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StatsToDo : Poisson Distribution : Explained

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Poisson Probability Comparing Two Counts References
Introduction Example 1 Example 2 Example 3
Poisson was a French mathematician, and amongst the many contributions he made, proposed the Poisson distribution, with the example of modelling the number of soldiers accidentally injured or killed from kicks by horses. This distribution became useful as it models events, particularly uncommon events.

Counts of events, based on the Poisson distribution, is a frequently encountered model in medical research. Examples of this are number of falls, asthma attacks, number of cells, and so on. The Poisson parameter Lambda (λ) is the total number of events (k) divided by the number of units (n) in the data (λ = k/n). The unit forms the basis or denominator for calculation of the average, and need not be individual cases or research subjects. For example, the number of asthma attacks may be based on the number of child months, or the number of pregnancies based on the number of women years in using a particular contraceptive.

This is different to the Binomial parameter of proportion or risk where proportion is the number of individuals classified as positive (p) divided by the total number of individuals in the data (r = p/n). Proportion or risk must always be a number between 0 and 1, while λ may be any positive number.

For examples, if we have 100 people, and only 90 of them go shopping in a week then the binomial risk of shopping is 90/100 = 0.9. However, some of the people will go shopping more than once in the week, and the total number of shopping trips between the 100 people may be 160, and the Poisson Lambda is 160/100 = 1.6 per 100 person week

Large Lambda (λ=k/n), say over 200, assumes an approximately normal or geometric distribution, and the count (or sqrt(count)) can be used as a Parametric measurement. If the events occur very few times per individual, so that individuals can be classified as positive or negative cases, then the Binomial distribution can be assumed and statistics related to proportions used. In between, or when events are infrequent, the Poisson distribution is used.

A detailed discussion of the use of Poisson related tests is in the reference listed below. Some clarification of nomenclature may be useful.

  • Counts of events (e.g. number of asthma attacks recorded) are represented by k (k1, and k2 for the 2 groups). These counts must be in terms of how many events over a defined period or environment (e.g. in 100 attacks in 300 children over 6 months, or 10 cells seen in 5 microlitres of fluid),
  • The denominator are represented by n (n1 and n2 for the 2 groups). e.g. 1800 children months, 5 microlitres.
  • The mean count, or count rate (k/n) is represented by λ for Lambda (λ1 and λ2 for the 2 groups). Λ and N are used in sample size and power calculations, while k and n are used in the test of statistical significance. e.g. 100 attacks(k) in 1800 children months (n) produces λ=100/1800 = 0.06 attacks per child month (λ)
  • Commonly, the one tail test is used for Poisson distribution, testing whether one group has more event than the other rather than whether the two groups are different without stating which one has more events.