# Poisson glm for rank or score data?

I have a question about analyzing a dataset that I'm currently working with. Each row of the dataset represents an individual songbird, and its reproductive success over the course of a breeding season. Reproductive success was recorded as a score or rank that was based on breeding activities that we observed for each bird. Scores were recorded as follows:

1 - unpaired
2 - paired
3 - successfully raised 1 brood of fledglings
4 - successfully raised 2 broods
5 - successfully raised 3 broods


In my analysis, these scores will be the response variables, and several environmental covariates will be used as predictor variables. Typically I see that ranked data is analyzed using ordinal logistic regression, but would it also be reasonable to model this data using a poisson glm/glmm? I have experience with poisson glm's, and this distribution is often used in my field (wildlife ecology) for count data or to model age structure. This reproductive index is not commonly used, so there are not many examples I've come across that attempted to do a similar analysis.

Thanks! Jay

There's no reason to think it would well modelled by a Poisson

• The Poisson has a non-zero probability of "0", your categories don't include 0

• the variable is bounded (it cannot exceed 5), the Poisson is not

• the categories are ordered but the values of your category labels are not meaningful; for example, there's no reason to regard three "1"'s and a "2" to be equivalent to a "5" -- the gaps between adjacent categories are nothing alike. So modelling the mean doesn't make sense.

• the Poisson cannot be bimodal (except in the sense that two adjacent values can both be modes). The categorical variable could theoretically be almost all 1's and all 5's in the population. In the Poisson, if the mean is 3.2, the mode must be 3.

• the Poisson specifies variance = mean; there's no reason to expect that this will be close to true in your data.

In short, this doesn't seem like you should expect this would work well.

What would the coefficients in your model mean? How would you interpret them?

• Hi Glen, Those points sound fair enough. I guess the only reason I ask is that I read a paper today that had used a poisson generalized linear mixed model to fit the relationship between time since fire and woodpecker age classes (ages were categorized as 2nd year, 3rd year, 4th year, after 4th year). That paper also does not seem to adhere to those points you laid out, and was in a well-regarded paper. Not arguing, just curious how the woodpecker example with age categories is acceptable. – Jason Dec 18 '15 at 23:07
• @Jason I assume that woodpecker age was the response? At least the gaps in age groups (aside from the last one) are equal and "mean age" is meaningful (if slightly screwed up by the last category -- if get woodpeckers were 6th year or older that would have little effect). So not everything in that analysis would be as meaningless is it would be for what you proposed -- at least a model for the mean has meaning, and the coefficients can (more or less) be interpreted in that light. I'd still list objections, but it has a somewhat better chance of not being completely terrible. – Glen_b Dec 18 '15 at 23:47