentially they want to test whether events follow a Poisson distribution in relation to time interval length.

I'm struggling to understand their rate estimation procedure. If we summed all the time intervals together and divided by the number of events to estimate the Poisson mean. We could then use that mean to generate an expected distribution and test the fit to the total dataset. However, they want to know if the distribution of events WITHIN each time interval follows a Poisson distribution.

My questions are then

1. can you calculate the mean events per unit time for each of the different time intervals. Then, use those means from all the different time intervals to calculate a global mean and generate an expected Poisson distribution for each time interval of different length to compare the observed events within each time interval and

2. is that what the authors possibly did?

Say you have observed time intervals $$T_i$$ (alpha male tenure lengths) and counts $$N_i$$ within those intervals. Then use a Poisson regression (with log link function) for $$N_i$$ with offset $$\log( T_i )$$ (here $$T_i$$ doubles as time interval and its length.) See for details Scaling vs Offsetting in Quasi-Poisson GLM or search this site.
mod0 <- glm(Ni ~ offset(log(Ti)) + <other variables>, family=poisson, data=your_data_frame)