scipy.stats.fisher_exact

scipy.stats.fisher_exact(table, alternative='two-sided')

Perform a Fisher exact test on a 2x2 contingency table.

Parameters

tablearray_like of ints

A 2x2 contingency table. Elements must be non-negative integers.

alternative{‘two-sided’, ‘less’, ‘greater’}, optional

Defines the alternative hypothesis. The following options are available (default is ‘two-sided’):

• ‘two-sided’

• ‘less’: one-sided

• ‘greater’: one-sided

See the Notes for more details.

Returns

oddsratiofloat

This is prior odds ratio and not a posterior estimate.

p_valuefloat

P-value, the probability of obtaining a distribution at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.

See also

chi2_contingency

Chi-square test of independence of variables in a contingency table. This can be used as an alternative to fisher_exact when the numbers in the table are large.

barnard_exact

Barnard’s exact test, which is a more powerful alternative than Fisher’s exact test for 2x2 contingency tables.

boschloo_exact

Boschloo’s exact test, which is a more powerful alternative than Fisher’s exact test for 2x2 contingency tables.

Notes

Null hypothesis and p-values

The null hypothesis is that the input table is from the hypergeometric distribution with parameters (as used in hypergeom) M = a + b + c + d, n = a + b and N = a + c, where the input table is [[a, b], [c, d]]. This distribution has support max(0, N + n - M) <= x <= min(N, n), or, in terms of the values in the input table, min(0, a - d) <= x <= a + min(b, c). x can be interpreted as the upper-left element of a 2x2 table, so the tables in the distribution have form:

[  x           n - x     ]
[N - x    M - (n + N) + x]

For example, if:

table = [6  2]
        [1  4]

then the support is 2 <= x <= 7, and the tables in the distribution are:

[2 6]   [3 5]   [4 4]   [5 3]   [6 2]  [7 1]
[5 0]   [4 1]   [3 2]   [2 3]   [1 4]  [0 5]

The probability of each table is given by the hypergeometric distribution hypergeom.pmf(x, M, n, N). For this example, these are (rounded to three significant digits):

x       2      3      4      5       6        7
p  0.0163  0.163  0.408  0.326  0.0816  0.00466

These can be computed with:

>>> from scipy.stats import hypergeom
>>> table = np.array([[6, 2], [1, 4]])
>>> M = table.sum()
>>> n = table[0].sum()
>>> N = table[:, 0].sum()
>>> start, end = hypergeom.support(M, n, N)
>>> hypergeom.pmf(np.arange(start, end+1), M, n, N)
array([0.01631702, 0.16317016, 0.40792541, 0.32634033, 0.08158508,
       0.004662  ])

The two-sided p-value is the probability that, under the null hypothesis, a random table would have a probability equal to or less than the probability of the input table. For our example, the probability of the input table (where x = 6) is 0.0816. The x values where the probability does not exceed this are 2, 6 and 7, so the two-sided p-value is 0.0163 + 0.0816 + 0.00466 ~= 0.10256:

>>> from scipy.stats import fisher_exact
>>> oddsr, p = fisher_exact(table, alternative='two-sided')
>>> p
0.10256410256410257

The one-sided p-value for alternative='greater' is the probability that a random table has x >= a, which in our example is x >= 6, or 0.0816 + 0.00466 ~= 0.08626:

>>> oddsr, p = fisher_exact(table, alternative='greater')
>>> p
0.08624708624708627

This is equivalent to computing the survival function of the distribution at x = 5 (one less than x from the input table, because we want to include the probability of x = 6 in the sum):

>>> hypergeom.sf(5, M, n, N)
0.08624708624708627

For alternative='less', the one-sided p-value is the probability that a random table has x <= a, (i.e. x <= 6 in our example), or 0.0163 + 0.163 + 0.408 + 0.326 + 0.0816 ~= 0.9949:

>>> oddsr, p = fisher_exact(table, alternative='less')
>>> p
0.9953379953379957

This is equivalent to computing the cumulative distribution function of the distribution at x = 6:

>>> hypergeom.cdf(6, M, n, N)
0.9953379953379957

Odds ratio

The calculated odds ratio is different from the one R uses. This SciPy implementation returns the (more common) “unconditional Maximum Likelihood Estimate”, while R uses the “conditional Maximum Likelihood Estimate”.

Examples

Say we spend a few days counting whales and sharks in the Atlantic and Indian oceans. In the Atlantic ocean we find 8 whales and 1 shark, in the Indian ocean 2 whales and 5 sharks. Then our contingency table is:

        Atlantic  Indian
whales     8        2
sharks     1        5

We use this table to find the p-value:

>>> from scipy.stats import fisher_exact
>>> oddsratio, pvalue = fisher_exact([[8, 2], [1, 5]])
>>> pvalue
0.0349...

The probability that we would observe this or an even more imbalanced ratio by chance is about 3.5%. A commonly used significance level is 5%–if we adopt that, we can therefore conclude that our observed imbalance is statistically significant; whales prefer the Atlantic while sharks prefer the Indian ocean.

scipy.stats.contingency.association scipy.stats.barnard_exact