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ANOVA

ANCOVA — Adjusting for a Covariate

Compare two teaching methods on post-test scores while statistically adjusting for pre-test differences. Pooled within-group slope, adjusted means, and the ANCOVA F-test.

📊 F(1,9) = 6.60, p = .030, η²p = .42
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📊 36 · ANCOVA — Adjusting for a Covariate

Comparing Two Teaching Methods, Adjusting for Pre-Test Score
ANOVA
Research Question: Two groups of 6 students (Method A, Method B) took a post-test after instruction. But students didn't start at the same level — each also has a pre-test score. Does teaching method affect the post-test score once we statistically adjust for pre-test differences?
Method A: PreMethod A: PostMethod B: PreMethod B: Post
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Adjusted group mean, using the pooled within-group regression slope b_w
Y ¯ j ′ = Y ¯ j − b w ( X ¯ j − X ¯ g r a n d ) , F = M S g r o u p . a d j M S e r r o r . a d j
  1. 1
    Pre-test means are nearly identical, post-test means are not:
    X ¯ A , p r e = 63.17 ,   X ¯ B , p r e = 62.83 ,   ( grand  X ¯ = 63.00 ) Y ¯ A , p o s t = 80.50 ,   Y ¯ B , p o s t = 71.83
    Because the groups started almost level (63.17 vs. 62.83), the unadjusted post-test gap is unlikely to be a pre-existing-ability artifact — but ANCOVA still gives the statistically correct adjustment.
  2. 2
    Pooled within-group regression slope (post on pre):
    b w = 0.6945
  3. 3
    Adjusted post-test means:
    Y ¯ A ′ = 80.50 − 0.6945 ( 63.17 − 63.00 ) = 80.38 Y ¯ B ′ = 71.83 − 0.6945 ( 62.83 − 63.00 ) = 71.95
  4. 4
    ANCOVA F-test:
    S S c o v a r i a t e = 73.16 ,   S S g r o u p . a d j = 225.33 ,   S S e r r o r . a d j = 307.18   ( d f = 9 ) F ( 1 , 9 ) = 225.33 / 1 307.18 / 9 = 6.60 , p = .030 ,   η p 2 = .423
  5. 5
    Decision rule: p = .030 < .05 → reject H₀ → Method A produces significantly higher adjusted post-test scores than Method B, controlling for pre-test.
80.38 / 71.95
Adjusted means (A/B)
6.60
F(1,9)
.030
p
.423
η²_p
🔴 Significant — F(1,9) = 6.60, p = .030, η²_p = .42. Method A's adjusted post-test mean (80.38) exceeds Method B's (71.95).
APA-7
A one-way ANCOVA was conducted, with pre-test score as the covariate. After adjusting for pre-test differences, there was a significant effect of teaching method on post-test scores, F(1, 9) = 6.60, p = .030, η²p = .42. Method A had a significantly higher adjusted mean (M = 80.38) than Method B (M = 71.95).

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