Lecture Notes: Analysis of Variance (ANOVA)

Overview of ANOVA

• ANOVA is used to analyze the contributions of different factors to variability in measurements.
• Factors: variables like gender, height, age, etc.
• Settings: gradations of a factor.
• Treatment: sometimes equivalent to settings; may involve combinations (e.g., gender and age).
• Response: the measured outcome.

Assumptions of ANOVA

• Each population is normally distributed with a common variance ((\sigma^2)).
• Populations can have different means.

Example Scenarios

• Scenario A: Low variability within groups, high variability between groups.
• Scenario B: High variability within groups, low variability between groups.

Applications of ANOVA

• Comparison of two or more means.
• Useful when multiple factors or treatments are involved.

Experimental Design: Randomized Design

• Samples are randomly selected from each of (k) populations.
• Number of factors = 1; levels of the factor = (k) (number of populations).

Hypothesis Testing in ANOVA

• Null hypothesis (H_0): all population means are equal (e.g., (\mu_1 = \mu_2 = \mu_k)).
• Alternate hypothesis: at least one mean is different.
• Student's t-test vs ANOVA: ANOVA performs a single test vs multiple t-tests.

ANOVA Calculations

• Total Sum of Squares (TSS): (\sum (x_{ij} - \bar{x})^2)
• Correction for the Mean (CM): (\text{CM} = \frac{\sum x_{ij}^2}{n})
• Sum of Squares for Treatments (SST): (\sum \frac{T_i^2}{n_i} - \text{CM})
• Sum of Squares for Error (SSE): (\text{TSS} - \text{SST})
• Degrees of Freedom:
  • TSS: (n - 1)
  • SST: (k - 1)
  • SSE: (n - k)
• Mean Squares:
  • MST (Mean Square for Treatments): (\frac{SST}{k-1})
  • MSE (Mean Square for Errors): (\frac{SSE}{n-k})

ANOVA Table Structure

• Source: Treatments, Errors
• Degrees of Freedom: (k-1) for treatments, (n-k) for errors
• Sum of Squares: SST, SSE
• Mean Squares: Corresponding MSE, MST
• F-value: Test statistic (F = \frac{MST}{MSE})

Example: Nutrition and Attention Span

• Study on the effect of nutrition on student attention spans.
• Treatments: no breakfast, light breakfast, heavy breakfast.
• Calculation of sample totals, corrections, and ANOVA table.

Conclusion

• ANOVA is an effective method to test the equality of more than two means without performing multiple t-tests.
• ANOVA results in an ANOVA table that summarizes the variance sources and corresponding statistics.