f-statistic

Goal. Test whether performance changes across four conditions measured on the same participants.

Design & Experiment

• Within-subjects factor: Condition with 4 levels (C1, C2, C3, C4).
• s = 8 participants measured in k = 4 conditions ⇒ total observations \(N = s \times k = 32\).
• Example context: the same students take four weekly quizzes after different study activities.

Figure 1: Profile plot (each subject as a line across the four conditions).

Data

Scores (rows = participants S1–S8; columns = conditions C1–C4):

Figure 2: Means ± SEM for C1–C4 (bar/line).

Step 1 — Condition Means (and sample variances)

\[ \begin{aligned} \bar X_{\mathrm{C1}} &= 573/8 = 71.625, \quad & s^2_{\mathrm{C1}} &= 4.8393 \\ \bar X_{\mathrm{C2}} &= 601/8 = 75.125, \quad & s^2_{\mathrm{C2}} &= 5.5536 \\ \bar X_{\mathrm{C3}} &= 621/8 = 77.625, \quad & s^2_{\mathrm{C3}} &= 7.6964 \\ \bar X_{\mathrm{C4}} &= 654/8 = 81.750, \quad & s^2_{\mathrm{C4}} &= 7.0714 \end{aligned} \]

Step 2 — Sums of Squares

Notation: \(s=8\) subjects, \(k=4\) conditions, grand mean \( \bar X = 76.53125\).

2A. Total

\[ SS_{\text{total}}=\sum_{i=1}^{s}\sum_{j=1}^{k}\bigl(X_{ij}-\bar X\bigr)^2 =\mathbf{611.96875}. \]

2B. Conditions (Treatment)

\[ SS_{\text{cond}}= s \sum_{j=1}^{k}\bigl(\bar X_{\cdot j}-\bar X\bigr)^2 = 8 \left[(71.625-76.53125)^2 + (75.125-76.53125)^2 + (77.625-76.53125)^2 + (81.75-76.53125)^2\right] =\mathbf{435.84375}. \]

2C. Subjects

\[ SS_{\text{subj}}= k \sum_{i=1}^{s}\bigl(\bar X_{i\cdot}-\bar X\bigr)^2 = 4 \sum_{i=1}^{8}\bigl(\bar X_{i\cdot}-76.53125\bigr)^2 =\mathbf{162.71875}. \]

2D. Error (Residual)

\[ SS_{\text{error}}= SS_{\text{total}} - SS_{\text{cond}} - SS_{\text{subj}} = 611.96875 - 435.84375 - 162.71875 =\mathbf{13.40625}. \]

Figure 3: Partitioning variance diagram (Total → Conditions + Subjects + Error).

Step 3 — Degrees of Freedom & Mean Squares

\[ \begin{aligned} df_{\text{cond}} &= k-1 = 3, \\ df_{\text{subj}} &= s-1 = 7, \\ df_{\text{error}} &= (s-1)(k-1) = 7\times3 = 21, \\ df_{\text{total}} &= sk-1 = 31. \end{aligned} \]

\[ MS_{\text{cond}} = \frac{SS_{\text{cond}}}{df_{\text{cond}}} =\frac{435.84375}{3}=\mathbf{145.28125},\qquad MS_{\text{error}} = \frac{SS_{\text{error}}}{df_{\text{error}}} =\frac{13.40625}{21}=\mathbf{0.6383928571}. \]

Step 4 — Test Statistic & p-value

\[ F = \frac{MS_{\text{cond}}}{MS_{\text{error}}} = \frac{145.28125}{0.6383928571} =\mathbf{227.5734}. \] With \(df_1=3\) and \(df_2=21\), this is extremely large. The right-tail p-value is effectively \(p \lt 10^{-12}\) (i.e., \(p \ll .001\)).

Figure 4: F distribution with observed F marked and right-tail region shaded.

Repeated-Measures ANOVA Summary Table

Interpretation

Mean performance increases steadily from C1 → C4, and the repeated-measures ANOVA shows a highly significant effect of Condition, \(F(3,21)=227.57,\, p\ll .001\). Follow-ups (e.g., paired t-tests with Bonferroni/Holm) can localize which pairs of conditions differ.

Assumptions (checklist)

• Sphericity (equal variances of the differences between condition pairs). If violated, apply Greenhouse–Geisser or Huynh–Feldt correction to \(df\).
• Approximately normal scores within each condition.
• No carryover/fatigue effects that confound order (counterbalancing helps).

Figure 5: Sphericity concept sketch (pairwise difference variances).

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Goal. Test the effects of Method (Lecture vs. Online) and Time (Early vs. Late) on exam scores, and whether there is an interaction between Method and Time.

Design & Experiment

• Factor A (Method): Lecture vs. Online
• Factor B (Time): Early vs. Late
• Balanced design: \(n=5\) per cell ⇒ total \(N=20\).

Students are randomly assigned to one of four cells (Method × Time). After a short module, all students take the same 100-point exam.

Figure 1: 2 × 2 layout (Method × Time).

Data

Scores by cell (five students per cell):

Within each cell the sample variance is 4 (SD = 2), so the within-cell sum of squares is \((n-1)s^2 = 4\times4 = 16\) per cell.

Figure 2: Means with SEM by Time, separate lines for Method.

Figure 3: Interaction plot (Lecture rises sharply; Online nearly flat).

Step 1 — Marginal Means and Grand Mean

Cell means: \[ \bar X_{\text{Lecture,Early}}=70,\; \bar X_{\text{Lecture,Late}}=78,\; \bar X_{\text{Online,Early}}=72,\; \bar X_{\text{Online,Late}}=73. \] Marginal means: \[ \bar X_{\text{Lecture}}=\frac{70+78}{2}=74,\quad \bar X_{\text{Online}}=\frac{72+73}{2}=72.5; \qquad \bar X_{\text{Early}}=\frac{70+72}{2}=71,\quad \bar X_{\text{Late}}=\frac{78+73}{2}=75.5. \] Grand mean: \[ \bar X=\frac{70+78+72+73}{4}=73.25. \]

Step 2 — Sums of Squares (Between)

Balanced design formulas (with \(n\) per cell, \(a=b=2\)):

• \(SS_A = nb \sum_a(\bar X_{a\cdot}-\bar X)^2\), here \(nb=10\).
• \(SS_B = na \sum_b(\bar X_{\cdot b}-\bar X)^2\), here \(na=10\).
• \(SS_{AB} = n \sum_{a,b}\big(\bar X_{ab}-\bar X_{a\cdot}-\bar X_{\cdot b}+\bar X\big)^2\), here \(n=5\).

Compute each term:

Factor A (Method): \[ \begin{aligned} SS_A &= 10\Big[(74-73.25)^2 + (72.5-73.25)^2\Big]\\ &= 10\big[0.75^2 + (-0.75)^2\big] = 10(0.5625+0.5625)=\mathbf{11.25}. \end{aligned} \]

Factor B (Time): \[ \begin{aligned} SS_B &= 10\Big[(71-73.25)^2 + (75.5-73.25)^2\Big]\\ &= 10\big[(-2.25)^2 + (2.25)^2\big] = 10(5.0625+5.0625)=\mathbf{101.25}. \end{aligned} \]

Interaction \(A\times B\): For each cell compute \(d_{ab}=\bar X_{ab}-\bar X_{a\cdot}-\bar X_{\cdot b}+\bar X\). Here each \(d_{ab}=\pm1.75\) so \(d_{ab}^2=3.0625\) and there are four cells: \[ SS_{AB}=5\times(4\times3.0625)=\mathbf{61.25}. \]

Step 3 — Within-Group (Error) and Total SS

Within each cell, \((n-1)s^2=16\). With 4 cells: \[ SS_{\text{within}}=\mathbf{64.00}. \]

Total: \[ SS_{\text{total}}=SS_A+SS_B+SS_{AB}+SS_{\text{within}} =11.25+101.25+61.25+64.00=\mathbf{238.75}. \]

Step 4 — Degrees of Freedom & Mean Squares

\[ \begin{aligned} &df_A=a-1=1,\quad df_B=b-1=1,\quad df_{AB}=(a-1)(b-1)=1,\\ &df_{\text{within}}=N-ab=20-4=\mathbf{16},\quad df_{\text{total}}=N-1=19. \end{aligned} \] \[ MS_A=\frac{11.25}{1}=11.25,\quad MS_B=\frac{101.25}{1}=101.25,\quad MS_{AB}=\frac{61.25}{1}=61.25,\quad MS_{\text{within}}=\frac{64.00}{16}=\mathbf{4.00}. \]

Step 5 — F Tests & p-values

\[ F_A=\frac{MS_A}{MS_{\text{within}}}=\frac{11.25}{4}= \mathbf{2.8125},\qquad F_B=\frac{MS_B}{MS_{\text{within}}}=\frac{101.25}{4}= \mathbf{25.3125},\qquad F_{AB}=\frac{MS_{AB}}{MS_{\text{within}}}=\frac{61.25}{4}= \mathbf{15.3125}. \] With \(df_1=1\), \(df_2=16\): \[ p_A \approx 0.11\;(\text{n.s.}),\quad p_B < 0.001,\quad p_{AB} \approx 0.001. \]

ANOVA Summary Table

Interpretation

Main effect of Time (B) is significant: Late > Early on average. Main effect of Method (A) is not significant at conventional levels. The interaction (A × B) is significant: Lecture improves markedly from Early→Late, while Online changes little—non-parallel lines in the interaction plot.

Figure 4: Interaction plot highlighting non-parallel lines.

Assumptions (checklist)

• Independence of observations within and across cells.
• Approximately normal scores within each cell.
• Homogeneity of variances across cells (here, each cell variance ≈ 4).

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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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