# What is the $\chi^2$ contingency test really showing?

I have a friend who used this function to compute a chi-squared contingency table in a paper they are writing, and I don't think I fully understand what the results of this test are.

Function docstring in case the link breaks:

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

The null hypothesis, as I understand it, is that the predictor variables are independent of the groups. More accurately, it is saying that the same distribution generated the data for all the groups.
What I don't understand is the alternative hypothesis. Is this saying that none of the variables are associated with the groups, or that they are collectively independent from the groups?

• Welcome to the forum. It is standard practice here to incorporate any relevant information you link to directly in the post, since links often disappear in time. So you might want to edit your post to explain the function you're referencing. Just a head's-up. :) Aug 4, 2022 at 15:18
• By definition, the alternative hypothesis is any of the probability models not included in the null hypothesis.
– whuber
Aug 4, 2022 at 16:06
• Wasn't that topic discussed many times on this site already? Did you try to search. Search posts with both your tags together. Aug 4, 2022 at 18:52

Let's start with a small example. Say we have two groups, smokers and non-smokers, and we want to know the proportion of those who developed lung cancer.

The data look like

m = np.array([[10, 29,], [90, 71]])



10 of the 100 non-smokers developed lung cancer, and 29 out of 100 of the smokers developed lung cancer.

The null hypothesis for the test can be restated as follows: The frequency of deaths within each group (smokers and non-smokers) is the same.

The alternative hypothesis sis that the frequencies are different within groups (or more precisely, that the frequency of deaths within each group is not the same). That can be interpreted in a bunch of ways, but what it really means is that there is at least once group with a different frequency. For two groups, that boils down to a difference between groups, but for more than 2 groups it means at least one is different.

Let's run the test.

from scipy.stats import chi2_contingency
import numpy as np

m = np.array([[10, 29,], [90, 71]])

Xi2, pval, df, expected = chi2_contingency(m)

print(pval)
>>>0.0013158811321789705


The p value is smaller than our nominal value (usually 0.05). This means that the probability of observing data at least as extreme as the ones we have observed is quite unlikely assuming the two did in reality have the same frequency of lung cancer. We would then conclude that the frequency of lung cancer between the two groups is different.

• If you were using multiple rows for various predictors, would you conclude that at least one predictor is dependent on the columns? Aug 5, 2022 at 13:32
• @RyanFolks Yes, that is correct Aug 5, 2022 at 14:03