The Binomial Intersection Distribution

Description

Density, distribution function, quantile function and random generation for the binomial intersection distribution.

Usage

dbint(n, A, range = NULL, log = FALSE)

pbint(n, A, vals, upper.tail = TRUE, log.p = FALSE)

qbint(p, n, A, upper.tail = TRUE, log.p = FALSE)

rbint(num = 5, n, A)

Arguments

Details

The binomial intersection distribution is given by

P(X = v|N) = {b \choose v} \left(\prod_{i=1}^{N-1} p_{i}\right)^{v} \left(1 - \prod_{i=1}^{N-1} p_{i}\right)^{b-v}

where b gives the sample size which is smallest. This is an approximation for the hypergeometric intersection distribution when n is large and b is small relative to the samples taken from the N-1 other urns.

Examples

## Generate the distribution of intersections sizes:
dd <- dbint(20, c(10, 12, 11, 14))
## Restrict the range of intersections.
dd <- dbint(20, c(10, 12), range = 0:5)
## Generate cumulative probabilities.
pp <- pbint(29, c(15, 8), vals = 5)
pp <- pbint(29, c(15, 8), vals = 2, upper.tail = FALSE)
## Extract quantiles:
qq <- qbint(0.15, 23, c(12, 10))
## Generate random samples from Binomial intersection distributions.
rr <- rbint(num = 10, 18, c(9, 14))

Drawing Distinct Categories from a Single Urn

Description

Density, distribution function, quantile function and random generation for the distribution of distinct categories drawn from a single urn in which there are duplicates in q of the categories.

Usage

dhydist(n, a, q, range = NULL, log = FALSE)

phydist(n, a, q, vals, upper.tail = TRUE, log.p = FALSE)

qhydist(p, n, a, q, upper.tail = TRUE, log.p = FALSE)

rhydist(num = 5, n, a, q)

Arguments

Examples

## Generate the distribution of distinct categories drawn from a single urn.
dd <- dhydist(20, 10, 12)
## Restrict the range of intersections.
dd <- dhydist(20, 10, 12, range = 5:10)
## Generate cumulative probabilities.
pp <- phydist(29, 15, 8, vals = 5)
pp <- phydist(29, 15, 8, vals = 2, upper.tail = FALSE)
## Extract quantiles:
qq <- qhydist(0.15, 23, 12, 10)
## Generate random samples based on this distribution.
rr <- rhydist(num = 10, 18, 9, 12)

The Hypergeometric Intersection Family of Distributions

Description

The Hypergeometric Intersection Family of Distributions

Usage

dhint(n, A, q = 0, range = NULL, approx = FALSE, log = FALSE, verbose = TRUE)

phint(n, A, q = 0, vals, upper.tail = TRUE, log.p = FALSE)

qhint(p, n, A, q = 0, upper.tail = TRUE, log.p = FALSE)

rhint(num = 5, n, A, q = 0)

Arguments

Details

The hypergeometric intersection distributions describe the distribution of intersection sizes when sampling without replacement from two separate urns in which reside balls belonging to the same n object categories. In the simplest case when there is exactly one ball in each category in each urn (symmetrical, singleton case), then the distribution is hypergeometric:

P(X=v)=\frac{{a \choose v}{n-a \choose b-v}}{{n \choose b}}

When there are three urns, the distribution is given by

P(X=v) = \frac{ {a \choose v} \sum_{i} {a-v \choose i} {n-a \choose b-v-i} {n-v-i \choose c-v} }{ {n \choose b} {n \choose c} }

If, however, we allow duplicates in q \leq n of the categories in the second urn, then the distribution of intersection sizes is described by the following variant of the hypergeometric:

P(X=v) = \sum_{m=0}^{\alpha} \sum_{l=0}^{\beta} \sum_{j=0}^{l} {n-q \choose v-l} {q \choose l} {q-l \choose m} {n-v-q+l \choose a-v-m} {l \choose j} {n+q-a-m-j \choose b-v} / {n \choose a}{n+q \choose b}

Value

'dhint', 'phint', and 'qhint' return a data frame with two columns: v, the intersection size, and p, the associated p-values. 'rhint' returns an integer vector of random samples based on the hypergeometric intersection distribution.

References

Kalinka, A. T. (2013). The probability of drawing intersections: extending the hypergeometric distribution. arXiv.1305.0717

Examples

## Generate the distribution of intersections sizes without duplicates:
dd <- dhint(20, c(10, 12))
## Restrict the range of intersections.
dd <- dhint(20, c(10, 12), range = 0:5)
## Allow duplicates in q of the categories in the second urn:
dd <- dhint(35, c(15, 11), 22, verbose = FALSE)
## Generate cumulative probabilities.
pp <- phint(29, c(15, 8), vals = 5)
pp <- phint(29, c(15, 8), vals = 2, upper.tail = FALSE)
pp <- phint(29, c(15, 8), 23, vals = 2)
## Extract quantiles:
qq <- qhint(0.15, 23, c(12, 10))
qq <- qhint(0.15, 23, c(12, 10), 18)
## Generate random samples from Hypergeometric intersection distributions.
rr <- rhint(num = 10, 18, c(9, 14))
rr <- rhint(num = 10, 22, c(11, 17), 12)

add.distr

Description

This function will add one or more distributions or hypothesis tests to an existing plot.

Usage

add.distr(..., cols = "blue", test.cols = "red")

Arguments

Value

Plots to the current device.

hint.dist.test

Description

Tests whether the absolute distance between two intersection sizes would be expected by chance, i.e. whether they fall into opposite tails of their respective Hypergeometric Intersection distributions.

Usage

hint.dist.test(d, n1, A1, n2, A2, q1 = 0, q2 = 0, alternative = "greater")

Arguments

Details

The distribution of absolute distances between two hypergeometric intersection sizes is given by

P(X=d) = \sum_{\{v_{1},v_{2}\}_{i} \in D_{d}}^{|D_{d}|} P(v_{1_i}|n_{1},a_{1},b_{1},...)\cdot P(v_{2_i}|n_{2},a_{2},b_{2},...)

where D_{d} is the set of pairs of intersection sizes, \{v_{1},v_{2}\}, with absolute differences of size d.

Value

An object of class hint.dist.test, which is a list containing the following components:

• parameters An integer vector giving the parameter values.

• p.value A numerical value giving the p-value associated with the test.

• alternative A character string naming the hypothesis that was tested.

hint.test

Description

Apply the hypergeometric intersection test to categorical data to test for enrichment or depletion of intersections between two samples.

Usage

hint.test(cats, draw1, draw2, alternative = "greater")

Arguments

Details

The hypergeometric intersection distributions describe the distribution of intersection sizes when sampling without replacement from two separate urns in which reside balls belonging to the same n object categories (see Hyperintersection).

Value

An object of class hint.test, which is a list containing the following components:

• parameters An integer vector giving the parameter values.

• p.value A numerical value giving the p-value associated with the test.

• alternative A character string naming the hypothesis that was tested.

References

Kalinka, A. T. (2013). The probability of drawing intersections: extending the hypergeometric distribution. arXiv.1305.0717

plot.hint.test

Description

This function visualises the results of a Hypergeometric Intersection test.

Usage

## S3 method for class 'hint.test'
plot(x, ...)

Arguments

Details

Plots the relevant Hypergeometric Intersection distribution as a segment plot, and highlights the region where the observed statistic falls, i.e. the region from which the probability is computed (two.sided tests are visualised in one tail, the one with the smallest density). This can be especially useful for pedagogical purposes.

Value

Plots to the current device.

plotDistr

Description

Plot a distribution or visualise the result of a hypothesis test.

Usage

plotDistr(
  distr,
  col = "black",
  test.col = "red",
  xlim = NULL,
  ylim = NULL,
  xlab = "Intersection size (v)",
  ylab = "Probability",
  add = FALSE,
  ...
)

Arguments

Details

Visualising the results of a hypothesis test may often be of interest, but can be especially useful for pedagogical purposes.

Value

Plots to the current device.

print.hint.test

Description

Prints the resuls of 'hint.test'.

Usage

## S3 method for class 'hint.test'
print(x, ...)

Arguments

Value

Prints output to the console.