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"""
Contingency table functions (:mod:`scipy.stats.contingency`)
============================================================
Functions for creating and analyzing contingency tables.
.. currentmodule:: scipy.stats.contingency
.. autosummary::
:toctree: generated/
chi2_contingency
relative_risk
odds_ratio
crosstab
association
expected_freq
margins
"""
from functools import reduce
import math
import numpy as np
from ._stats_py import power_divergence, _untabulate
from ._relative_risk import relative_risk
from ._crosstab import crosstab
from ._odds_ratio import odds_ratio
from scipy._lib._bunch import _make_tuple_bunch
from scipy import stats
__all__ = ['margins', 'expected_freq', 'chi2_contingency', 'crosstab',
'association', 'relative_risk', 'odds_ratio']
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
--------
>>> import numpy as np
>>> from scipy.stats.contingency import margins
>>> 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
--------
>>> import numpy as np
>>> from scipy.stats.contingency import expected_freq
>>> 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
Chi2ContingencyResult = _make_tuple_bunch(
'Chi2ContingencyResult',
['statistic', 'pvalue', 'dof', 'expected_freq'], []
)
def chi2_contingency(observed, correction=True, lambda_=None, *, method=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
`scipy.stats.power_divergence` for details.
method : ResamplingMethod, optional
Defines the method used to compute the p-value. Compatible only with
`correction=False`, default `lambda_`, and two-way tables.
If `method` is an instance of `PermutationMethod`/`MonteCarloMethod`,
the p-value is computed using
`scipy.stats.permutation_test`/`scipy.stats.monte_carlo_test` with the
provided configuration options and other appropriate settings.
Otherwise, the p-value is computed as documented in the notes.
Note that if `method` is an instance of `MonteCarloMethod`, the ``rvs``
attribute must be left unspecified; Monte Carlo samples are always drawn
using the ``rvs`` method of `scipy.stats.random_table`.
.. versionadded:: 1.15.0
Returns
-------
res : Chi2ContingencyResult
An object containing attributes:
statistic : float
The test statistic.
pvalue : float
The p-value of the test.
dof : int
The degrees of freedom. NaN if `method` is not ``None``.
expected_freq : ndarray, same shape as `observed`
The expected frequencies, based on the marginal sums of the table.
See Also
--------
scipy.stats.contingency.expected_freq
scipy.stats.fisher_exact
scipy.stats.chisquare
scipy.stats.power_divergence
scipy.stats.barnard_exact
scipy.stats.boschloo_exact
:ref:`hypothesis_chi2_contingency` : Extended example
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 `scipy.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 `scipy.stats.chisquare`. That is, if one calls::
res = chi2_contingency(obs, correction=False)
then the following is true::
(res.statistic, res.pvalue) == 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):
>>> import numpy as np
>>> from scipy.stats import chi2_contingency
>>> obs = np.array([[10, 10, 20], [20, 20, 20]])
>>> res = chi2_contingency(obs)
>>> res.statistic
2.7777777777777777
>>> res.pvalue
0.24935220877729619
>>> res.dof
2
>>> res.expected_freq
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.
>>> res = chi2_contingency(obs, lambda_="log-likelihood")
>>> res.statistic
2.7688587616781319
>>> res.pvalue
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]]]])
>>> res = chi2_contingency(obs)
>>> res.statistic
8.7584514426741897
>>> res.pvalue
0.64417725029295503
When the sum of the elements in a two-way table is small, the p-value
produced by the default asymptotic approximation may be inaccurate.
Consider passing a `PermutationMethod` or `MonteCarloMethod` as the
`method` parameter with `correction=False`.
>>> from scipy.stats import PermutationMethod
>>> obs = np.asarray([[12, 3],
... [17, 16]])
>>> res = chi2_contingency(obs, correction=False)
>>> ref = chi2_contingency(obs, correction=False, method=PermutationMethod())
>>> res.pvalue, ref.pvalue
(0.0614122539870913, 0.1074) # may vary
For a more detailed example, see :ref:`hypothesis_chi2_contingency`.
"""
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 "
f"frequencies has a zero element at {zeropos}.")
if method is not None:
return _chi2_resampling_methods(observed, expected, correction, lambda_, method)
# 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.
# Magnitude of correction no bigger than difference; see gh-13875
diff = expected - observed
direction = np.sign(diff)
magnitude = np.minimum(0.5, np.abs(diff))
observed = observed + magnitude * direction
chi2, p = power_divergence(observed, expected,
ddof=observed.size - 1 - dof, axis=None,
lambda_=lambda_)
return Chi2ContingencyResult(chi2, p, dof, expected)
def _chi2_resampling_methods(observed, expected, correction, lambda_, method):
if observed.ndim != 2:
message = 'Use of `method` is only compatible with two-way tables.'
raise ValueError(message)
if correction:
message = f'`{correction=}` is not compatible with `{method=}.`'
raise ValueError(message)
if lambda_ is not None:
message = f'`{lambda_=}` is not compatible with `{method=}.`'
raise ValueError(message)
if isinstance(method, stats.PermutationMethod):
res = _chi2_permutation_method(observed, expected, method)
elif isinstance(method, stats.MonteCarloMethod):
res = _chi2_monte_carlo_method(observed, expected, method)
else:
message = (f'`{method=}` not recognized; if provided, `method` must be an '
'instance of `PermutationMethod` or `MonteCarloMethod`.')
raise ValueError(message)
return Chi2ContingencyResult(res.statistic, res.pvalue, np.nan, expected)
def _chi2_permutation_method(observed, expected, method):
x, y = _untabulate(observed)
# `permutation_test` with `permutation_type='pairings' permutes the order of `x`,
# which pairs observations in `x` with different observations in `y`.
def statistic(x):
# crosstab the resample and compute the statistic
table = crosstab(x, y)[1]
return np.sum((table - expected)**2/expected)
return stats.permutation_test((x,), statistic, permutation_type='pairings',
alternative='greater', **method._asdict())
def _chi2_monte_carlo_method(observed, expected, method):
method = method._asdict()
if method.pop('rvs', None) is not None:
message = ('If the `method` argument of `chi2_contingency` is an '
'instance of `MonteCarloMethod`, its `rvs` attribute '
'must be unspecified. Use the `MonteCarloMethod` `rng` argument '
'to control the random state.')
raise ValueError(message)
rng = np.random.default_rng(method.pop('rng', None))
# `random_table.rvs` produces random contingency tables with the given marginals
# under the null hypothesis of independence
rowsums, colsums = stats.contingency.margins(observed)
X = stats.random_table(rowsums.ravel(), colsums.ravel(), seed=rng)
def rvs(size):
n_resamples = size[0]
return X.rvs(size=n_resamples).reshape(size)
expected = expected.ravel()
def statistic(table, axis):
return np.sum((table - expected)**2/expected, axis=axis)
return stats.monte_carlo_test(observed.ravel(), rvs, statistic,
alternative='greater', **method)
def association(observed, method="cramer", correction=False, lambda_=None):
"""Calculates degree of association between two nominal variables.
The function provides the option for computing one of three measures of
association between two nominal variables from the data given in a 2d
contingency table: Tschuprow's T, Pearson's Contingency Coefficient
and Cramer's V.
Parameters
----------
observed : array-like
The array of observed values
method : {"cramer", "tschuprow", "pearson"} (default = "cramer")
The association test statistic.
correction : bool, optional
Inherited from `scipy.stats.contingency.chi2_contingency()`
lambda_ : float or str, optional
Inherited from `scipy.stats.contingency.chi2_contingency()`
Returns
-------
statistic : float
Value of the test statistic
Notes
-----
Cramer's V, Tschuprow's T and Pearson's Contingency Coefficient, all
measure the degree to which two nominal or ordinal variables are related,
or the level of their association. This differs from correlation, although
many often mistakenly consider them equivalent. Correlation measures in
what way two variables are related, whereas, association measures how
related the variables are. As such, association does not subsume
independent variables, and is rather a test of independence. A value of
1.0 indicates perfect association, and 0.0 means the variables have no
association.
Both the Cramer's V and Tschuprow's T are extensions of the phi
coefficient. Moreover, due to the close relationship between the
Cramer's V and Tschuprow's T the returned values can often be similar
or even equivalent. They are likely to diverge more as the array shape
diverges from a 2x2.
References
----------
.. [1] "Tschuprow's T",
https://en.wikipedia.org/wiki/Tschuprow's_T
.. [2] Tschuprow, A. A. (1939)
Principles of the Mathematical Theory of Correlation;
translated by M. Kantorowitsch. W. Hodge & Co.
.. [3] "Cramer's V", https://en.wikipedia.org/wiki/Cramer's_V
.. [4] "Nominal Association: Phi and Cramer's V",
http://www.people.vcu.edu/~pdattalo/702SuppRead/MeasAssoc/NominalAssoc.html
.. [5] Gingrich, Paul, "Association Between Variables",
http://uregina.ca/~gingrich/ch11a.pdf
Examples
--------
An example with a 4x2 contingency table:
>>> import numpy as np
>>> from scipy.stats.contingency import association
>>> obs4x2 = np.array([[100, 150], [203, 322], [420, 700], [320, 210]])
Pearson's contingency coefficient
>>> association(obs4x2, method="pearson")
0.18303298140595667
Cramer's V
>>> association(obs4x2, method="cramer")
0.18617813077483678
Tschuprow's T
>>> association(obs4x2, method="tschuprow")
0.14146478765062995
"""
arr = np.asarray(observed)
if not np.issubdtype(arr.dtype, np.integer):
raise ValueError("`observed` must be an integer array.")
if len(arr.shape) != 2:
raise ValueError("method only accepts 2d arrays")
chi2_stat = chi2_contingency(arr, correction=correction,
lambda_=lambda_)
phi2 = chi2_stat.statistic / arr.sum()
n_rows, n_cols = arr.shape
if method == "cramer":
value = phi2 / min(n_cols - 1, n_rows - 1)
elif method == "tschuprow":
value = phi2 / math.sqrt((n_rows - 1) * (n_cols - 1))
elif method == 'pearson':
value = phi2 / (1 + phi2)
else:
raise ValueError("Invalid argument value: 'method' argument must "
"be 'cramer', 'tschuprow', or 'pearson'")
return math.sqrt(value)