Chapter 9: Tests of Independence | CFA Level 1

1

Introduction to Tests of Independence

Tests of independence are statistical methods used to determine whether two categorical variables are related to each other. In finance and investment analysis, these tests help us understand relationships between different qualitative factors that might affect investment decisions.

Key Concepts

Independence: Two variables are independent if the occurrence of one does not affect the probability of the other
Categorical Variables: Variables that can be divided into distinct categories or groups
Association: A relationship between two variables where they tend to occur together in predictable patterns

Important Note

Tests of independence help investors understand whether factors like company size, sector, or credit rating are related to investment outcomes such as default rates or performance categories.

2

Chi-Square Test of Independence

The chi-square test of independence is the most commonly used test for determining whether two categorical variables are independent. It compares observed frequencies with expected frequencies under the assumption of independence.

Hypotheses

Null Hypothesis (H₀): The two variables are independent
Alternative Hypothesis (H₁): The two variables are not independent (they are associated)

When to Use Chi-Square Test

Both variables are categorical
Data consists of frequencies or counts
Sample size is sufficiently large
Expected frequencies in each cell are at least 5

3

Contingency Tables

A contingency table (or cross-tabulation) displays the frequency distribution of observations for two categorical variables. It forms the basis for chi-square tests.

Structure of a Contingency Table

Variable A \ Variable B	Category B₁	Category B₂	Category B₃	Row Total
Category A₁	O₁₁	O₁₂	O₁₃	R₁
Category A₂	O₂₁	O₂₂	O₂₃	R₂
Column Total	C₁	C₂	C₃	N

Where:

Oᵢⱼ = Observed frequency in cell (i,j)
Rᵢ = Row total for row i
Cⱼ = Column total for column j
N = Grand total (total sample size)

Expected Frequencies

Under the assumption of independence, the expected frequency for each cell is calculated as:

$$E_{ij} = \frac{R_i \times C_j}{N}$$

Investment Example

Suppose we want to test whether company size (Small, Medium, Large) is independent of investment performance (Poor, Average, Good). We collect data from 300 companies:

Company Size	Poor Performance	Average Performance	Good Performance	Total
Small	45	55	50	150
Medium	25	35	40	100
Large	10	20	20	50
Total	80	110	110	300

4

Chi-Square Test Statistic

The chi-square test statistic measures the difference between observed and expected frequencies. It follows a chi-square distribution under the null hypothesis of independence.

Formula for Chi-Square Statistic

$$\chi^2 = \sum_{i=1}^{r} \sum_{j=1}^{c} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$$

Where:

χ² = Chi-square test statistic
Oᵢⱼ = Observed frequency in cell (i,j)
Eᵢⱼ = Expected frequency in cell (i,j)
r = Number of rows
c = Number of columns

Properties of Chi-Square Statistic

Always non-negative (χ² ≥ 0)
Large values indicate greater deviation from independence
Small values suggest the variables might be independent
Follows chi-square distribution with appropriate degrees of freedom

Calculating Expected Frequencies

Using our investment example, let's calculate expected frequencies:

For Small companies with Poor performance:

$$E_{11} = \frac{150 \times 80}{300} = 40$$

For Medium companies with Average performance:

$$E_{22} = \frac{100 \times 110}{300} = 36.67$$

Complete expected frequencies table:

Company Size	Poor	Average	Good
Small	40.00	55.00	55.00
Medium	26.67	36.67	36.67
Large	13.33	18.33	18.33

5

Degrees of Freedom

The degrees of freedom for a chi-square test of independence depend on the size of the contingency table. This determines which chi-square distribution to use for finding critical values and p-values.

Formula for Degrees of Freedom

$$df = (r - 1) \times (c - 1)$$

Where:

df = degrees of freedom
r = number of rows in the contingency table
c = number of columns in the contingency table

Understanding Degrees of Freedom

Degrees of freedom represent the number of values that can vary freely when calculating the test statistic, given the constraint that row and column totals are fixed.

Degrees of Freedom Examples

2×2 table: df = (2-1) × (2-1) = 1
3×3 table: df = (3-1) × (3-1) = 4
3×4 table: df = (3-1) × (4-1) = 6
Our investment example (3×3): df = (3-1) × (3-1) = 4

Important Consideration

As the degrees of freedom increase, the chi-square distribution becomes more symmetric and approaches a normal distribution. This affects the critical values and the shape of the rejection region.

6

Critical Values and Decision Rule

The decision to reject or fail to reject the null hypothesis is based on comparing the calculated chi-square statistic with the critical value from the chi-square distribution.

Decision Rule

If χ² > χ²critical: Reject H₀ (variables are not independent)
If χ² ≤ χ²critical: Fail to reject H₀ (variables may be independent)

Common Critical Values

Degrees of Freedom	α = 0.10	α = 0.05	α = 0.01
1	2.706	3.841	6.635
2	4.605	5.991	9.210
3	6.251	7.815	11.345
4	7.779	9.488	13.277
5	9.236	11.071	15.086

P-value Approach

Alternatively, you can calculate the p-value and compare it with the significance level:

If p-value < α: Reject H₀
If p-value ≥ α: Fail to reject H₀

7

Practical Examples

Example 1: Company Size and Performance

Let's complete our investment example by calculating the chi-square statistic:

Step 1: Calculate chi-square statistic

$$\chi^2 = \frac{(45-40)^2}{40} + \frac{(55-55)^2}{55} + \frac{(50-55)^2}{55} + \frac{(25-26.67)^2}{26.67} + \frac{(35-36.67)^2}{36.67} + \frac{(40-36.67)^2}{36.67} + \frac{(10-13.33)^2}{13.33} + \frac{(20-18.33)^2}{18.33} + \frac{(20-18.33)^2}{18.33}$$

$$\chi^2 = 0.625 + 0 + 0.455 + 0.104 + 0.076 + 0.303 + 0.833 + 0.152 + 0.152 = 2.70$$

Step 2: Compare with critical value

With df = 4 and α = 0.05, χ²critical = 9.488

Since 2.70 < 9.488, we fail to reject H₀

Conclusion: There is insufficient evidence to conclude that company size and performance are associated.

Example 2: Investment Sector and Risk Rating

An investment firm wants to test whether investment sector is independent of risk rating. Data from 200 investments:

Sector	Low Risk	Medium Risk	High Risk	Total
Technology	15	25	35	75
Healthcare	20	30	25	75
Financial	25	15	10	50
Total	60	70	70	200

Solution:

1. Calculate expected frequencies for each cell

2. Compute χ² = 16.67 (detailed calculation omitted for brevity)

3. df = (3-1) × (3-1) = 4

4. At α = 0.05, χ²critical = 9.488

5. Since 16.67 > 9.488, reject H₀

Conclusion: Investment sector and risk rating are not independent.

8

Interpretation of Results

Proper interpretation of chi-square test results is crucial for making informed investment decisions and understanding relationships between categorical variables.

When Variables are Independent

The occurrence of one category does not affect the probability of another
Observed frequencies are close to expected frequencies
Chi-square statistic is small
Investment implications: Factors can be considered separately in analysis

When Variables are Associated

Certain combinations occur more or less frequently than expected
Patterns in the data suggest relationships
Chi-square statistic is large
Investment implications: Factors should be considered together

Practical Implications for Investors

Investment Applications

Portfolio Construction: Understanding which factors are independent helps in diversification
Risk Assessment: Associated variables may indicate correlated risks
Performance Analysis: Identifying factors that influence returns
Due Diligence: Testing relationships between company characteristics and outcomes

Effect Size and Practical Significance

Statistical significance doesn't always imply practical importance. Consider:

Cramér's V for measuring association strength
Residual analysis to identify which cells contribute most to the chi-square statistic
Business context and economic significance

$$\text{Cramér's V} = \sqrt{\frac{\chi^2}{N \times \min(r-1, c-1)}}$$

9

Assumptions and Limitations

Chi-square tests have several important assumptions that must be met for valid results. Understanding these limitations is crucial for proper application in financial analysis.

Key Assumptions

Independence of Observations: Each observation must be independent of others
Expected Frequency Requirement: Expected frequency in each cell should be at least 5
Mutually Exclusive Categories: Each observation belongs to exactly one category for each variable
Random Sampling: Data should be collected through random sampling

When Assumptions are Violated

Violation	Consequence	Solution
Small expected frequencies	Test statistic distribution is inaccurate	Fisher's exact test or combine categories
Non-independent observations	Inflated Type I error rate	Use appropriate clustering methods
Non-random sampling	Results may not be generalizable	Acknowledge limitations in interpretation

Limitations in Financial Applications

Market Dependencies: Financial data often exhibits temporal dependencies
Sample Size Issues: Small samples in specialized investment categories
Category Definition: Subjective categorization of continuous variables
Time-Varying Relationships: Relationships may change over different market conditions

Best Practices

Always check expected frequencies before conducting the test
Consider the business context when interpreting results
Use complementary analysis methods when possible
Be cautious about causal interpretations

10

Chapter Summary

Key Learning Points

Tests of Independence: Statistical methods to determine if categorical variables are related
Chi-Square Test: Most common test using observed vs. expected frequencies
Contingency Tables: Cross-tabulations that organize categorical data
Test Statistic: χ² = Σ[(O-E)²/E] follows chi-square distribution
Degrees of Freedom: df = (r-1) × (c-1) determines critical values

Investment Applications

Testing relationships between company characteristics and performance
Analyzing associations between market sectors and risk levels
Understanding dependencies in portfolio construction
Validating assumptions about factor independence

Critical Formulas

$$E_{ij} = \frac{R_i \times C_j}{N}$$

Expected Frequency

$$\chi^2 = \sum_{i=1}^{r} \sum_{j=1}^{c} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$$

Chi-Square Test Statistic

$$df = (r - 1) \times (c - 1)$$

Degrees of Freedom

Next Steps

In the next chapter, we'll explore Simple Linear Regression, which allows us to model relationships between continuous variables and make predictions based on linear associations.