treatment-effects

A mixed design combines a between-subjects factor (different groups of participants) with a within-subjects factor (the same participants measured repeatedly).
It is also called a split-plot design.

This design is common in psychology, education, and medicine.
Example: groups of patients (between factor) measured at different time points (within factor).

Structure of the Design

• Between-subjects factor: separate groups of participants (e.g., Drug vs. Placebo).
• Within-subjects factor: repeated measures on each participant (e.g., Week 1, Week 2, Week 3).
• Interaction: tests whether the effect of the within factor depends on the between factor.

Degrees of Freedom

For a design with:

• $$a$$ levels of the between-subjects factor
• $$b$$ levels of the within-subjects factor
• $$n$$ subjects in total
• Between: $$df_{\text{between}} = a - 1$$
• Subjects (within groups): $$df_{\text{subjects}} = N - a$$
• Within: $$df_{\text{within}} = b - 1$$
• Interaction: $$df_{A \times B} = (a-1)(b-1)$$
• Error terms depend on design partitioning.

Example

Two groups of students (Drug, Placebo) are tested across three weeks.

• Between factor (Group): Drug vs. Placebo
• Within factor (Time): Weeks 1–3
• Interaction: Drug improves over time, Placebo stays flat

Symbolic Formula

$$F = \frac{MS_{\text{effect}}}{MS_{\text{error}}}$$

Where $$\text{effect}$$ may be between, within, or interaction, depending on the hypothesis.

Definition

• Mixed (split-plot) ANOVA: combines a between factor (different groups) and a within factor (repeated measures).
• Use: tests real-world designs where groups are compared across time or conditions.

Visuals

Figure L9.1 — Mixed ANOVA Layout. Two groups (Drug, Placebo) × three repeated measures (Weeks 1–3).

Figure L9.2 — Mixed ANOVA Interaction Plot. Drug group line rises sharply; Placebo line flat.

Figure L9.3 — ANOVA Summary Table for mixed design.

Why This Matters

Mixed designs are realistic and powerful.
They reflect how experiments are often run: groups compared across time.
This design unites the logic of between- and within-subjects testing.

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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:

• 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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