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RM-ANOVA
Repeated-Measures ANOVA — Memory Scores at Three Time Points
Within-subjects ANOVA for participants measured at Baseline, Week 4, and Week 8. SS decomposition.
📊 F(2,14) significant, η² large
Step-by-step solution
🔁 9 · Repeated-Measures ANOVA
Memory Scores at Three Time Points
RM-ANOVA
Research Question:
Do memory scores change across three time points (Baseline, Week 4, Week 8) for the same 5 participants?
| Subject | Baseline | Week 4 | Week 8 | Row Mean |
|---|---|---|---|---|
| 1 | 4 | 6 | 8 | 6.0 |
| 2 | 5 | 7 | 9 | 7.0 |
| 3 | 3 | 5 | 7 | 5.0 |
| 4 | 6 | 8 | 10 | 8.0 |
| 5 | 2 | 4 | 6 | 4.0 |
| T̄ⱼ | 4.0 | 6.0 | 8.0 | 6.0 (Grand) |
SS Decomposition
$$SS_{Time} = n\sum_j(\bar{T}_j - \bar{x}_{..})^2 = 5[(4{-}6)^2+(6{-}6)^2+(8{-}6)^2] = 5[4+0+4] = 40$$
$$SS_{Subjects} = k\sum_i(\bar{P}_i - \bar{x}_{..})^2 = 3[(6{-}6)^2+(7{-}6)^2+(5{-}6)^2+(8{-}6)^2+(4{-}6)^2] = 3[0+1+1+4+4] = 30$$
$$SS_{Error} = SS_{Total} - SS_{Time} - SS_{Subjects}$$
$$SS_{Total} = \sum(x_{ij}-\bar{x}_{..})^2 = 70, \quad SS_{Error} = 70 - 40 - 30 = 0$$
$$F = \frac{MS_{Time}}{MS_{Error}} = \frac{40/2}{0/8} \rightarrow \infty \quad \text{(perfect linear trend — theoretical)}$$
$$\eta^2_{partial} = \frac{SS_{Time}}{SS_{Time}+SS_{Error}} = \frac{40}{40+0} = 1.000$$
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This example shows a perfect linear trend. In practice, use MindStat to compute exact F, df, p with Greenhouse-Geisser/Huynh-Feldt corrections when Mauchly's sphericity test is violated.
APA-7 (template)
A one-way repeated-measures ANOVA indicated a significant effect of time on memory scores, F(2, 8) = XX.XX, p < .001, η²_p = .XX. Pairwise comparisons with Bonferroni correction revealed significant increases from T1 to T2 (p = .XXX) and T2 to T3 (p = .XXX).