contingency.py

"""Some functions for working with contingency tables (i.e. cross tabulations).
"""


from __future__ import division, print_function, absolute_import

from functools import reduce
import numpy as np
from .stats import power_divergence


__all__ = ['margins', 'expected_freq', 'chi2_contingency']


def margins(a):
    """Return a list of the marginal sums of the array `a`.

    Parameters
    ----------
    a : ndarray
        The array for which to compute the marginal sums.

    Returns
    -------
    margsums : list of ndarrays
        A list of length `a.ndim`.  `margsums[k]` is the result
        of summing `a` over all axes except `k`; it has the same
        number of dimensions as `a`, but the length of each axis
        except axis `k` will be 1.

    Examples
    --------
    >>> a = np.arange(12).reshape(2, 6)
    >>> a
    array([[ 0,  1,  2,  3,  4,  5],
           [ 6,  7,  8,  9, 10, 11]])
    >>> from scipy.stats.contingency import margins
    >>> m0, m1 = margins(a)
    >>> m0
    array([[15],
           [51]])
    >>> m1
    array([[ 6,  8, 10, 12, 14, 16]])

    >>> b = np.arange(24).reshape(2,3,4)
    >>> m0, m1, m2 = margins(b)
    >>> m0
    array([[[ 66]],
           [[210]]])
    >>> m1
    array([[[ 60],
            [ 92],
            [124]]])
    >>> m2
    array([[[60, 66, 72, 78]]])
    """
    margsums = []
    ranged = list(range(a.ndim))
    for k in ranged:
        marg = np.apply_over_axes(np.sum, a, [j for j in ranged if j != k])
        margsums.append(marg)
    return margsums


def expected_freq(observed):
    """
    Compute the expected frequencies from a contingency table.

    Given an n-dimensional contingency table of observed frequencies,
    compute the expected frequencies for the table based on the marginal
    sums under the assumption that the groups associated with each
    dimension are independent.

    Parameters
    ----------
    observed : array_like
        The table of observed frequencies.  (While this function can handle
        a 1-D array, that case is trivial.  Generally `observed` is at
        least 2-D.)

    Returns
    -------
    expected : ndarray of float64
        The expected frequencies, based on the marginal sums of the table.
        Same shape as `observed`.

    Examples
    --------
    >>> observed = np.array([[10, 10, 20],[20, 20, 20]])
    >>> from scipy.stats.contingency import expected_freq
    >>> expected_freq(observed)
    array([[ 12.,  12.,  16.],
           [ 18.,  18.,  24.]])

    """
    # Typically `observed` is an integer array. If `observed` has a large
    # number of dimensions or holds large values, some of the following
    # computations may overflow, so we first switch to floating point.
    observed = np.asarray(observed, dtype=np.float64)

    # Create a list of the marginal sums.
    margsums = margins(observed)

    # Create the array of expected frequencies.  The shapes of the
    # marginal sums returned by apply_over_axes() are just what we
    # need for broadcasting in the following product.
    d = observed.ndim
    expected = reduce(np.multiply, margsums) / observed.sum() ** (d - 1)
    return expected


def chi2_contingency(observed, correction=True, lambda_=None):
    """Chi-square test of independence of variables in a contingency table.

    This function computes the chi-square statistic and p-value for the
    hypothesis test of independence of the observed frequencies in the
    contingency table [1]_ `observed`.  The expected frequencies are computed
    based on the marginal sums under the assumption of independence; see
    `scipy.stats.contingency.expected_freq`.  The number of degrees of
    freedom is (expressed using numpy functions and attributes)::

        dof = observed.size - sum(observed.shape) + observed.ndim - 1


    Parameters
    ----------
    observed : array_like
        The contingency table. The table contains the observed frequencies
        (i.e. number of occurrences) in each category.  In the two-dimensional
        case, the table is often described as an "R x C table".
    correction : bool, optional
        If True, *and* the degrees of freedom is 1, apply Yates' correction
        for continuity.  The effect of the correction is to adjust each
        observed value by 0.5 towards the corresponding expected value.
    lambda_ : float or str, optional.
        By default, the statistic computed in this test is Pearson's
        chi-squared statistic [2]_.  `lambda_` allows a statistic from the
        Cressie-Read power divergence family [3]_ to be used instead.  See
        `power_divergence` for details.

    Returns
    -------
    chi2 : float
        The test statistic.
    p : float
        The p-value of the test
    dof : int
        Degrees of freedom
    expected : ndarray, same shape as `observed`
        The expected frequencies, based on the marginal sums of the table.

    See Also
    --------
    contingency.expected_freq
    fisher_exact
    chisquare
    power_divergence

    Notes
    -----
    An often quoted guideline for the validity of this calculation is that
    the test should be used only if the observed and expected frequencies
    in each cell are at least 5.

    This is a test for the independence of different categories of a
    population. The test is only meaningful when the dimension of
    `observed` is two or more.  Applying the test to a one-dimensional
    table will always result in `expected` equal to `observed` and a
    chi-square statistic equal to 0.

    This function does not handle masked arrays, because the calculation
    does not make sense with missing values.

    Like stats.chisquare, this function computes a chi-square statistic;
    the convenience this function provides is to figure out the expected
    frequencies and degrees of freedom from the given contingency table.
    If these were already known, and if the Yates' correction was not
    required, one could use stats.chisquare.  That is, if one calls::

        chi2, p, dof, ex = chi2_contingency(obs, correction=False)

    then the following is true::

        (chi2, p) == stats.chisquare(obs.ravel(), f_exp=ex.ravel(),
                                     ddof=obs.size - 1 - dof)

    The `lambda_` argument was added in version 0.13.0 of scipy.

    References
    ----------
    .. [1] "Contingency table",
           https://en.wikipedia.org/wiki/Contingency_table
    .. [2] "Pearson's chi-squared test",
           https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test
    .. [3] 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
    --------
    A two-way example (2 x 3):

    >>> from scipy.stats import chi2_contingency
    >>> obs = np.array([[10, 10, 20], [20, 20, 20]])
    >>> chi2_contingency(obs)
    (2.7777777777777777,
     0.24935220877729619,
     2,
     array([[ 12.,  12.,  16.],
            [ 18.,  18.,  24.]]))

    Perform the test using the log-likelihood ratio (i.e. the "G-test")
    instead of Pearson's chi-squared statistic.

    >>> g, p, dof, expctd = chi2_contingency(obs, lambda_="log-likelihood")
    >>> g, p
    (2.7688587616781319, 0.25046668010954165)

    A four-way example (2 x 2 x 2 x 2):

    >>> obs = np.array(
    ...     [[[[12, 17],
    ...        [11, 16]],
    ...       [[11, 12],
    ...        [15, 16]]],
    ...      [[[23, 15],
    ...        [30, 22]],
    ...       [[14, 17],
    ...        [15, 16]]]])
    >>> chi2_contingency(obs)
    (8.7584514426741897,
     0.64417725029295503,
     11,
     array([[[[ 14.15462386,  14.15462386],
              [ 16.49423111,  16.49423111]],
             [[ 11.2461395 ,  11.2461395 ],
              [ 13.10500554,  13.10500554]]],
            [[[ 19.5591166 ,  19.5591166 ],
              [ 22.79202844,  22.79202844]],
             [[ 15.54012004,  15.54012004],
              [ 18.10873492,  18.10873492]]]]))
    """
    observed = np.asarray(observed)
    if np.any(observed < 0):
        raise ValueError("All values in `observed` must be nonnegative.")
    if observed.size == 0:
        raise ValueError("No data; `observed` has size 0.")

    expected = expected_freq(observed)
    if np.any(expected == 0):
        # Include one of the positions where expected is zero in
        # the exception message.
        zeropos = list(zip(*np.nonzero(expected == 0)))[0]
        raise ValueError("The internally computed table of expected "
                         "frequencies has a zero element at %s." % (zeropos,))

    # The degrees of freedom
    dof = expected.size - sum(expected.shape) + expected.ndim - 1

    if dof == 0:
        # Degenerate case; this occurs when `observed` is 1D (or, more
        # generally, when it has only one nontrivial dimension).  In this
        # case, we also have observed == expected, so chi2 is 0.
        chi2 = 0.0
        p = 1.0
    else:
        if dof == 1 and correction:
            # Adjust `observed` according to Yates' correction for continuity.
            observed = observed + 0.5 * np.sign(expected - observed)

        chi2, p = power_divergence(observed, expected,
                                   ddof=observed.size - 1 - dof, axis=None,
                                   lambda_=lambda_)

    return chi2, p, dof, expected