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Fonction fisher_exact - module scipy.stats

Signature de la fonction fisher_exact

def fisher_exact(table, alternative='two-sided') 

Description

fisher_exact.__doc__

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

    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'
        * 'less': one-sided
        * 'greater': one-sided

        See the Notes for more details.

    Returns
    -------
    oddsratio : float
        This is prior odds ratio and not a posterior estimate.
    p_value : float
        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.