Study Notes: Associations between Categorical Variables (Chapter 10)

Associations between Categorical Variables

Expected Counts in Contingency Tables

When analyzing categorical data, expected counts help determine whether observed frequencies differ from what would be expected under independence.

• Expected Count: The number of cases expected in a cell of a contingency table if the row and column variables are independent.

• Formula: $\text{Expected Count} = \frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}$

• Example: If a survey divides respondents by TV commercial version and purchase, expected counts for each cell are calculated using the formula above.

Comparing Observed and Expected Counts

Comparing observed counts to expected counts allows us to assess whether there is an association between categorical variables.

• Observed Count: The actual number of cases in each cell.

• Weighted Average: Used to compare effectiveness across groups. $\text{Weighted Average} = \frac{\sum (\text{Group Size} \times \text{Group Mean})}{\text{Total Size}}$

• Example: If Version 1 and Version 2 of a commercial have different purchase rates, calculate the weighted average to compare overall effectiveness.

Distribution of Sample Proportions

When sampling proportions, the distribution can be approximated as normal if certain conditions are met.

• Normal Approximation: The sampling distribution of proportions is approximately normal if $np \geq 10$ and $n(1-p) \geq 10$.

• Mean and Standard Deviation: $\mu = p$ $\sigma = \sqrt{\frac{p(1-p)}{n}}$

• Example: For 1000 books, if 24.8% are hardcover, $\mu = 0.248$, $\sigma = \sqrt{\frac{0.248 \times 0.752}{1000}}$.

• Note: If the distribution is not normal, normal approximation should not be used.

Types of Association in Contingency Tables

Associations between categorical variables can be positive, negative, or non-existent.

• Positive Association: Higher values of one variable are associated with higher values of another.

• Negative Association: Higher values of one variable are associated with lower values of another.

• No Association: Variables are independent; observed counts are close to expected counts.

• Example Table:

	Positive	Negative	None
Observed	Higher than expected	Lower than expected	Close to expected
Interpretation	Variables are associated	Variables are inversely associated	No association

Chi-Square Test for Association

The Chi-Square Test is used to determine if there is a significant association between categorical variables.

• Null Hypothesis ($H_0$): The variables are independent.

• Alternative Hypothesis ($H_A$): The variables are associated.

• Test Statistic: $\chi^2 = \sum \frac{(O - E)^2}{E}$ where $O$ is the observed count and $E$ is the expected count.

• Decision Rule: If $p$-value < significance level, reject $H_0$.

• Example: If the $p$-value is 0.03 and the significance level is 0.05, reject the null hypothesis and conclude there is an association.

Interpreting Statistical Significance

Statistical significance indicates whether the observed association is likely due to chance.

• Significant Result: Suggests a real association between variables.

• Non-Significant Result: Suggests no evidence of association.

• Example: If the test statistic is large and the $p$-value is small, the result is significant.

Summary Table: Steps in Analyzing Categorical Association

Step	Description
1. State Hypotheses	Define $H_0$ and $H_A$
2. Calculate Expected Counts	Use formula for each cell
3. Compute Test Statistic	Calculate $\chi^2$
4. Find $p$-value	Compare to significance level
5. Draw Conclusion	Interpret result in context

Additional info: These notes expand on brief review answers by providing definitions, formulas, and structured steps for analyzing associations between categorical variables, as covered in Chapter 10 of a college statistics course.