## Friday 3 November 2017

### Multivariate hypergeometric distribution in R

A hypergeometric distribution can be used where you are sampling coloured balls from an urn without replacement.

A univariate hypergeometric distribution can be used when there are two colours of balls in the urn, and a multivariate hypergeometric distribution can be used when there are more than two colours of balls.

The multivariate hypergeometric distribution can be used to ask questions such as: what is the probability that, if I have 80 distinct colours of balls in the urn, and sample 100 balls from the urn with replacement, that I will have at least one ball of each colour?

I recently have a similar problem in bioinformatics: if there are 6329 distinct copies of a gene in our eppendorph, and (after some PCRs) we sequence 10,000 distinct reads, what is the probability that we sequenced at least one read from each of the 6329 distinct copies of the gene? In other words, if there are 6329 distinct colours of balls in an urn, and we pick out 10,000 balls (without replacement), what is the probability that we chose at least one ball of each colour?

In R the dhyper function could be used in the case where there are two colours of ball in an urn, that is a univariate hypergeometric distribution. But what about where there are 6329 colours of ball, that is, a multivariate hypergeometric distribution? Then I came across the extraDistr R package, which has a multivariate hypergeometric distribution. The function is called "MultiHypergeometric".
I installed and load this library in R using:

If two cards are drawn from a deck of 52 cards, what is the probability that one is a spade and one is a heart?
I found a nice example of using the MultiHypergeometric distribution here by Jonathan Fivelsdal, thank you! He asked the question: if two cards are drawn from a deck of 52 cards, what is the probability that one is a spade and one is a heart? In R:
> dmvhyper(x = c('heart' = 1, 'spade' = 1, 'other' = 0), n = c('heart' = 13, 'spade' = 13, 'other' = 26), k=2)
[1] 0.127451
Here x is a vector that has the frequency of hearts and spades and other cards in our sample of interest,
n is a vector that has the frequency of hearts and spades and other cards in the deck of cards,
k is the number of cards in our sample of interest.
[Note: the manual for this package says k is 'the number of balls drawn from the urn'.]

Given a huge bag containing scrabble letters, if know the number of each letter tile in the bag, what is the probability of drawing a particular word of length N if we select out N letter tiles without replacement?

This is another nice example that I found here by Herb Susmann.
The answer is that we can do something like this:
> dmvhyper(x = letterfreqs, n = freqs, k = sum(letterfreqs))
where x = letterfreqs is a vector of length 26, with the frequency of each letter in the word of interest,
n is a vector of length 26 with the frequency of each letter tile in the bag,
k is the number of letters in the word of interest (of length N), ie. it is N.

If I have 6329*2=12,658 balls of 6329 distinct colours in an urn (two of each colour), and pick out 10,000 balls (without replacement), what is the probability that I chose at least one ball of each colour?
> dmvhyper(x = rep(1, 6329), n = rep(2, 6329), k = 10000, log = FALSE)
[1] 0
x is an m-column matrix of quantiles. From the example above by Jonathan Fivelsdal using a deck of cards, I think that this gives the counts of each of the colours of ball in the sample that you're asking about. In our case we are asking about having at least one ball of each colour.
The vector n is an m-length vector and has the number of balls in each of the m different colours. In our case m is 6329, so it is a 6329-length vector. Here we just assume we have 2 balls of each colour in the urn.
In this case k is the number of balls drawn from the urn, which is 10,000 in our case.

Notes to myself:
- in the case of our sequencing problem, we need to take into account the number of sequences in the urn, ie. the total number of sequences after the PCR.
- is sequencing molecules a sampling without replacement problem? Need to check...