Poisson or Binomial distribution for modeling?

Question

I want to model the change in rate of an event in a sampled population. The populations are grouped according to some grouping variable. Each individual observation is of one organism, and the sample can either return as positive or negative (+/-). My resulting parameters are:

-number of positive samples (+)

-total number tested per grouping variable

Should this be modeled using a Poisson or binomial distribution? My sample parameters are meant to estimate the true rate value in an overall larger population.

Answer 1

As in this question, in typical count data situations, the rule of thumb is:

• if the count has a ceiling/maximum value set by the experimental design, model the response as binomial (or some over/underdispersed variant: quasibinomial, beta-binomial, observation-level random effect ...)
• if there is no well-defined limit (e.g. the number of trees in a 1-hectare plot can't be infinite, but we can't typically quantify the number of available "tree sites" that are available) then use a Poisson response (or some variant: quasi-Poisson, negative binomial, generalized Poisson, COM-Poisson ...)
• if the count has a maximum value but the proportion of the maximum is always small (e.g. the number of cancer cases in a county), then a binomial and a Poisson with a log-offset term to scale the maximum value will give nearly identical results, and it's a matter of computational convenience.

If you want a literature reference I have one for you :-): Bolker, Benjamin M. Ecological Models and Data in R Princeton University Press, 2008:

p. 161:

You should only use the binomial in fitting data when there is an upper limit to the number of possible successes. When $N$ is large and $p$ is small, so that the probability of getting N successes is small, the binomial approaches the Poisson distribution, which is covered in the next section (Figure 4.17). Examples: number of surviving individuals/nests out of an initial sample; number of infested/infected animals, fruits, etc. in a sample; number of a particular class (haplotype, subspecies, etc.) in a larger population.

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p. 163:

The Poisson distribution has no upper limit, although values much larger than the mean value are highly improbable. This characteristic provides a rule for choosing between the binomial and Poisson. If you expect to observe a “ceiling” on the number of counts, you should use the binomial; if you expect the number of counts to be effectively unlimited, even if it is theoretically bounded (e.g. there can’t really be an infinite number of plants in your sampling quadrat), use the Poisson.

Examples: number of seeds/seedlings falling in a gap; number of offspring produced in a season (although this might be better fit by a binomial if the number of breeding attempts is fixed); number of prey caught per unit time.

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PS it turns out I am quite consistent: I wrote this answer without looking at the examples I used in 2008, and came up with almost the same wording and examples ...

Answer 2

Poisson and Binomial distributions both model the counts of events.

The Binomial distribution is about the number of counts of events in a sequence of experiments that measure a boolean valued outcome. The events can only occur on a fixed number of cases/experiments.

The Poisson distribution is about the number of counts of events in an interval of time or space. The events can occur continuously.

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See also on Wikipedia:

https://en.wikipedia.org/wiki/Poisson_distribution

In probability theory and statistics, the Poisson distribution (/ˈpwɑːsɒn/; French pronunciation: ​[pwasɔ̃]), named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

https://en.wikipedia.org/wiki/Binomial_distribution

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).