scipy.stats.

# fisher_exact#

scipy.stats.fisher_exact(table, alternative='two-sided')[source]#

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:
resSignificanceResult

An object containing attributes:

statisticfloat

This is the prior odds ratio, not a posterior estimate.

pvaluefloat

The probability under the null hypothesis of obtaining a table at least as extreme as the one that was actually observed.

`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.

`contingency.odds_ratio`

Compute the odds ratio (sample or conditional MLE) for a 2x2 contingency table.

`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.

Fisher’s exact test

Extended example

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:

```>>> import numpy as np
>>> 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
>>> res = fisher_exact(table, alternative='two-sided')
>>> res.pvalue
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`:

```>>> res = fisher_exact(table, alternative='greater')
>>> res.pvalue
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`:

```>>> res = fisher_exact(table, alternative='less')
>>> res.pvalue
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 value computed by the R function `fisher.test`. This implementation returns the “sample” or “unconditional” maximum likelihood estimate, while `fisher.test` in R uses the conditional maximum likelihood estimate. To compute the conditional maximum likelihood estimate of the odds ratio, use `scipy.stats.contingency.odds_ratio`.

References

[1]

Fisher, Sir Ronald A, “The Design of Experiments: Mathematics of a Lady Tasting Tea.” ISBN 978-0-486-41151-4, 1935.

[2]

“Fisher’s exact test”, https://en.wikipedia.org/wiki/Fisher’s_exact_test

Examples

```>>> from scipy.stats import fisher_exact
>>> res = fisher_exact([[8, 2], [1, 5]])
>>> res.statistic
20.0
>>> res.pvalue
0.034965034965034975
```

For a more detailed example, see Fisher’s exact test.