scipy.stats.chisquare#

scipy.stats.chisquare(f_obs, f_exp=None, ddof=0, axis=0)[source]#

Calculate a one-way chi-square test.

The chi-square test tests the null hypothesis that the categorical data has the given frequencies.

Parameters

f_obsarray_like: Observed frequencies in each category.
f_exparray_like, optional: Expected frequencies in each category. By default the categories are assumed to be equally likely.
ddofint, optional: “Delta degrees of freedom”: adjustment to the degrees of freedom for the p-value. The p-value is computed using a chi-squared distribution with k - 1 - ddof degrees of freedom, where k is the number of observed frequencies. The default value of ddof is 0.
axisint or None, optional: The axis of the broadcast result of f_obs and f_exp along which to apply the test. If axis is None, all values in f_obs are treated as a single data set. Default is 0.

Returns

chisqfloat or ndarray: The chi-squared test statistic. The value is a float if axis is None or f_obs and f_exp are 1-D.
pfloat or ndarray: The p-value of the test. The value is a float if ddof and the return value chisq are scalars.

See also

scipy.stats.power_divergence
scipy.stats.fisher_exact: Fisher exact test on a 2x2 contingency table.
scipy.stats.barnard_exact: An unconditional exact test. An alternative to chi-squared test for small sample sizes.

Notes

This test is invalid when the observed or expected frequencies in each category are too small. A typical rule is that all of the observed and expected frequencies should be at least 5. According to [3], the total number of samples is recommended to be greater than 13, otherwise exact tests (such as Barnard’s Exact test) should be used because they do not overreject.

Also, the sum of the observed and expected frequencies must be the same for the test to be valid; chisquare raises an error if the sums do not agree within a relative tolerance of 1e-8.

The default degrees of freedom, k-1, are for the case when no parameters of the distribution are estimated. If p parameters are estimated by efficient maximum likelihood then the correct degrees of freedom are k-1-p. If the parameters are estimated in a different way, then the dof can be between k-1-p and k-1. However, it is also possible that the asymptotic distribution is not chi-square, in which case this test is not appropriate.

References

1: Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 8. https://web.archive.org/web/20171022032306/http://vassarstats.net:80/textbook/ch8pt1.html
2: “Chi-squared test”, https://en.wikipedia.org/wiki/Chi-squared_test
3: Pearson, Karl. “On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling”, Philosophical Magazine. Series 5. 50 (1900), pp. 157-175.

Examples

When just f_obs is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies.

>>> from scipy.stats import chisquare
>>> chisquare([16, 18, 16, 14, 12, 12])
(2.0, 0.84914503608460956)

With f_exp the expected frequencies can be given.

>>> chisquare([16, 18, 16, 14, 12, 12], f_exp=[16, 16, 16, 16, 16, 8])
(3.5, 0.62338762774958223)

When f_obs is 2-D, by default the test is applied to each column.

>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
>>> obs.shape
(6, 2)
>>> chisquare(obs)
(array([ 2.        ,  6.66666667]), array([ 0.84914504,  0.24663415]))

By setting axis=None, the test is applied to all data in the array, which is equivalent to applying the test to the flattened array.

>>> chisquare(obs, axis=None)
(23.31034482758621, 0.015975692534127565)
>>> chisquare(obs.ravel())
(23.31034482758621, 0.015975692534127565)

ddof is the change to make to the default degrees of freedom.

>>> chisquare([16, 18, 16, 14, 12, 12], ddof=1)
(2.0, 0.73575888234288467)

The calculation of the p-values is done by broadcasting the chi-squared statistic with ddof.

>>> chisquare([16, 18, 16, 14, 12, 12], ddof=[0,1,2])
(2.0, array([ 0.84914504,  0.73575888,  0.5724067 ]))

f_obs and f_exp are also broadcast. In the following, f_obs has shape (6,) and f_exp has shape (2, 6), so the result of broadcasting f_obs and f_exp has shape (2, 6). To compute the desired chi-squared statistics, we use axis=1:

>>> chisquare([16, 18, 16, 14, 12, 12],
...           f_exp=[[16, 16, 16, 16, 16, 8], [8, 20, 20, 16, 12, 12]],
...           axis=1)
(array([ 3.5 ,  9.25]), array([ 0.62338763,  0.09949846]))

scipy.stats.ttest_rel

scipy.stats.cramervonmises