Introduction
The chi-square test for independence is a statistical test used to determine whether there is a significant association between categorical variables. It assesses whether the observed frequencies of categorical variables differ significantly from the expected frequencies under the assumption that there is no relationship between the variables. This lesson covers the definition, calculation, interpretation, assumptions, and practical applications of the chi-square test for independence in data science.
Definition
The chi-square test for independence evaluates the null hypothesis, which states that there is no association between the categorical variables X and Y. The alternative hypothesis suggests that there is a significant association between the variables.
Calculation
To perform the chi-square test for independence:
Construct a Contingency Table: Organize the data into a contingency table (also known as a cross-tabulation or contingency table) that summarizes the frequencies of observations for each combination of categories of X and Y.
Calculate Expected Frequencies: Compute the expected frequencies for each cell under the assumption of independence using the formula:
\[
Eij = \frac{(\text{row sum}_i \times \text{column sum}_j)}{\text{total sum}}
\]
Compute the Chi-Square Statistic: Calculate the chi-square statistic χ2 using the formula:
\[
χ2 = \sum \frac{(O_{ij} – E_{ij})^2}{E_{ij}}χ2 = ∑Eij(Oij−Eij)2
\]
Where Oij is the observed frequency and Eij is the expected frequency.
Degrees of Freedom: Determine the degrees of freedom df for the chi-square distribution, calculated as:
\[
df=(r−1)(c−1)
\]
Compare with Critical Value: Compare the calculated chi-square statistic χ2 with the critical value from the chi-square distribution table at a specified significance level (e.g., α=0.05). If χ2 exceeds the critical value, reject the null hypothesis, indicating a significant association between variables X and Y.
Interpretation
- Significance Level: Choose a significance level α to assess the statistical significance of the test.
- Strength of Association: The larger the chi-square statistic χ2, the stronger the association between variables X and Y.
Assumptions of Chi-Square Test for Independence
- Independence: Observations must be independent within and between categories.
- Sample Size: Each expected frequency should be sufficiently large (typically, no more than 20% of cells should have expected frequencies less than 5).
Practical Applications
- Market Research: Analyze associations between demographic variables (e.g., age group and product preference) in consumer surveys.
- Healthcare: Assess relationships between risk factors (e.g., smoking status and disease outcomes) in epidemiological studies.
- Education: Evaluate the effectiveness of teaching methods by comparing student performance across different instructional strategies.
Example Calculation
import numpy as np
from scipy.stats import chi2_contingency
# Example data
observed = np.array([[10, 20, 30], [15, 25, 35]])
# Perform chi-square test for independence
chi2, p_value, dof, expected = chi2_contingency(observed)
print(f"Chi-square statistic: {chi2}")
print(f"P-value: {p_value}")
print(f"Degrees of freedom: {dof}")
print("Expected frequencies:")
print(expected)
Conclusion
The chi-square test for independence is a powerful statistical tool for analyzing relationships between categorical variables. By understanding how to calculate and interpret the chi-square statistic, data scientists can determine whether observed associations are statistically significant, derive insights from categorical data, and make informed decisions in various fields of data science and research. Mastery of the chi-square test supports rigorous hypothesis testing, enhances the validity of analytical conclusions, and facilitates effective communication of findings in both academic and practical applications.