Poisson was a French mathematician, and amongst the many contributions he made, proposed the Poisson distribution, using the example of 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 observation 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.

Poisson 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) values, 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.

Some clarification of nomenclature may be useful.

• Counts of events (e.g. number of asthma attacks recorded) are represented by k. This count 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 mean count, or count rate (k/n) is represented by λ. e.g. 100 attacks(k) in 1800 children months (n) produces λ=100/1800 = 0.06 attacks per child month (λ)

The following are examples of how the Poisson Model and test are used

• Example 1: Looking back, the number of patient complaints per month in a hospital ward was 2,5,4, and 3 in the last 4 months, averaged to 3.5 per month (λ). Since a new ward manager arrived the number of complaint last month was 6 (k). We want to know what was the probability that this count was no different to 3.5.

  Using the Poisson Test, we found that the probability of 6 or more complaints being the same as λ=3.5 is 0.14. If we use p< 0.05 as a decision criteria, then we cannot confidently conclude that 6 is greater than the mean of 3.5.

• Example 2: Following the events in Example 1, in the same ward, the total number of complains over the following 6 months was 30 (averaged to 5 per month). We now want to know whether this should be considered as higher than the previous average of 3.5.

  If the mean (λ) is 3.5 per month, then over 6 months, the number of complaints expected is 6 x 3.5 = 21. Using the Poisson Test, the probability of having 30 or more complaints when the expected was 21 is 0.0374. If we use p<0.05 as the decision criteria, then we can conclude that 30 complaints over 6 months is significantly greater than the expected average of 3.5 per month.

• Example 3: We wish to establish some standards for detecting blood in the urine by microscopic examination. If the urine is normal, we expect, on average, 3 or less red blood cells per high power field under the microscope. We wish to establish the maximum total red cell count over 5 high power fields acceptable for normality.

  If the average (λ) is 3/hpf, then the expected total count over 5 fields is 15. Using a sequence of Poisson Probability Tests, the probabilities and cumulative probabilities of observing counts from 0 to 25 are shown in the following table.

  Expected λ	Observed k	Probability <=k	Probability >=k
  15	0	<0.0001	>0.9999
  15	1	<0.0001	>0.9999
  15	2	<0.0001	>0.9999
  15	3	0.0002	>0.9999
  15	4	0.0009	0.9998
  15	5	0.0028	0.9991
  15	6	0.0076	0.9972
  15	7	0.018	0.9924
  15	8	0.0374	0.982
  15	9	0.0699	0.9626
  15	10	0.1185	0.9301
  15	11	0.1848	0.8815
  15	12	0.2676	0.8152
  15	13	0.3632	0.7324
  15	14	0.4657	0.6368
  15	15	0.5681	0.5343
  15	16	0.6641	0.4319
  15	17	0.7489	0.3359
  15	18	0.8195	0.2511
  15	19	0.8752	0.1805
  15	20	0.917	0.1248
  15	21	0.9469	0.083
  15	22	0.9673	0.0531
  15	23	0.9805	0.0327
  15	24	0.9888	0.0195
  15	25	0.9938	0.0112

  From column 4, the probability of observing 22 or more cells is 0.0531, and 23 or more cells 0.0327. If we use p<0.05 as unlikely, we can conclude that we are unlikely to see a total of 23 or more cells in 5 high power fields if we expect an average of 3 per high power field. We can therefore use observing a total of 23 or more red blood cells in 5 high power fields as the diagnostic criteria for detecting blood in the urine.

References

Poisson Distribution by Wikipedia

Steel RGD, Torrie JH Dickey DA (1997) Principles and Procedures of Statistics. A Biomedical Approach. The McGraw-Hill Companies, Inc New York. p. 558

Data	Data Entry The data is a 2 column table of numbers, each row a separate test   - The columns are separated by white spaces (space or tab)   - Column 1 is the Reference or expected mean count λ (Must be +ve value)   - Column 2 is the observed count k (must be a +ve integer value)

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