scipy.stats.fisher_exact

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

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

The null hypothesis is that the true odds ratio of the populations underlying the observations is one, and the observations were sampled from these populations under a condition: the marginals of the resulting table must equal those of the observed table. The statistic returned is the unconditional maximum likelihood estimate of the odds ratio, and the p-value is the probability under the null hypothesis of obtaining a table at least as extreme as the one that was actually observed. There are other possible choices of statistic and two-sided p-value definition associated with Fisher’s exact test; please see the Notes for more information.

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’: the odds ratio of the underlying population is not one

• ‘less’: the odds ratio of the underlying population is less than one

• ‘greater’: the odds ratio of the underlying population is greater than one

See the Notes for more details.

Returns

oddsratiofloat

This is prior odds ratio and not a posterior estimate.

p_valuefloat

P-value, the probability under the null hypothesis of obtaining a table at least as extreme as the one that was actually observed.

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 true odds ratio of the populations underlying the observations is one, and the observations were sampled at random from these populations under a condition: the marginals of the resulting table must equal those of the observed table. Equivalently, 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.