# 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:
    table array_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:
    res SignificanceResult

      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.

  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.

  `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.

  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

  [3]

  Emma V. Low et al. “Identifying the lowest effective dose of acetazolamide for the prophylaxis of acute mountain sickness: systematic review and meta-analysis.” BMJ, 345, DOI:10.1136/bmj.e6779, 2012.

  Examples

  In [3], the effective dose of acetazolamide for the prophylaxis of acute mountain sickness was investigated. The study notably concluded:

  > Acetazolamide 250 mg, 500 mg, and 750 mg daily were all efficacious for preventing acute mountain sickness. Acetazolamide 250 mg was the lowest effective dose with available evidence for this indication.

  The following table summarizes the results of the experiment in which some participants took a daily dose of acetazolamide 250 mg while others took a placebo. Cases of acute mountain sickness were recorded:

                                  Acetazolamide   Control/Placebo
      Acute mountain sickness            7           17
      No                                15            5

  Is there evidence that the acetazolamide 250 mg reduces the risk of acute mountain sickness? We begin by formulating a null hypothesis \(H_0\):

  > The odds of experiencing acute mountain sickness are the same with the acetazolamide treatment as they are with placebo.

  Let’s assess the plausibility of this hypothesis with Fisher’s test.

      >>> from scipy.stats import fisher_exact
      >>> res = fisher_exact([[7, 17], [15, 5]], alternative='less')
      >>> res.statistic
      0.13725490196078433
      >>> res.pvalue
      0.0028841933752349743

  Using a significance level of 5%, we would reject the null hypothesis in favor of the alternative hypothesis: “The odds of experiencing acute mountain sickness with acetazolamide treatment are less than the odds of experiencing acute mountain sickness with placebo.”

  Note

  Because the null distribution of Fisher’s exact test is formed under the assumption that both row and column sums are fixed, the result of the test are conservative when applied to an experiment in which the row sums are not fixed.

  In this case, the column sums are fixed; there are 22 subjects in each group. But the number of cases of acute mountain sickness is not (and cannot be) fixed before conducting the experiment. It is a consequence.

  Boschloo’s test does not depend on the assumption that the row sums are fixed, and consequently, it provides a more powerful test in this situation.

      >>> from scipy.stats import boschloo_exact
      >>> res = boschloo_exact([[7, 17], [15, 5]], alternative='less')
      >>> res.statistic
      0.0028841933752349743
      >>> res.pvalue
      0.0015141406667567101

  We verify that the p-value is less than with `fisher_exact`.