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ANOVA: Analysis of Variance

Instructor: Adlyn Perez-Figueroa
Course: PSY3010

Overview of ANOVA

• ANOVA stands for Analysis of Variance.
• It includes tests that compare two or more means.
  

Types of ANOVA Tests

• One-sample t-test: Tests the mean of a single sample against a known mean.
• Dependent t-test: Compares means from the same group at different times.
• Independent t-test: Compares means between two independent groups.
• Analysis of Variance (ANOVA):
    - Utilizes the F-statistic to compare three or more means.
    - Avoids multiple t-tests to reduce the risk of Type I error.

Assumptions of ANOVA

1. Normality:
      - Samples should be drawn from a normally distributed population.
2. Independence:
      - Observations must be independent of each other.
3. Homogeneity of Variance:
      - Group variances should be approximately equal.

Experimental Design in ANOVA

Independent Variable

• Represents the groups being compared in the study.
• Levels: The number of groups for comparison.

Dependent Variable

• Must be continuous and measure outcomes of interest.

Testing Assumptions

• Independence:
     - Ensure study design prevents overlap among participants (e.g., participants should not participate in the survey more than once).
• Normality:
     - Utilize a Q-Q plot for visual assessment.
     - Shapiro-Wilk Test:
       - A non-significant p-value suggests that the assumption of normality is met.
• Homogeneity of Variance:
     - Conduct Levene’s Test:
       - A non-significant p-value indicates that this assumption is satisfied.

ANOVA Calculations

1. Group Means: Denoted as M1, M2, M3 for each group being compared.
2. Grand Mean (Mg):
      - The mean of all participants across all groups.
3. Sample Sizes:
      - Total sample size denoted by N.    - Group sample sizes as n1, n2, n3.

Hypothesis Testing in ANOVA

Hypotheses

• Null Hypothesis (H0):
     - States that there are no differences among the group means:
  $H_0: ext{µ}_1 = ext{µ}_2 = ext{µ}_3$
• Alternative Hypothesis (HA):
     - Suggests that at least one mean is different among the groups.

Finding the F-Statistic

Steps to Calculate F

1. Calculate the variance or differences between groups.
2. Calculate the variance within each group.
3. Compare both variances:
      - If the between-group variance is significantly larger than the within-group variance, the F-statistic is statistically significant.
4. The formula for F-statistic is:
   $F = \frac{ ext{between group variance}}{ ext{within group variance}}$

Detailed Calculations for F-Statistic

2A: Sum of Squares Between (SSB)

• Reflects systematic variability.
• To calculate SSB:
     - Subtract each group mean from the grand mean and then square and sum them.
  

2B: Sum of Squares Within (SSW)

• Reflects random error within groups.
• To calculate SSW:
     - Take each observation, subtract it from its group mean, square the result, and sum them.
• If SSW is too large, F may not be significant.
  

2C: Degrees of Freedom

• Degrees of freedom between:
     - $df_B = k - 1$
     - where k is the number of groups.
• Degrees of freedom within:
     - $df_W = N - k$
• Degrees of freedom total:
     - $df_T = N - 1$
  

2D: Mean Square Calculation

• Represent overall variability:
     - Mean Square Between (MSB):
       - $MSB = \frac{SSB}{df_B}$
     - Mean Square Within (MSW):
       - $MSW = \frac{SSW}{df_W}$
  

2E: F-Statistic Calculation

• To compute the F-statistic:
     - $F = \frac{MSB}{MSW}$
• A significantly larger MSB compared to MSW indicates statistical significance, suggesting real differences exist.

Finding the P-value

• The p-value interpretation follows the same principles as it does in t-tests.
• A statistically significant p-value indicates at least one group differs from the others.

Post-Hoc Testing

• If a significant F is found, use post-hoc tests to identify which groups differ.
• Tukey Test:
     - The most commonly used post-hoc test in ANOVA for multiple comparisons.

Additional Resources

• JAMOVI: Software tool for statistical analysis and practice with ANOVA.