Simple Enrichment Test -- calculate hypergeometric p-values in R

Hypergeometric test are useful for enrichment analysis. For example, having a gene list in hand, people might want to tell which functions (GO terms) are enriched among these genes. Hypergeometric test (or its equivalent: one-tailed Fisher's exact test) will give you statistical confidence in \(p\)-values.

R software provids function phyper and fisher.test for Hypergeometric and Fisher's exact test accordingly. However, it is tricky to get it right. I spent some time to make it clear.


Here is a simple example:
Five cards were chosen from a well shuffled deck
x = the number of diamonds selected.
We use a 2x2 table to represent the case:

                Diamond     Non-Diamond
selected        x                     5-x               total 5 sampled cards
left               13-x                 34+x             total 47 left cards after sampling
                 13 Dia          39 Non-Dia         total 52 cards

We 're asking if diamond enriched or depleted in our selected cards, comparing to the background.

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Here are the different parameters used by phyper and fisher.test:

phyper(x, 13, 39, 5, lower.tail=TRUE);

# Numerical parameters in order:

# (success-in-sample, success-in-bkgd, failure-in-bkgd, sample-size).

fisher.test(matrix(c(x, 13-x, 5-x, 34+x), 2, 2), alternative='less');

# Numerical parameters in order:

# (success-in-sample, success-in-left-part, failure-in-sample, failure-in-left-part).

It's obvious that hypergeometric test compares sample to bkgd, while fisher's exact test compares sample to the left part of bkgd after sampling without replacement. They will give the same p-value (because they assume the same distribution).

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Here is the results:

x=1; # x could be 0~5 

hitInSample = 1  # could be 0~5

hitInPop = 13 

failInPop = 52 - hitInPop 

sampleSize = 5

• Test for under-representation (depletion)

phyper(hitInSample, hitInPop, failInPop, sampleSize, lower.tail= TRUE);

## [1] 0.6329532

fisher.test(matrix(c(hitInSample, hitInPop-hitInSample, sampleSize-hitInSample, failInPop-sampleSize +hitInSample), 2, 2), alternative='less')$p.value; 

## [1] 0.6329532

• Test for over-representation (enrichment)

phyper(hitInSample-1, hitInPop, failInPop, sampleSize, lower.tail= FALSE);

## [1] 0.7784664

fisher.test(matrix(c(hitInSample, hitInPop-hitInSample, sampleSize-hitInSample, failInPop-sampleSize +hitInSample), 2, 2), alternative='greater')$p.value; 

## [1] 0.7784664

•  Why hitInSample-1 when testing over-representation?

Because if lower.tail is TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. We subtract x by 1, when P[X ≥ x] is needed.

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So are there any advantages fisher.test has over phyper, as they give the same p-values?
Yes, fisher.test can do two other jobs: two-side test, and giving confidence intervals of odds ratio. Please refer to its manual for details. For one-side p-value calculating, they don't have any difference if correct parameters were used.