scipy.stats.chisquare

scipy.stats.chisquare(f_obs, f_exp=None, ddof=0, axis=0)

Calculates a one-way chi square test.

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

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.

See also

power_divergence, mstats.chisquare

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.

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 a chisquare, in which case this test is not appropriate.

References

[R261]

Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html

[R262]

“Chi-squared test”, http://en.wikipedia.org/wiki/Chi-squared_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.

>>> 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]))