scipy.stats.contingency.chi2_contingency#

scipy.stats.contingency.chi2_contingency(observed, correction=True, lambda_=None)[source]#

Chi-square test of independence of variables in a contingency table.

This function computes the chi-square statistic and p-value for the hypothesis test of independence of the observed frequencies in the contingency table [1] observed. The expected frequencies are computed based on the marginal sums under the assumption of independence; see scipy.stats.contingency.expected_freq. The number of degrees of freedom is (expressed using numpy functions and attributes):

dof = observed.size - sum(observed.shape) + observed.ndim - 1

Parameters

observedarray_like: The contingency table. The table contains the observed frequencies (i.e. number of occurrences) in each category. In the two-dimensional case, the table is often described as an “R x C table”.
correctionbool, optional: If True, and the degrees of freedom is 1, apply Yates’ correction for continuity. The effect of the correction is to adjust each observed value by 0.5 towards the corresponding expected value.
lambda_float or str, optional: By default, the statistic computed in this test is Pearson’s chi-squared statistic [2]. lambda_ allows a statistic from the Cressie-Read power divergence family [3] to be used instead. See scipy.stats.power_divergence for details.

Returns

chi2float: The test statistic.
pfloat: The p-value of the test
dofint: Degrees of freedom
expectedndarray, same shape as observed: The expected frequencies, based on the marginal sums of the table.

See also

scipy.stats.contingency.expected_freq
scipy.stats.fisher_exact
scipy.stats.chisquare
scipy.stats.power_divergence
scipy.stats.barnard_exact
scipy.stats.boschloo_exact

Notes

An often quoted guideline for the validity of this calculation is that the test should be used only if the observed and expected frequencies in each cell are at least 5.

This is a test for the independence of different categories of a population. The test is only meaningful when the dimension of observed is two or more. Applying the test to a one-dimensional table will always result in expected equal to observed and a chi-square statistic equal to 0.

This function does not handle masked arrays, because the calculation does not make sense with missing values.

Like stats.chisquare, this function computes a chi-square statistic; the convenience this function provides is to figure out the expected frequencies and degrees of freedom from the given contingency table. If these were already known, and if the Yates’ correction was not required, one could use stats.chisquare. That is, if one calls:

chi2, p, dof, ex = chi2_contingency(obs, correction=False)

then the following is true:

(chi2, p) == stats.chisquare(obs.ravel(), f_exp=ex.ravel(),
                             ddof=obs.size - 1 - dof)

The lambda_ argument was added in version 0.13.0 of scipy.

References

1: “Contingency table”, https://en.wikipedia.org/wiki/Contingency_table
2: “Pearson’s chi-squared test”, https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test
3: Cressie, N. and Read, T. R. C., “Multinomial Goodness-of-Fit Tests”, J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984), pp. 440-464.

Examples

A two-way example (2 x 3):

>>> from scipy.stats import chi2_contingency
>>> obs = np.array([[10, 10, 20], [20, 20, 20]])
>>> chi2_contingency(obs)
(2.7777777777777777,
 0.24935220877729619,
 2,
 array([[ 12.,  12.,  16.],
        [ 18.,  18.,  24.]]))

Perform the test using the log-likelihood ratio (i.e. the “G-test”) instead of Pearson’s chi-squared statistic.

>>> g, p, dof, expctd = chi2_contingency(obs, lambda_="log-likelihood")
>>> g, p
(2.7688587616781319, 0.25046668010954165)

A four-way example (2 x 2 x 2 x 2):

>>> obs = np.array(
...     [[[[12, 17],
...        [11, 16]],
...       [[11, 12],
...        [15, 16]]],
...      [[[23, 15],
...        [30, 22]],
...       [[14, 17],
...        [15, 16]]]])
>>> chi2_contingency(obs)
(8.7584514426741897,
 0.64417725029295503,
 11,
 array([[[[ 14.15462386,  14.15462386],
          [ 16.49423111,  16.49423111]],
         [[ 11.2461395 ,  11.2461395 ],
          [ 13.10500554,  13.10500554]]],
        [[[ 19.5591166 ,  19.5591166 ],
          [ 22.79202844,  22.79202844]],
         [[ 15.54012004,  15.54012004],
          [ 18.10873492,  18.10873492]]]]))