# scipy.stats.chi2_contingency#

scipy.stats.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:
resChi2ContingencyResult

An object containing attributes:

statisticfloat

The test statistic.

pvaluefloat

The p-value of the test.

dofint

The degrees of freedom.

expected_freqndarray, same shape as observed

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

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 scipy.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 scipy.stats.chisquare. That is, if one calls:

res = chi2_contingency(obs, correction=False)


then the following is true:

(res.statistic, res.pvalue) == 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

[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.

[4]

Berger, Jeffrey S. et al. “Aspirin for the Primary Prevention of Cardiovascular Events in Women and Men: A Sex-Specific Meta-analysis of Randomized Controlled Trials.” JAMA, 295(3):306-313, DOI:10.1001/jama.295.3.306, 2006.

Examples

In [4], the use of aspirin to prevent cardiovascular events in women and men was investigated. The study notably concluded:

…aspirin therapy reduced the risk of a composite of cardiovascular events due to its effect on reducing the risk of ischemic stroke in women […]

The article lists studies of various cardiovascular events. Let’s focus on the ischemic stoke in women.

The following table summarizes the results of the experiment in which participants took aspirin or a placebo on a regular basis for several years. Cases of ischemic stroke were recorded:

                  Aspirin   Control/Placebo
Ischemic stroke     176           230
No stroke         21035         21018


Is there evidence that the aspirin reduces the risk of ischemic stroke? We begin by formulating a null hypothesis $$H_0$$:

The effect of aspirin is equivalent to that of placebo.

Let’s assess the plausibility of this hypothesis with a chi-square test.

>>> import numpy as np
>>> from scipy.stats import chi2_contingency
>>> table = np.array([[176, 230], [21035, 21018]])
>>> res = chi2_contingency(table)
>>> res.statistic
6.892569132546561
>>> res.pvalue
0.008655478161175739


Using a significance level of 5%, we would reject the null hypothesis in favor of the alternative hypothesis: “the effect of aspirin is not equivalent to the effect of placebo”. Because scipy.stats.contingency.chi2_contingency performs a two-sided test, the alternative hypothesis does not indicate the direction of the effect. We can use stats.contingency.odds_ratio to support the conclusion that aspirin reduces the risk of ischemic stroke.

Below are further examples showing how larger contingency tables can be tested.

A two-way example (2 x 3):

>>> obs = np.array([[10, 10, 20], [20, 20, 20]])
>>> res = chi2_contingency(obs)
>>> res.statistic
2.7777777777777777
>>> res.pvalue
0.24935220877729619
>>> res.dof
2
>>> res.expected_freq
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.

>>> res = chi2_contingency(obs, lambda_="log-likelihood")
>>> res.statistic
2.7688587616781319
>>> res.pvalue
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]]]])
>>> res = chi2_contingency(obs)
>>> res.statistic
8.7584514426741897
>>> res.pvalue
0.64417725029295503