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Explanations
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.
Javascript Program
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.
ReferencesPoisson Distribution by WikipediaSteel RGD, Torrie JH Dickey DA (1997) Principles and Procedures of Statistics. A Biomedical Approach. The McGraw-Hill Companies, Inc New York. p. 558
R Codes for Poisson Test
The probabilities of obstaining a count >=k or <=k, given the average count (lambda) Function PoissonP: The probability of observing count k given average count lambda # function PoissonP <- function(lambda,k) # lambda is expected count, k is observed count { return (exp(log(exp(-lambda)) + log(lambda) * k - log(factorial(k)))) }The main sequence of the program myDat = (" Lambda K 2.2 4 6.7 5 10.9 15 ") df <- read.table(textConnection(myDat),header=TRUE) # conversion to data frame df # display input data (lambda and k) PLtEq <- vector() # probability of a count <= k PGtEq <- vector() # probability of a count >= k for(i in 1:nrow(df)) { c = 0 for(j in 0:df$K[i]) { p = PoissonP(df$Lambda[i],j) c = c + p } PLtEq = append(PLtEq, c) PGtEq = append(PGtEq, 1 - c + p) } df$PLtEq <- PLtEq df$PGtEq <- PGtEq df # display data frame including Probabilities of a count <= k and >=KThe results are > df # display input data (lambda and k) Lambda K 1 2.2 4 2 6.7 5 3 10.9 15 > df # display data frame including Probabilities of a count <= k and >=K Lambda K PLtEq PGtEq 1 2.2 4 0.9275037 0.1806476 2 6.7 5 0.3406494 0.7978410 3 10.9 15 0.9126344 0.1387804 |