# Poisson regression on the means of count data

I just finished a small research project about hummingbirds and the effect of hummingbird feeders. I am a bit unsure about how to proceed with the statistics.

We placed 15 points in a distance gradient away from the feeders, where we sampled visitation rates, pollination and bird / floral abundance (all count data). We sampled all points twice, what I would like to do is to use Poisson regression on the mean of the two samples. Does that makes statistical sense? First, I would guess that using the means would push the data distribution towards normality. Second, can you use Poisson regression on non whole numbers?

• (1) If you have replicates why not analyze them as such? Why do you want to take means first? (2) How are counts not whole numbers? Are you dividing by the duration of the observation period to get rates? Or is it just that the means are fractions? – Scortchi Jun 23 '14 at 9:53
• Is there a particular reason you need to average the data before modelling it? – Glen_b Jun 23 '14 at 12:12
• @Scortchi, I think he means that for unit $i$, he measured twice (eg, 4 & 5), so the mean of $i$'s measurements is 4.5--no longer a whole number. – gung Jul 3 '14 at 15:42
• Can you say what the two measurements are? Are they before & after? If so, you could simply use the before data as a covariate, & not even have to worry about using a GLMM. – gung Jul 3 '14 at 15:43

First, I would guess that using the means would push the data distribution towards normality.

Well, there is no assumption about normality is Poisson regression, so no need.

Second, can you use Poisson regression on non whole numbers?

Well, yes, (in R) it is called quasi-Poisson regression. But if you have the original counts, as you should, there is no need. Just use Poisson regression on the original count. For further advice we would need more context.

Firstly, it is almost certainly better to use the original counts (Poisson regression assumes whole numbers). If you divided by time (i.e. counts per hour or day), you can use a log-time offset (in fact, you would want to do that for animal counts).

Secondly, if you really must analyze the mean rates (e.g. you did not retain the underlying counts), you would probably want to log-transform the mean rates. However, the annoying thing would be that higher values should have higher variance (the sampling distribution would for the observed log rate under a Poisson distribution is asymptotically normal with mean true log rate and standard deviation 1/sqrt(observed count)).

Thirdly, observations from the same spot are almost certainly correlated, you could either just sum them up to a total count, which would be fine if there are no differences between the two separate observations per spot that you want/can take into account. Or you model them as separate count data observations, but at a minimum you would want to take the correlation into account by e.g. adding a location random effect on the intercept of a Poisson regression model. You can go further by modeling the spatial correlation (presumably counts at spots that are less far apart are correlated, because the same animals are more likely to wander from one spot to the other, plants are more likely to spread there etc.).

Finally, how do you put distance into the model? E.g. a non-linear function like a spline function might be worth considering.