scipy.stats.

chisquare#

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

Perform Pearson’s chi-squared test.

Pearson’s chi-squared test [1] is a goodness-of-fit test for a multinomial distribution with given probabilities; that is, it assesses the null hypothesis that the observed frequencies (counts) are obtained by independent sampling of N observations from a categorical distribution with given expected 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 categories. 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.
sum_checkbool, optional: Whether to perform a check that sum(f_obs) - sum(f_exp) == 0. If True, (default) raise an error when the relative difference exceeds the square root of the precision of the data type. See Notes for rationale and possible exceptions.

Returns:

res: Power_divergenceResult

An object containing attributes:

statisticfloat 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.
pvaluefloat or ndarray: The p-value of the test. The value is a float if ddof and the result attribute statistic 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.
Chi-square test: Extended example

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 [2], the total number of observations is recommended to be greater than 13, otherwise exact tests (such as Barnard’s Exact test) should be used because they do not overreject.

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.

For Pearson’s chi-squared test, the total observed and expected counts must match for the p-value to accurately reflect the probability of observing such an extreme value of the statistic under the null hypothesis. This function may be used to perform other statistical tests that do not require the total counts to be equal. For instance, to test the null hypothesis that f_obs[i] is Poisson-distributed with expectation f_exp[i], set ddof=-1 and sum_check=False. This test follows from the fact that a Poisson random variable with mean and variance f_exp[i] is approximately normal with the same mean and variance; the chi-squared statistic standardizes, squares, and sums the observations; and the sum of n squared standard normal variables follows the chi-squared distribution with n degrees of freedom.

References

[1]

“Pearson’s chi-squared test”. Wikipedia. https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test

[2]

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 only the mandatory f_obs argument is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies:

>>> import numpy as np
>>> from scipy.stats import chisquare
>>> chisquare([16, 18, 16, 14, 12, 12])
Power_divergenceResult(statistic=2.0, pvalue=0.84914503608460956)

The optional f_exp argument gives the expected frequencies.

>>> chisquare([16, 18, 16, 14, 12, 12], f_exp=[16, 16, 16, 16, 16, 8])
Power_divergenceResult(statistic=3.5, pvalue=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)
Power_divergenceResult(statistic=array([2.        , 6.66666667]), pvalue=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)
Power_divergenceResult(statistic=23.31034482758621, pvalue=0.015975692534127565)
>>> chisquare(obs.ravel())
Power_divergenceResult(statistic=23.310344827586206, pvalue=0.01597569253412758)

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

>>> chisquare([16, 18, 16, 14, 12, 12], ddof=1)
Power_divergenceResult(statistic=2.0, pvalue=0.7357588823428847)

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])
Power_divergenceResult(statistic=2.0, pvalue=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)
Power_divergenceResult(statistic=array([3.5 , 9.25]), pvalue=array([0.62338763, 0.09949846]))

For a more detailed example, see Chi-square test.