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272 lines
9.3 KiB
Python
272 lines
9.3 KiB
Python
"""Some functions for working with contingency tables (i.e. cross tabulations).
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"""
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from __future__ import division, print_function, absolute_import
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from functools import reduce
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import numpy as np
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from .stats import power_divergence
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__all__ = ['margins', 'expected_freq', 'chi2_contingency']
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def margins(a):
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"""Return a list of the marginal sums of the array `a`.
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Parameters
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----------
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a : ndarray
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The array for which to compute the marginal sums.
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Returns
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-------
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margsums : list of ndarrays
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A list of length `a.ndim`. `margsums[k]` is the result
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of summing `a` over all axes except `k`; it has the same
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number of dimensions as `a`, but the length of each axis
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except axis `k` will be 1.
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Examples
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--------
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>>> a = np.arange(12).reshape(2, 6)
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>>> a
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array([[ 0, 1, 2, 3, 4, 5],
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[ 6, 7, 8, 9, 10, 11]])
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>>> m0, m1 = margins(a)
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>>> m0
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array([[15],
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[51]])
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>>> m1
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array([[ 6, 8, 10, 12, 14, 16]])
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>>> b = np.arange(24).reshape(2,3,4)
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>>> m0, m1, m2 = margins(b)
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>>> m0
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array([[[ 66]],
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[[210]]])
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>>> m1
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array([[[ 60],
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[ 92],
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[124]]])
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>>> m2
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array([[[60, 66, 72, 78]]])
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"""
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margsums = []
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ranged = list(range(a.ndim))
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for k in ranged:
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marg = np.apply_over_axes(np.sum, a, [j for j in ranged if j != k])
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margsums.append(marg)
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return margsums
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def expected_freq(observed):
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"""
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Compute the expected frequencies from a contingency table.
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Given an n-dimensional contingency table of observed frequencies,
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compute the expected frequencies for the table based on the marginal
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sums under the assumption that the groups associated with each
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dimension are independent.
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Parameters
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----------
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observed : array_like
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The table of observed frequencies. (While this function can handle
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a 1-D array, that case is trivial. Generally `observed` is at
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least 2-D.)
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Returns
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-------
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expected : ndarray of float64
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The expected frequencies, based on the marginal sums of the table.
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Same shape as `observed`.
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Examples
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--------
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>>> observed = np.array([[10, 10, 20],[20, 20, 20]])
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>>> expected_freq(observed)
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array([[ 12., 12., 16.],
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[ 18., 18., 24.]])
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"""
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# Typically `observed` is an integer array. If `observed` has a large
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# number of dimensions or holds large values, some of the following
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# computations may overflow, so we first switch to floating point.
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observed = np.asarray(observed, dtype=np.float64)
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# Create a list of the marginal sums.
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margsums = margins(observed)
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# Create the array of expected frequencies. The shapes of the
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# marginal sums returned by apply_over_axes() are just what we
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# need for broadcasting in the following product.
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d = observed.ndim
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expected = reduce(np.multiply, margsums) / observed.sum() ** (d - 1)
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return expected
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def chi2_contingency(observed, correction=True, lambda_=None):
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"""Chi-square test of independence of variables in a contingency table.
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This function computes the chi-square statistic and p-value for the
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hypothesis test of independence of the observed frequencies in the
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contingency table [1]_ `observed`. The expected frequencies are computed
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based on the marginal sums under the assumption of independence; see
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`scipy.stats.contingency.expected_freq`. The number of degrees of
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freedom is (expressed using numpy functions and attributes)::
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dof = observed.size - sum(observed.shape) + observed.ndim - 1
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Parameters
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----------
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observed : array_like
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The contingency table. The table contains the observed frequencies
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(i.e. number of occurrences) in each category. In the two-dimensional
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case, the table is often described as an "R x C table".
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correction : bool, optional
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If True, *and* the degrees of freedom is 1, apply Yates' correction
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for continuity. The effect of the correction is to adjust each
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observed value by 0.5 towards the corresponding expected value.
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lambda_ : float or str, optional.
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By default, the statistic computed in this test is Pearson's
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chi-squared statistic [2]_. `lambda_` allows a statistic from the
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Cressie-Read power divergence family [3]_ to be used instead. See
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`power_divergence` for details.
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Returns
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-------
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chi2 : float
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The test statistic.
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p : float
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The p-value of the test
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dof : int
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Degrees of freedom
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expected : ndarray, same shape as `observed`
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The expected frequencies, based on the marginal sums of the table.
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See Also
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--------
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contingency.expected_freq
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fisher_exact
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chisquare
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power_divergence
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Notes
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-----
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An often quoted guideline for the validity of this calculation is that
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the test should be used only if the observed and expected frequency in
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each cell is at least 5.
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This is a test for the independence of different categories of a
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population. The test is only meaningful when the dimension of
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`observed` is two or more. Applying the test to a one-dimensional
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table will always result in `expected` equal to `observed` and a
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chi-square statistic equal to 0.
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This function does not handle masked arrays, because the calculation
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does not make sense with missing values.
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Like stats.chisquare, this function computes a chi-square statistic;
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the convenience this function provides is to figure out the expected
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frequencies and degrees of freedom from the given contingency table.
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If these were already known, and if the Yates' correction was not
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required, one could use stats.chisquare. That is, if one calls::
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chi2, p, dof, ex = chi2_contingency(obs, correction=False)
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then the following is true::
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(chi2, p) == stats.chisquare(obs.ravel(), f_exp=ex.ravel(),
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ddof=obs.size - 1 - dof)
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The `lambda_` argument was added in version 0.13.0 of scipy.
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References
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----------
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.. [1] "Contingency table", http://en.wikipedia.org/wiki/Contingency_table
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.. [2] "Pearson's chi-squared test",
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http://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test
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.. [3] Cressie, N. and Read, T. R. C., "Multinomial Goodness-of-Fit
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Tests", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984),
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pp. 440-464.
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Examples
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--------
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A two-way example (2 x 3):
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>>> obs = np.array([[10, 10, 20], [20, 20, 20]])
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>>> chi2_contingency(obs)
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(2.7777777777777777,
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0.24935220877729619,
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2,
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array([[ 12., 12., 16.],
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[ 18., 18., 24.]]))
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Perform the test using the log-likelihood ratio (i.e. the "G-test")
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instead of Pearson's chi-squared statistic.
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>>> g, p, dof, expctd = chi2_contingency(obs, lambda_="log-likelihood")
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>>> g, p
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(2.7688587616781319, 0.25046668010954165)
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A four-way example (2 x 2 x 2 x 2):
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>>> obs = np.array(
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... [[[[12, 17],
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... [11, 16]],
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... [[11, 12],
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... [15, 16]]],
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... [[[23, 15],
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... [30, 22]],
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... [[14, 17],
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... [15, 16]]]])
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>>> chi2_contingency(obs)
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(8.7584514426741897,
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0.64417725029295503,
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11,
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array([[[[ 14.15462386, 14.15462386],
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[ 16.49423111, 16.49423111]],
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[[ 11.2461395 , 11.2461395 ],
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[ 13.10500554, 13.10500554]]],
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[[[ 19.5591166 , 19.5591166 ],
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[ 22.79202844, 22.79202844]],
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[[ 15.54012004, 15.54012004],
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[ 18.10873492, 18.10873492]]]]))
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"""
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observed = np.asarray(observed)
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if np.any(observed < 0):
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raise ValueError("All values in `observed` must be nonnegative.")
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if observed.size == 0:
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raise ValueError("No data; `observed` has size 0.")
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expected = expected_freq(observed)
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if np.any(expected == 0):
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# Include one of the positions where expected is zero in
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# the exception message.
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zeropos = list(zip(*np.where(expected == 0)))[0]
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raise ValueError("The internally computed table of expected "
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"frequencies has a zero element at %s." % (zeropos,))
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# The degrees of freedom
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dof = expected.size - sum(expected.shape) + expected.ndim - 1
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if dof == 0:
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# Degenerate case; this occurs when `observed` is 1D (or, more
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# generally, when it has only one nontrivial dimension). In this
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# case, we also have observed == expected, so chi2 is 0.
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chi2 = 0.0
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p = 1.0
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else:
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if dof == 1 and correction:
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# Adjust `observed` according to Yates' correction for continuity.
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observed = observed + 0.5 * np.sign(expected - observed)
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chi2, p = power_divergence(observed, expected,
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ddof=observed.size - 1 - dof, axis=None,
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lambda_=lambda_)
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return chi2, p, dof, expected
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