SciPy Chi-Square Test

Chi-Square Test tests whether the variables in a crosstab are independent based on the observed frequencies. SciPy’s chi2_contingency() can help us quickly calculate chi-square statistic and p-value.

Contingency Table or Crosstab

A contingency table, also called crosstab, shows the distribution of times between two variables. If you are not familiar with crosstab, you can read the following article first.

• • Machine Learning

Pandas Crosstab

• ByWayne
• 14/12/2020

The crosstab below is obtained from Women Entrepreneurship and Labor Force.

Chi-Square Test

Chi-square test verifies whether the observed frequencies are consistent with the expected frequencies. We define the following hypothesis:

• Null hypothesis (H0) : There is no correlation between two variables (independent).
• Alternative hypothesis (H1) : Two variables are dependent.

Next, we will calculate the chi-square value and p-value. Then, use chi-square value or p-value to test whether H0 holds true.

When testing with chi-square value, if chi-square value is less than or equal to the critical value of chi-square, H0 holds true.

When testing with p-value, we will define a significance factor to determine whether there is significant correlation between variables. Generally speaking, this significance factor is taken as 0.05. Therefore, if p-value is greater than or equal to the significance factor, then H0 holds true.

Expected Value Table

The formula for calculating the expected frequencies is:

After adding the total in the above crosstab, it will be like the following table.

Using the formula, we can calculate the following expected frequency table.

Chi-Square Table

Use the following formula to calculate the chi-square table.

The finally calculated chi-square table is as follows.

According to the chi-square table above, we add up all the values ​​in the table to get the chi-square value ().

Degree of Freedom

In order to find the critical value of chi-square, we must first calculate the degrees of freedom. The degree of freedom can be calculated using the following formula.

Therefore, the degree of freedom of the above crosstab is (2 – 1) x (2 – 1) = 1.

After that, according to the degree of freedom is 1 and the significance factor is 0.05, we find the value is 3.841 in the following table.

If the chi-square value is less than or equal to the critical value of chi-square, then H0 is true. However, our chi-square value is 29.2472 and the critical value is 3.841, so 29.2472 is greater than 3.841. Therefore, H0 is rejected, that is, the two variables are dependent.

P-Value

In addition to using chi-square value to verify H0, you can also use p-value to test. When p-value is greater than or equal to the significance factor, H0 is true. scipy.stats.chi2_contingency() will return a p-value. Generally speaking, the significance factor is 0.05, so if the p-value is greater than or equal to 0.05, H0 is true.

Python SciPy

With SciPy’s chi2_contingency(), we can get all the values described above easily.

chi2, p, dof, expected = scipy.stats.chi2_contingency(observed)

• observed: The observed frequencies, the type is array_like.
• chi2: chi-square value, the type is float.
• p: p-value, the type is float.
• dof: degree of freedom, the type is int.
• expected: The expected frequencies, the type is ndarray. It has the same shape as observed.

Example

The following code shows how to use SciPy’s chi2_contingency() to verify H0. Among them, observed can also be directly filled in the return value of pandas.crosstab().

from scipy.stats import chi2_contingency
observed = [
    [20, 7],
    [0, 24],
]
chi2, p, dof, expected = chi2_contingency(observed)
print('chi-square:', chi2)
print('p-value:', p)
print('degree of freedom:', dof)
print('expected value table:', expected)
if p >= 0.05:
    print('H0 is accepted')
else:
    print('H0 is rejected')

The output is as follows.

chi-square: 29.247311827956988
p-value: 6.370454442050726e-08
degree of freedom: 1
expected value table: [[10.58823529 16.41176471]
 [ 9.41176471 14.58823529]]
H0 is rejected