Vous êtes un professionnel et vous avez besoin d'une formation ? Machine Learning
avec Scikit-Learn Voir le programme détaillé

Module « scipy.stats »

Fonction fisher_exact - module scipy.stats

Signature de la fonction fisher_exact

def fisher_exact(table, alternative=None, *, method=None) 

Description

help(scipy.stats.fisher_exact)

Perform a Fisher exact test on a contingency table.

For a 2x2 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 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.

For other table sizes, or if `method` is provided, the null hypothesis
is that the rows and columns of the tables have fixed sums and are
independent; i.e., the table was sampled from a `scipy.stats.random_table`
distribution with the observed marginals. The statistic is the
probability mass of this distribution evaluated at `table`, and the
p-value is the percentage of the population of tables with statistic at
least as extreme (small) as that of `table`. There is only one alternative
hypothesis available: the rows and columns are not independent.

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 contingency table.  Elements must be non-negative integers.
alternative : {'two-sided', 'less', 'greater'}, optional
    Defines the alternative hypothesis for 2x2 tables; unused for other
    table sizes.
    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.

method : ResamplingMethod, optional
    Defines the method used to compute the p-value.
    If `method` is an instance of `PermutationMethod`/`MonteCarloMethod`,
    the p-value is computed using
    `scipy.stats.permutation_test`/`scipy.stats.monte_carlo_test` with the
    provided configuration options and other appropriate settings.
    Note that if `method` is an instance of `MonteCarloMethod`, the ``rvs``
    attribute must be left unspecified; Monte Carlo samples are always drawn
    using the ``rvs`` method of `scipy.stats.random_table`.
    Otherwise, the p-value is computed as documented in the notes.

    .. versionadded:: 1.15.0

Returns
-------
res : SignificanceResult
    An object containing attributes:

    statistic : float
        For a 2x2 table with default `method`, this is the odds ratio - the
        prior odds ratio not a posterior estimate. In all other cases, this
        is the probability density of obtaining the observed table under the
        null hypothesis of independence with marginals fixed.
    pvalue : float
        The probability under the null hypothesis of obtaining a
        table at least as extreme as the one that was actually observed.

Raises
------
ValueError
    If `table` is not two-dimensional or has negative entries.

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.
:ref:`hypothesis_fisher_exact` : 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 tables with shape other than ``(2, 2)``, provide an instance of
`scipy.stats.MonteCarloMethod` or `scipy.stats.PermutationMethod` for the
`method` parameter:

>>> import numpy as np
>>> from scipy.stats import MonteCarloMethod
>>> rng = np.random.default_rng(4507195762371367)
>>> method = MonteCarloMethod(rng=rng)
>>> fisher_exact([[8, 2, 3], [1, 5, 4]], method=method)
SignificanceResult(statistic=np.float64(0.005782), pvalue=np.float64(0.0603))

For a more detailed example, see :ref:`hypothesis_fisher_exact`.

Vous êtes un professionnel et vous avez besoin d'une formation ? Sensibilisation à
l'Intelligence Artificielle Voir le programme détaillé