1.10. Non-Parametric test for Hypothesis Testing: Chi-Square Goodness-of-fit

The Chi-Square test is a statistical procedure for determining the difference between observed and expected data. This test can also be used to determine whether it correlates to the categorical variables in our data. It helps to find out whether a difference between two categorical variables is due to chance or a relationship between them [1].

There are two main types of Chi-Square tests [2]: 

• Chi-Square Goodness-of-Fit Use the goodness-of-fit test to decide whether a population with an unknown distribution “fits” a known distribution.

• Use the test for independence to decide whether two variables (factors) are independent or dependent, i.e. whether these two variables have a significant association relationship between them or not .

Chi-Square Goodness-of-Fit Test - Step-by-Step Manual Example

A factory tracks the number of machinery breakdowns each day of the week. The data is as follows:

χobs 2 = 0.25+2.25+0+0.25+1+0.5625+1.5625 = 5.875

4. Determine the Degrees of Freedom:

Degrees of freedom (df) = 𝑛−1, where: 𝑛  is the number of categories (days).

df = 7 − 1 = 6

5. Find the Critical Value:

At a 5% level of significance (α = 0.05) and 6 degrees of freedom, the critical value from the Chi-Square distribution table is approximately χcrit 2 = 12.592.

import scipy.stats as stats

# Observed frequencies

observed = [14, 22, 16, 18, 12, 19, 11]

# Expected frequencies (uniform distribution)

total_breakdowns = sum(observed)

expected = [total_breakdowns / 7] * 7

# Calculate Chi-Square statistic

chi_square_statistic, p_value = stats.chisquare(observed, expected)

# Degrees of freedom

df = len(observed) - 1

# Critical value at 5% significance level

critical_value = stats.chi2.ppf(0.95, df)

# Print the results

print(f"Chi-Square Statistic: {chi_square_statistic}")

print(f"Degrees of Freedom: {df}")

print(f"Critical Value: {critical_value}")

print(f"P-Value: {p_value}")

if chi_square_statistic < critical_value:

    print("Accept the null hypothesis: Breakdowns are uniformly distributed.")

else:

    print("Reject the null hypothesis: Breakdowns are not uniformly distributed.")

Chi-Square Statistic: 5.875

Degrees of Freedom: 6

Critical Value: 12.591587243743977

P-Value: 0.43733749366050445

Accept the null hypothesis: Breakdowns are uniformly distributed. 

1. State the Hypotheses: 

• Null Hypothesis (H0):  Gender and preferred shirt color are independent.

• Alternative Hypothesis (Ha): Gender and preferred shirt color are not independent.

2. Calculate the Expected Frequency: 

For calculating Chi-squared test statistics we need to calculate the expected value. So, add all the rows and columns and overall totals:

import pandas as pd

from scipy.stats import chi2_contingency

from scipy.stats import chi2

# Given dataset

df_dict = {

   'Black': [48, 34],

   'White': [12, 46],

   'Red': [33, 42],

   'Blue': [57, 26]

}

dataset_table = pd.DataFrame(df_dict, index=['Male', 'Female'])

print("Dataset Table:")

print(dataset_table)

print()

# Observed Values

Observed_Values = dataset_table.values

print("Observed Values:")

print(Observed_Values)

print()

# Perform chi-square test

val = chi2_contingency(dataset_table)

Expected_Values = val[3]

print("Expected Values:")

print(Expected_Values)

print()

# Degree of Freedom

no_of_rows = len(dataset_table.iloc[0:2, 0])

no_of_columns = 4

ddof = (no_of_rows - 1) * (no_of_columns - 1)

print("Degree of Freedom:", ddof)

print()

# Chi-square statistic

chi_square = sum([(o - e) ** 2. / e for o, e in zip(Observed_Values, Expected_Values)])

chi_square_statistic = chi_square[0] + chi_square[1]

print("Chi-square statistic:", chi_square_statistic)

print()

# Critical value

alpha = 0.05

# p-value

p_value = 1 - chi2.cdf(x=chi_square_statistic, df=ddof)

print('p-value:', p_value)

print()

# Significance level

print('Significance level:', alpha)

print('p-value:', p_value)

if p_value <= alpha:

   print("Reject H0, Gender and preferred shirt color are independent")

else:

   print("Fail to reject H0, Gender and preferred shirt color are not independent")

Dataset Table:

        Black  White  Red  Blue

Male       48     12   33    57

Female     34     46   42    26

Observed Values:

[[48 12 33 57]

 [34 46 42 26]]

Expected Values:

[[41.27516779 29.19463087 37.75167785 41.77852349]

 [40.72483221 28.80536913 37.24832215 41.22147651]]

Degree of Freedom: 3

Chi-square statistic: 22.597058622962738

p-value: 4.899565434945963e-05

Significance level: 0.05

p-value: 4.899565434945963e-05

Reject H0, Gender and preferred shirt color are independent