scipy.stats.chi2_contingency

scipy.stats.chi2_contingency(observed, correction=True, lambda_=None)

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 [R221] 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:

observed : array_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”.

correction : bool, 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 [R222]. lambda_ allows a statistic from the Cressie-Read power divergence family [R223] to be used instead. See power_divergence for details.

Returns:

chi2 : float

The test statistic.

p : float

The p-value of the test

dof : int

Degrees of freedom

expected : ndarray, same shape as observed

The expected frequencies, based on the marginal sums of the table.

See also

contingency.expected_freq, fisher_exact, chisquare, power_divergence

Notes

An often quoted guideline for the validity of this calculation is that the test should be used only if the observed and expected frequency in each cell is 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

[R221]

(1, 2) “Contingency table”, http://en.wikipedia.org/wiki/Contingency_table

[R222]

(1, 2) “Pearson’s chi-squared test”, http://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test

[R223]

(1, 2) 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):

>>> 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]]]]))