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_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 -------- scipy.stats.power_divergence
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 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
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))