In a repeated-measures design, the same participants are tested under multiple conditions.
This reduces error, because each person serves as their own control.
It is more powerful than a one-way ANOVA with independent groups.
Structure of the Design
- Rows (subjects): variation due to individual differences
- Columns (conditions): variation due to treatments
- Error: leftover variability after accounting for rows and columns
Degrees of Freedom
- $$df_{\text{rows}} = n - 1$$
- $$df_{\text{columns}} = k - 1$$
- $$df_{\text{error}} = (n - 1)(k - 1)$$
Where:
- $$n$$ = number of subjects
- $$k$$ = number of conditions
Example
Five students are tested under three conditions:
| Subject | Cond 1 | Cond 2 | Cond 3 |
|---|---|---|---|
| S1 | 70 | 75 | 80 |
| S2 | 68 | 74 | 79 |
| S3 | 72 | 77 | 83 |
| S4 | 69 | 73 | 78 |
| S5 | 71 | 76 | 82 |
- Means increase steadily across conditions.
- ANOVA will partition the variance into Rows, Columns (treatments), and Error.
Symbolic Formula
$$F = \frac{MS_{\text{columns}}}{MS_{\text{error}}}$$
Formula in words:
$$F = \frac{\text{mean square for conditions}}{\text{mean square for error}}$$
Definition
- Repeated-measures ANOVA: compares means of the same group measured under different conditions.
- Advantage: controls for subject differences, increases statistical power.
Visuals
Figure L8.1 — Repeated-Measures Profile Plot. Each subject shown as a line across conditions.
Figure L8.2 — ANOVA Summary Table for repeated measures. Rows | Columns | Error.
Why This Matters
Repeated-measures designs are common in psychology, neuroscience, and medicine.
They allow researchers to detect changes over time or across treatments with fewer subjects and greater sensitivity.
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