Understanding the Chi-Square Test
The chi-square test of independence examines whether two categorical variables are independent of each other or related in some way. This test is fundamental in statistical analysis when working with categorical data.
For example, researchers might use this test to determine whether:
- Education level is related to voting preference
- Gender is related to product preference
- Treatment type is related to recovery outcomes
How the Test Works
The chi-square test compares observed frequencies with expected frequencies calculated under the assumption of independence. The test evaluates whether any differences between observed and expected values are due to chance or a real relationship.
Χ² = Σ((O-E)²/E)
Where O = observed frequency and E = expected frequency
Key Components
- Observed frequencies (O): The actual count in each category
- Expected frequencies (E): What we would expect if variables were independent
- Degrees of freedom: (r-1)(c-1) where r = number of rows and c = number of columns
- p-value: Probability of obtaining the observed results by chance
Null Hypothesis (H₀)
The two categorical variables are independent (not related).
Alternative Hypothesis (H₁)
The two categorical variables are dependent (related).
If the p-value is less than your significance level (typically 0.05), you reject the null hypothesis and conclude that the variables are related.
Practical Example
Imagine a researcher wants to know if education level (high school, college, graduate) is related to adoption of a new technology (adopted, not adopted).
The steps would be:
- Collect data on both variables for a sample population
- Create a contingency table of observed frequencies
- Calculate expected frequencies for each cell
- Compute the chi-square statistic
- Determine degrees of freedom: (3-1)(2-1) = 2
- Find the p-value and compare to significance level
- Draw a conclusion about the relationship
Important Considerations
For reliable results, each expected frequency should be at least 5. If this assumption is violated, Fisher's exact test may be more appropriate for small samples.
Knowledge Check
What does the chi-square test of independence determine?