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Fonction power_divergence - module scipy.stats

Signature de la fonction power_divergence

def power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None) 

Description

power_divergence.__doc__

Cressie-Read power divergence statistic and goodness of fit test.

    This function tests the null hypothesis that the categorical data
    has the given frequencies, using the Cressie-Read power divergence
    statistic.

    Parameters
    ----------
    f_obs : array_like
        Observed frequencies in each category.
    f_exp : array_like, optional
        Expected frequencies in each category.  By default the categories are
        assumed to be equally likely.
    ddof : int, optional
        "Delta degrees of freedom": adjustment to the degrees of freedom
        for the p-value.  The p-value is computed using a chi-squared
        distribution with ``k - 1 - ddof`` degrees of freedom, where `k`
        is the number of observed frequencies.  The default value of `ddof`
        is 0.
    axis : int or None, optional
        The axis of the broadcast result of `f_obs` and `f_exp` along which to
        apply the test.  If axis is None, all values in `f_obs` are treated
        as a single data set.  Default is 0.
    lambda_ : float or str, optional
        The power in the Cressie-Read power divergence statistic.  The default
        is 1.  For convenience, `lambda_` may be assigned one of the following
        strings, in which case the corresponding numerical value is used::

            String              Value   Description
            "pearson"             1     Pearson's chi-squared statistic.
                                        In this case, the function is
                                        equivalent to `stats.chisquare`.
            "log-likelihood"      0     Log-likelihood ratio. Also known as
                                        the G-test [3]_.
            "freeman-tukey"      -1/2   Freeman-Tukey statistic.
            "mod-log-likelihood" -1     Modified log-likelihood ratio.
            "neyman"             -2     Neyman's statistic.
            "cressie-read"        2/3   The power recommended in [5]_.

    Returns
    -------
    statistic : float or ndarray
        The Cressie-Read power divergence test statistic.  The value is
        a float if `axis` is None or if` `f_obs` and `f_exp` are 1-D.
    pvalue : float or ndarray
        The p-value of the test.  The value is a float if `ddof` and the
        return value `stat` are scalars.

    See Also
    --------
    chisquare

    Notes
    -----
    This test is invalid when the observed or expected frequencies in each
    category are too small.  A typical rule is that all of the observed
    and expected frequencies should be at least 5.

    Also, the sum of the observed and expected frequencies must be the same
    for the test to be valid; `power_divergence` raises an error if the sums
    do not agree within a relative tolerance of ``1e-8``.

    When `lambda_` is less than zero, the formula for the statistic involves
    dividing by `f_obs`, so a warning or error may be generated if any value
    in `f_obs` is 0.

    Similarly, a warning or error may be generated if any value in `f_exp` is
    zero when `lambda_` >= 0.

    The default degrees of freedom, k-1, are for the case when no parameters
    of the distribution are estimated. If p parameters are estimated by
    efficient maximum likelihood then the correct degrees of freedom are
    k-1-p. If the parameters are estimated in a different way, then the
    dof can be between k-1-p and k-1. However, it is also possible that
    the asymptotic distribution is not a chisquare, in which case this
    test is not appropriate.

    This function handles masked arrays.  If an element of `f_obs` or `f_exp`
    is masked, then data at that position is ignored, and does not count
    towards the size of the data set.

    .. versionadded:: 0.13.0

    References
    ----------
    .. [1] Lowry, Richard.  "Concepts and Applications of Inferential
           Statistics". Chapter 8.
           https://web.archive.org/web/20171015035606/http://faculty.vassar.edu/lowry/ch8pt1.html
    .. [2] "Chi-squared test", https://en.wikipedia.org/wiki/Chi-squared_test
    .. [3] "G-test", https://en.wikipedia.org/wiki/G-test
    .. [4] Sokal, R. R. and Rohlf, F. J. "Biometry: the principles and
           practice of statistics in biological research", New York: Freeman
           (1981)
    .. [5] Cressie, N. and Read, T. R. C., "Multinomial Goodness-of-Fit
           Tests", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984),
           pp. 440-464.

    Examples
    --------
    (See `chisquare` for more examples.)

    When just `f_obs` is given, it is assumed that the expected frequencies
    are uniform and given by the mean of the observed frequencies.  Here we
    perform a G-test (i.e. use the log-likelihood ratio statistic):

    >>> from scipy.stats import power_divergence
    >>> power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood')
    (2.006573162632538, 0.84823476779463769)

    The expected frequencies can be given with the `f_exp` argument:

    >>> power_divergence([16, 18, 16, 14, 12, 12],
    ...                  f_exp=[16, 16, 16, 16, 16, 8],
    ...                  lambda_='log-likelihood')
    (3.3281031458963746, 0.6495419288047497)

    When `f_obs` is 2-D, by default the test is applied to each column.

    >>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
    >>> obs.shape
    (6, 2)
    >>> power_divergence(obs, lambda_="log-likelihood")
    (array([ 2.00657316,  6.77634498]), array([ 0.84823477,  0.23781225]))

    By setting ``axis=None``, the test is applied to all data in the array,
    which is equivalent to applying the test to the flattened array.

    >>> power_divergence(obs, axis=None)
    (23.31034482758621, 0.015975692534127565)
    >>> power_divergence(obs.ravel())
    (23.31034482758621, 0.015975692534127565)

    `ddof` is the change to make to the default degrees of freedom.

    >>> power_divergence([16, 18, 16, 14, 12, 12], ddof=1)
    (2.0, 0.73575888234288467)

    The calculation of the p-values is done by broadcasting the
    test statistic with `ddof`.

    >>> power_divergence([16, 18, 16, 14, 12, 12], ddof=[0,1,2])
    (2.0, array([ 0.84914504,  0.73575888,  0.5724067 ]))

    `f_obs` and `f_exp` are also broadcast.  In the following, `f_obs` has
    shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting
    `f_obs` and `f_exp` has shape (2, 6).  To compute the desired chi-squared
    statistics, we must use ``axis=1``:

    >>> power_divergence([16, 18, 16, 14, 12, 12],
    ...                  f_exp=[[16, 16, 16, 16, 16, 8],
    ...                         [8, 20, 20, 16, 12, 12]],
    ...                  axis=1)
    (array([ 3.5 ,  9.25]), array([ 0.62338763,  0.09949846]))