module 3 stat section B2

Overview of Chi Squared Test of Independence

• Purpose: Analyze associations between two categorical variables.

• Statistical Test: Chi Squared Test of Independence.

• Compared to: Chi Squared Goodness of Fit Test (used for single categorical variable).

Example Study Design

• Research Question: Is it better to give up smoking by going cold turkey or joining a smoking support group?

• Sample Size: 320 people total.

  • 200 participants in smoking support group.

  • 120 participants going cold turkey.

• Design Type: Quasi-experimental (non-random allocation to groups).

Potential Biases

• Concern: Age as a confounding factor.

  • Younger individuals may be more likely to achieve cessation, if more young people attend one method over the other, results may be skewed towards that demographic.

Hypothesis Formulation

• Null Hypothesis (H0): Smoking support groups do not increase the proportion of people giving up smoking.

• Alternative Hypothesis (H1): Smoking support groups increase the proportion of people that give up smoking.

  • Type: One-sided hypothesis (only interested in increase).

Results Summary

• Example data shows higher quit rate in the support group compared to the cold turkey group.

• Notable Observations:

  • 50% of support group participants quit smoking.

  • <50% of cold turkey participants quit smoking.

  • High number (48 participants) in cold turkey group showed no change in smoking habits.

Calculation of Expected Frequencies

• Under null hypothesis, calculate expected outcomes for each category.

• Example Calculation:

  • Total who gave up smoking: 184.

  • Expected in smoking support group (n=200): 115.

  • Expected in cold turkey group (n=120): 69.

• Repeat for other categories: smoking less and no change.

Test Statistic

• Calculation method:

  • For each cell: (Observed - Expected)^2 / Expected

  • Sum all results for total chi squared value (χ²).

• Example Result: Overall chi squared value = 34.3.

Interpretation of Test Statistic

• High chi squared value indicates significant discrepancy between observed and expected results, suggesting relationship exists.

• Check significance using the degrees of freedom (df).

  • df for a 2x3 table: (rows-1)(columns-1) = 2.

• Typical threshold for df=2: 5.9; our chi squared = 34.3 implies significant association.

P-value Calculation

• Determine the p-value for χ² = 34.3 with df=2.

• Result: p-value < 0.001; highly significant.

• Conclusion: Reject null hypothesis; smoking support groups likely positively affect cessation rates.

Assumptions of the Chi Squared Test

1. Observational units (participants) are independent.

2. Expected counts in each cell should be ≥ 5.

   • If counts fall below this threshold, consider combining categories.

Visual Representation

• Percentages can effectively display results.

  • Smoking Support Group: 64% quit.

  • Cold Turkey: 47% quit.

• Graphs (e.g. side-by-side bar chart) visualize differences in quit rates effectively.

Future Topics

• Continuing analysis of association measures between categorical variables.

• Explore more techniques for summarizing differences in groups.