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ANOVA

Two-Way ANOVA — Teaching Method × Class Size Interaction

Full 2×2 factorial ANOVA: main effects and significant interaction F(1,8)=5.16. SS decomposition and η².

📊 F_A=16.27, F_B=5.25, F_A×B=5.16
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🔀 12 · Two-Way ANOVA (Factorial Design)

Teaching Method × Class Size on Exam Scores
ANOVA
Research Question: Does the effect of teaching method (Traditional vs. Active Learning) on exam scores depend on class size (Small ≤15 vs. Large >30)? N=24, n=6 per cell.
Small Class (≤15)Large Class (>30)Row Mean
Traditional68.070.069.0
Active Learning82.074.078.0
Column Mean75.072.073.0 (GM)
SS Formulas (balanced design, n=6 per cell)
S S A = n ⋅ k B ⋅ ∑ ( M ¯ A i − GM ) 2 , S S B = n ⋅ k A ⋅ ∑ ( M ¯ B j − GM ) 2 S S A × B = n ⋅ ∑ ( M i j − M ¯ A i − M ¯ B j + GM ) 2
  1. 1
    Main Effect A (Teaching Method):
    S S A = 6 × 2 × [ ( 69.0 − 73.0 ) 2 + ( 78.0 − 73.0 ) 2 ] = 12 × [ 16 + 25 ] = 492 F A = 492 / 1 30.25 = 16.27 , p = .001
  2. 2
    Main Effect B (Class Size):
    S S B = 6 × 2 × [ ( 75.0 − 73.0 ) 2 + ( 72.0 − 73.0 ) 2 ] = 12 × [ 4 + 1 ] = 60 F B = 60 / 1 30.25 = 1.98 , p = .174
  3. 3
    Interaction A×B — the key test:
    Residuals:  ( 68 − 69 − 75 + 73 ) = − 3 , ( 70 − 69 − 72 + 73 ) = 2 , ( 82 − 78 − 75 + 73 ) = 2 , ( 74 − 78 − 72 + 73 ) = − 3 S S A × B = 6 × [ ( − 3 ) 2 + 2 2 + 2 2 + ( − 3 ) 2 ] = 6 × 26 = 156 , F A × B = 156 30.25 = 5.16 , p = .034
  4. 4
    Interpret the interaction: Active Learning produces a much larger gain in small classes (+14 pts) than in large classes (+4 pts). Simple main effect analysis confirms the interaction.
SourceSSdfMSFpη²p
Method (A)492149216.27.001.448
Class Size (B)601601.98.174.090
A × B Interaction15611565.16.034.205
Error (Within)6052030.25
Total131323
✅ Significant interaction A×B, F(1,20)=5.16, p=.034 — the benefit of Active Learning is significantly larger in small classes.
APA-7
A 2×2 between-subjects ANOVA revealed a significant main effect of teaching method, F(1, 20) = 16.27, p = .001, η²p = .45, and a significant Method × Class Size interaction, F(1, 20) = 5.16, p = .034, η²p = .21. Active Learning produced substantially larger gains in small (M = 82.0) than large classes (M = 74.0), whereas Traditional instruction showed minimal size effects.

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