Python – Pearson’s Chi-Square Test

The Pearson’s Chi-Square statistical hypothesis is a test for independence between categorical variables. In this article, we will perform the test using a mathematical approach and then using Python’s SciPy module.
First, let us see the mathematical approach :

The Contingency Table :
A Contingency table (also called crosstab) is used in statistics to summarise the relationship between several categorical variables. Here, we take a table that shows the number of men and women buying different types of pets.

  dog cat bird total
men 207 282 241 730
women 234 242 232 708
total 441 524 473 1438

The aim of the test is to conclude whether the two variables( gender and choice of pet ) are related to each other.

Null hypothesis:
We start by defining the null hypothesis (H0) which states that there is no relation between the variables. An alternate hypothesis would state that there is a significant relation between the two.

We can verify the hypothesis by these methods:

  • Using p-value:

We define a significance factor to determine whether the relation between the variables is of considerable significance. Generally a significance factor or alpha value of 0.05 is chosen. This alpha value denotes the probability of erroneously rejecting H0 when it is true. A lower alpha value is chosen in cases where we expect more precision. If the p-value for the test comes out to be strictly greater than the alpha value, then H0 holds true.

  • Using chi-square value:

If our calculated value of chi-square is less or equal to the tabular(also called critical) value of chi-square, then H0 holds true.

Expected Values Table :

Next, we prepare a similar table of calculated(or expected) values. To do this we need to calculate each item in the new table as :

\frac{row\ total\ *\ column\ total}{grand\ total}

The expected values table :

  dog cat bird total
men 223.87343533 266.00834492 240.11821975 730
women 217.12656467 257.99165508 232.88178025 708
total 441 524 473 1438

Chi-Square Table :

We prepare this table by calculating for each item the following:

\frac{( Observed\_value\ -\ Calculated\_value)^2 }{ Calculated\_value}

The chi-square table:

  observed (o) calculated (c) (o-c)^2 / c
  207 223.87343533 1.2717579435607573
  282 266.00834492 0.9613722161954465
  241 240.11821975 0.003238139990850831
  234 217.12656467 1.3112758457617977
  242 257.99165508 0.991245364156322
  232 232.88178025 0.0033387601600580606
Total     4.542228269825232

From this table, we obtain the total of the last column, which gives us the calculated value of chi-square.  Hence the calculated value of chi-square is 4.542228269825232

Now, we need to find the critical value of chi-square. We can obtain this from a table. To use this table, we need to know the degrees of freedom for the dataset.  The degrees of freedom is defined as : (no. of rows – 1) * (no. of columns – 1).
Hence, the degrees of freedom is (2-1) * (3-1) = 2

Now, let us look at the table and find the value corresponding to 2 degrees of freedom and 0.05 significance factor :

The tabular or critical value of chi-square here is  5.991


critical\ value\ of\ \chi^2\ >=\ calculated\ value\ of\ \chi^2

Therefore, H0 is accepted, that is, the variables do not have a significant relation.

Next, let us see how to perform the test in Python.

Performing the test using Python (scipy.stats) :

SciPy is an Open Source Python library, which is used in mathematics, engineering, scientific and technical computing. 


pip install scipy

The chi2_contingency() function of scipy.stats module takes as input, the contingency table in 2d array format. It returns a tuple containing test statistics, the p-value, degrees of freedom and expected table(the one we created from the calculated values) in that order. 

Hence, we need to compare the obtained p-value with alpha value of 0.05.





from scipy.stats import chi2_contingency
# defining the table
data = [[207, 282, 241], [234, 242, 232]]
stat, p, dof, expected = chi2_contingency(data)
# interpret p-value
alpha = 0.05
print("p value is " + str(p))
if p <= alpha:
    print('Dependent (reject H0)')
    print('Independent (H0 holds true)')


Output : 

p value is 0.1031971404730939
Independent (H0 holds true)


p-value > alpha 

Therefore, we accept H0, that is, the variables do not have a significant relation.

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