scipy.stats.

chi2_contingency#

scipy.stats.chi2_contingency(observed, correction=True, lambda_=None, *, method=None)[source]#

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:

observedarray_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”.
correctionbool, 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.
methodResamplingMethod, 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.

Added in version 1.15.0.

Returns:

resChi2ContingencyResult

An object containing attributes:

statisticfloat: The test statistic.
pvaluefloat: The p-value of the test.
dofint: The degrees of freedom. NaN if method is not None.
expected_freqndarray, 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
Chi-square test of independence of variables in a contingency table: 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 Chi-square test of independence of variables in a contingency table.