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"""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 scipy import special
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]])
>>> 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]])
>>> 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 frequency in
each cell is 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", http://en.wikipedia.org/wiki/Contingency_table
.. [2] "Pearson's chi-squared test",
http://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):
>>> 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(np.where(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
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