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chisquare(f_obs, f_exp=None, ddof=0, axis=0)

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

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 , 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.

Parameters

f_obs : array_like: Observed frequencies in each category.
f_exp : array_like, optional: Expected frequencies in each category. By default the categories are assumed to be equally likely.
ddof : int, 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.
axis : int 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

chisq : float 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.
p : float or ndarray: The p-value of the test. The value is a float if ddof and the return value chisq are scalars.

Calculate a one-way chi-square test.

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.

import numpy as np
from scipy.stats import chisquare
chisquare([16, 18, 16, 14, 12, 12])

With `f_exp` the expected frequencies can be given.

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

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

chisquare(obs)

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)

chisquare(obs.ravel())

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

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

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])

`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)

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.stats.contingency:chi2_contingency

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