Hypergeometric test

Definition

$$p=\sum_{i=E}^{n}\frac{\binom{M}{i}\binom{N-M}{n-i}}{\binom{N}{n}}$$

• N = size of population
• M = number of items in population with property 'E'
• N-M = number of items in population without property 'E'
• n = number of items sampled
• i = number of items in the sample with property 'E'

Example

R script

• package: Hypergeometric {stats}

• original function prototype

# dhyper gives the density
dhyper(x, m, n, k, log = FALSE)

# phyper gives the distribution function
phyper(q, m, n, k, lower.tail = TRUE, log.p = FALSE)

# qhyper gives the quantile function
qhyper(p, m, n, k, lower.tail = TRUE, log.p = FALSE)

# rhyper generates random deviates.
rhyper(nn, m, n, k)

• parameters

x,q: vector of quantiles representing the number of white balls drawn without replacement from an urn which contains both black and white balls.
m: the number of white balls in the urn.
n: the number of black balls in the urn.
k: the number of balls drawn from the urn.

• Example.1 in R

# prepare data
m <- 10; n <- 7; k <- 8
x <- 0:(k+1)

# start to calculate hypergeometric test
phyper(x, m, n, k)    # test result
dhyper(x, m, n, k)    # distribution density

• Example.2 in R

# prepare data
# pop size : 5260
# sample size : 131
# Number of items in the pop that are classified as successes : 1998
# Number of items in the sample that are classified as successes : 62

# data
#            population            
# condition        sample        others        total
#    success        62        1936        1998
#    failure        69        3193        3262
# total            131        5129        5260

# start calculate hypergeometric test (phyper)
# with description
# phyper(white balls drawn, total white balls, total black balls, totally drawn)
phyper(62, 1998, 5260-1998, 131)