KIN 610 · Quantitative Methods in Kinesiology

Factorial Analysis
of Variance

Chapter 16

Ovande Furtado Jr., PhD · Cal State Northridge

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Context

Chapter Roadmap

Previous chapters answered one question at a time: Does training affect strength? Does group predict aerobic capacity?

These are incomplete. In practice, researchers ask:

Who benefits from training?
By how much do gains differ?
When do gains emerge?

Three design types covered

Between-subjects factorial — two between-groups factors, one cell per person
Mixed factorial — between + within factors (most common in training research)
Within-within factorial — all factors within-subjects

Running example

Did a 12-week training program produce strength gains, and did those gains differ by sex or across time?

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Chapter 16

Learning Objectives

Define a factorial design; explain how it differs from multiple one-way ANOVAs
Distinguish between main effects and interaction effects; explain what an interaction means
Describe variance partitioning in between-subjects, mixed, and within-within factorial ANOVAs
Identify the correct error term for each F-ratio in a mixed ANOVA
Interpret an interaction plot; distinguish qualitative from quantitative interactions
Conduct simple effects analysis to decompose a significant interaction
Compute and interpret η²_p and ω²_p for each effect
Report factorial ANOVA results in APA format

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Procedure

Workflow for Factorial ANOVA

1Identify the design — how many factors? Is each between-subjects or within-subjects?

2Check assumptions — normality, homogeneity of variance (between-subjects), sphericity (within-subjects ≥ 3 levels)

3Run the factorial ANOVA — inspect Factor A, Factor B, and A × B interaction

4Interpret the interaction first — if significant, main effects cannot be interpreted in isolation

5If interaction significant — run simple effects: test each factor at each level of the other

6If interaction not significant — interpret main effects; apply post hoc tests for factors with ≥ 3 levels

7Calculate effect sizes — η²_p and ω²_p for every source

8Report in APA format — interaction first, then main effects, with descriptives

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Definitions

Factors, Levels, and Cells

Factor — each independent variable in the design (e.g., Sex, Group, Time)
Level — a discrete category of a factor (Female/Male; Control/Training)
Cell — unique intersection of factor levels; each cell has its own mean — the raw material of factorial ANOVA
Design notation — 2 × 3 means Factor A has 2 levels, Factor B has 3 → 6 cells
Balanced — equal n per cell → factors are orthogonal, SS partitions cleanly
Unbalanced — unequal n → SPSS uses Type III SS (default), which handles this correctly

2(Sex) × 2(Group) — 4 cells

	Control	Training
Female	Female-Control	Female-Training
Male	Male-Control	Male-Training

Complexity grows fast

A 3 × 3 design has 9 cells. Adding a third 2-level factor creates 2 × 3 × 2 = 12 cells. Most applied research uses two-factor designs.

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🎯 Knowledge Check

In a 2(Group: Control/Training) × 2(Sex: Female/Male) factorial design, what does the cell mean for "Female-Training" specifically represent?

C is correct. A cell mean represents one unique combination of factor levels — in this case, only the Female participants assigned to Training. Options A and B describe marginal means, which collapse across the other factor. Option D describes a contrast, not a mean.

KC · 5 / 10

Motivation

Why Not Run Separate One-Way ANOVAs?

Three Fatal Flaws

Inflated Type I error — each test at α = .05 adds ~5% false positive risk; two tests ≈ 10% familywise error
Cannot detect interactions — the effect of Group depending on Sex simply does not exist in separate analyses; the test is absent
Larger error term — separate ANOVAs cannot remove variance from other factors before estimating error → less power

The factorial ANOVA solves all three simultaneously

One model, controlled error rate, smaller error term, and — crucially — the ability to detect and estimate interactions.

The interaction is the exclusive contribution

Whether a training program benefits males and females differently is a question that lives only inside a factorial design. It cannot be answered any other way.

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🎯 Knowledge Check

A researcher runs two separate one-way ANOVAs — one for Group, one for Sex — instead of a single factorial ANOVA. Which problem is impossible to fix within that approach?

B is correct. Separate one-way ANOVAs test each main effect in isolation but have no mechanism to estimate an interaction term — the question of whether Group's effect depends on Sex simply has no answer in that analysis. Factorial ANOVA is the only design that can test it.

KC · 1 / 10

Core Concepts

Main Effects

A main effect is the overall effect of one factor, averaging — collapsing — across all levels of the other factor(s).

Main effect of Group: Training (85 kg) vs. Control (77 kg), averaged across both sexes
Main effect of Sex: Male vs. Female, averaged across both training conditions

Main effects tell the simple, unconditional story — and they are perfectly interpretable when no interaction is present.

When an interaction is significant

The main effect averages two different patterns into one number that accurately describes neither. It is technically correct but substantively misleading.

Collapsing to a Main Effect

To get the main effect of Group, average across Sex ↓

	Control	Training
Female	77.54	87.18
Male	76.22	83.64
⬇ average across Sex
Marginal M	77.14	85.06

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Core Concepts

Interactions: Quantitative vs. Qualitative

An interaction occurs when the effect of one factor depends on the level of another factor.

Quantitative (Ordinal) Interaction

Both groups change in the same direction, but the magnitude differs. Lines diverge — do not cross. Example: training adds 12 kg for males, 4 kg for females — both improve.

Qualitative (Disordinal) Interaction

The direction reverses across levels — training helps one group but impairs another. Lines cross. Rare but scientifically dramatic.

Cardinal rule — permanent

Always interpret the interaction before the main effects. A significant interaction means the main effects are incomplete summaries.

Type	Pattern	Lines
Quantitative	Same direction, different magnitude	Diverge
Qualitative	Direction reverses	Cross
None	Identical effect across groups	Parallel

Quantitative interactions are common in movement science; qualitative interactions are rare but most compelling.

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🎯 Knowledge Check

A 2×2 factorial ANOVA shows a significant main effect of Group (p = .003) and a non-significant interaction (p = .54). What is the correct next step?

B is correct. When the interaction is non-significant, the main effect averages across levels of the other factor without misrepresenting any subgroup — it can be reported and interpreted directly. Simple effects analysis is only required after a significant interaction.

KC · 6 / 10

Visualization

Reading Interaction Plots

No Interaction

Parallel → interpret main effects directly

Quantitative Interaction

Diverge — same direction, different magnitude

Qualitative Interaction

Cross — direction reverses between groups

The one-question visual test

Are the lines parallel? Yes → no interaction, interpret main effects. No → interaction present, run simple effects before interpreting anything else.

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🎯 Knowledge Check

The plot shows training improved both groups, but Females gained far more than Males. What type of interaction is this?

B is correct. Both groups improve (same direction), but the magnitude differs — lines diverge without crossing. That's the defining signature of a quantitative/ordinal interaction. A qualitative interaction requires the lines to cross.

KC · 2 / 10

Between-Subjects Factorial ANOVA

Variance Partitioning

Every participant belongs to exactly one cell. Total variance splits into four sources:

SS_total = SS_A + SS_B + SS_A×B + SS_error

All three effects share the same within-cell error term:

F_A = MS_A / MS_error | F_B = MS_B / MS_error | F_A×B = MS_A×B / MS_error

Source	df	Denominator
Factor A	a − 1	MS_error
Factor B	b − 1	MS_error
A × B Interaction	(a−1)(b−1)	MS_error
Error (within-cell)	N − ab	—

a = levels of A · b = levels of B · N = total sample size · ab = number of cells

SPSS handles all computation

Focus on what each source tests and which error term it uses. You must understand this to interpret the output correctly.

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Worked Example

2(Sex) × 2(Group) Between-Subjects ANOVA

Research question

Did a 12-week training program improve post-test strength, and did this effect differ between male and female participants? N = 60

Cell Means (M ± SD)

Sex	Group	n	M (kg)	SD
Female	Control	21	77.54	14.70
Female	Training	12	87.18	8.21
Male	Control	9	76.22	12.92
Male	Training	18	83.64	14.72

Marginal Means

Group	n	M (kg)	SD
Control (total)	30	77.14	13.98
Training (total)	30	85.06	12.48

Unbalanced design (unequal cell n) → SPSS uses Type III SS automatically.

Before reading the table

Look at the cell means visually first. Do the female and male training gains look similar in magnitude? That will tell you what the interaction test should show.

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Worked Example

2 × 2 Results and Interpretation

Source	SS	df	MS	F	p	η²_p
Sex	0.22	1	0.22	0.00	.974	.000
Group	939.31	1	939.31	5.22	.026	.085
Sex × Group	101.14	1	101.14	0.56	.457	.010
Error	10,084.70	56	180.08	—

Step 1: Interaction first

Sex × Group: F(1,56) = 0.56, p = .457, η²_p = .010 — not significant, negligible effect. The training benefit did not differ between males and females. ✓ Main effects can be interpreted.

Step 2: Main effects

Group: F(1,56) = 5.22, p = .026, η²_p = .085 — significant, medium effect. Training (85.06 kg) outperformed Control (77.14 kg).

Sex: F(1,56) = 0.00, p = .974 — not significant. Males and females had similar post-test strength.

Null interaction = substantive finding

The same training prescription was equally effective for both sexes. No sex-specific program modifications are indicated. Null interaction results are not failures.

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Mixed Factorial ANOVA

The Workhorse of Training Research

The mixed factorial ANOVA (also: split-plot design) combines:

≥ 1 between-subjects factor (e.g., Group: Control vs. Training — different people)
≥ 1 within-subjects factor (e.g., Time: Pre, Mid, Post — same people measured repeatedly)

The question it answers

Do groups differ in their trajectory of change over time? Neither a between-subjects ANOVA nor a repeated-measures ANOVA alone can answer this.

The Group × Time interaction is the scientific heart of most longitudinal intervention studies in movement science.

Why two separate error terms?

Between-subjects variation is large — people differ enormously in baseline strength.

Within-subjects variation is much smaller — it captures only how inconsistently each person responds across time, stripped of their individual baseline.

Critical asymmetry

Within-subjects F-ratios are not comparable in magnitude to between-subjects F-ratios. Use η²_p to compare effect sizes across sources.

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🎯 Knowledge Check

A 2(Group: Control/Training) × 2(Sex: Female/Male) ANOVA yields: main effect of Group p = .001, main effect of Sex p = .04, interaction p = .71. Which statement is best supported by these results?

C is correct. A non-significant interaction means the effect of training did not depend on sex — both main effects can be interpreted directly and independently. Option A implies a significant interaction. Option B is wrong because simple effects are only needed after a significant interaction. Option D is wrong because p-values reflect evidence against the null, not the size or importance of an effect — effect sizes do that.

KC · 7 / 10

Mixed Factorial ANOVA

Two Error Terms — One Model

Effect	Type	Error Term	Why?
Group	Between-subjects	MS_{Subjects/Group}	Captures stable individual differences in baseline strength
Time	Within-subjects	MS_{Time×Subj/Group}	Individual inconsistency across time only, after removing baselines
Group × Time	Within-subjects	MS_{Time×Subj/Group}	Same within-subjects error as Time

Three F-Ratios

F_Group = MS_Group / MS_{Subjects/Group}
F_Time = MS_Time / MS_{Time×Subj/Group}
F_Group×Time = MS_Group×Time / MS_{Time×Subj/Group}

Common mistake — wrong error term

Using the within-subjects error for the Group effect produces an artificially large F, because MS_within is far smaller than MS_{Subjects/Group}. SPSS prevents this automatically in Repeated Measures GLM.

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🎯 Knowledge Check

A researcher is puzzled: "Why does SPSS report a different error term for the Group effect than for the Time effect in my mixed ANOVA?" Which explanation is correct?

C is correct. In a mixed ANOVA, comparing groups requires accounting for the fact that people differ from each other — that person-to-person variability is the relevant noise for the Group effect. But when looking at change over time, each participant serves as their own baseline, so those individual differences subtract out. Two fundamentally different sources of variability mean two different error terms. SPSS handles the calculation automatically.

KC · 3 / 10

Mixed Factorial ANOVA

Variance Partitioning — 2(Group) × 3(Time)

Between-Subjects Partition

SS_{between subjects} = SS_Group + SS_Subj/Group

SS_Subj/Group = between-subjects error → denominator for F_Group

Within-Subjects Partition

SS_{within subjects} = SS_Time + SS_Group×Time + SS_{Time×Subj/Group}

SS_{Time×Subj/Group} = within-subjects error → denominator for F_Time and F_Group×Time

Why within-subjects error is smaller

The between-subjects error reflects all stable individual differences in absolute strength — a large, noisy term.

The within-subjects error captures only how inconsistently each person responds across time, after removing individual baselines — typically much smaller.

Power asymmetry

Time and Group×Time effects are tested against the small within-subjects error → much higher statistical power than the Group effect, which uses the large between-subjects error.

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Assumption Check

Sphericity in the Mixed ANOVA

Sphericity requires that the variances of all pairwise differences among within-subjects levels be approximately equal. In the mixed design, this applies to:

Time main effect — uses within-subjects error
Group × Time interaction — uses the same within-subjects error
Group (between-subjects) — not affected by sphericity; uses between-subjects error

Mauchly's Result	Action
p > .05	Sphericity Assumed — use those rows for Time and Group×Time
p < .05, ε_GG < .75	Greenhouse-Geisser correction — apply to Time and Group×Time
p < .05, ε_GG ≥ .75	Huynh-Feldt correction — apply to Time and Group×Time

Always report in APA write-up

Report Mauchly's W, df, and p. If a correction was applied, report the epsilon (ε) value and state which correction was used — even when sphericity is not violated.

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Worked Example

2(Group) × 3(Time) Mixed ANOVA — Setup

Research question

Did strength change over a 12-week training program, and did the trajectory of change differ between the training and control groups? n = 30 per group; outcome: strength (kg)

Cell Means (SD)

Group	Pre	Mid	Post
Control	76.34 (13.70)	76.85 (13.87)	77.14 (13.98)
Training	79.67 (12.26)	81.69 (12.26)	85.06 (12.48)

Before reading ANOVA output: inspect the pattern. Control is essentially flat. Training climbs steadily. Expect a significant Group×Time interaction.

Sphericity Check

Mauchly's W(2) = .93, p = .059

Not violated (p > .05) → use Sphericity Assumed rows for Time and Group×Time

Why check this first?

If you read the Sphericity Assumed F-values when sphericity is violated, your p-values will be too small and you'll over-reject H₀. Always check Mauchly's before reading Time and interaction results.

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🎯 Knowledge Check

Mauchly's test of sphericity is significant (W = .68, p = .02). What is the correct next step?

C is correct. A significant Mauchly's test means the sphericity assumption is violated, inflating the F-ratio if left uncorrected. The standard remedy is to adjust the degrees of freedom using Greenhouse-Geisser (more conservative) or Huynh-Feldt (less conservative when ε > .75). Removing the factor or switching to MANOVA are not appropriate responses to a sphericity violation.

KC · 8 / 10

Worked Example

2 × 3 Mixed ANOVA Results

Source	df	MS	F	p	η²_p	ω²_p
Group	1	1,294.44	2.52	.117	.042	.008
Subjects/Group (error B)	58	513.32	—
Time	2	145.24	108.55	<.001	.652	.544
Group × Time	2	81.63	61.00	<.001	.513	.400
Time × Subj/Group (error W)	116	1.34	—

Step 1: Interaction — significant

Group × Time: F(2,116) = 61.00, p < .001, η²_p = .513, ω²_p = .400 — a large effect. The groups followed different strength trajectories. → Run simple effects; interpret main effects in context.

Why is Group not significant?

Groups start at similar levels and diverge over time. Their time-averaged means aren't dramatically different — so the Group main effect (p = .117) misses the pattern the interaction reveals. The interaction is more sensitive to diverging trajectories.

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Visualization

Group × Time Interaction — Diverging Trajectories

Key Observations

Control: flat across 12 weeks — Δ < 1 kg total; Time effect non-significant within this group
Training: progressive gains — +2.02 kg by Mid, +5.39 kg by Post
Lines diverge without crossing → quantitative interaction; training benefits are larger, not reversed

Next step

Significant interaction → decompose with simple effects analysis before drawing conclusions about Time or Group.

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Decomposing the Interaction

Simple Effects Analysis

Simple effects test the effect of one factor separately at each level of the other factor. Run only after a significant omnibus interaction.

Perspective 1: Time within each Group

Does strength change across time in Control? → No (76.34, 76.85, 77.14 — <1 kg, non-significant)
Does strength change across time in Training? → Yes, strongly (79.67 → 85.06, +5.39 kg, p < .001)

Perspective 2: Group at each Time point

Pre: groups comparable (76.34 vs. 79.67 kg) — randomization check ✓
Mid: emerging difference (76.85 vs. 81.69 kg)
Post: significant divergence (77.14 vs. 85.06 kg, Δ = 7.92 kg)

When to run simple effects

Only after a statistically significant interaction. Running them after a non-significant interaction capitalizes on chance and inflates Type I error — the omnibus test is your gate.

In SPSS

General Linear Model → Repeated Measures → Options → Compare main effects → Bonferroni. Or: split file by Group, run separate one-way repeated-measures ANOVAs.

Examine the plot first

The interaction plot tells you where the interaction lives. Run only theoretically motivated simple effects — not all possible combinations.

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Quantifying Practical Significance

Effect Sizes: η²_p and ω²_p

Partial Eta-Squared (η²_p)

Reported automatically by SPSS

η²_p = SS_effect / (SS_effect + SS_{error for that effect})

The "partial" means each effect is sized relative to its own error term, not total variance. This correctly isolates each effect.

Effect	Error in denominator
Group	SS_{Subjects/Group} (between)
Time	SS_{Time×Subj/Group} (within)
Group × Time	SS_{Time×Subj/Group} (within)

Partial Omega-Squared (ω²_p)

Less biased — recommended for small-to-moderate samples

ω²_p = (SS_effect − df_effect · MS_error) / (SS_effect + (N·p − df_effect) · MS_error)

Cohen's benchmarks (apply cautiously)

Small ≈ .01 | Medium ≈ .06 | Large ≥ .14

Well-controlled lab training studies routinely produce η²_p > .50 for within-subjects effects. That's not unusual — it reflects the favorable signal-to-noise ratio of within-subject designs.

Report for every source

η²_p AND ω²_p for Group, Time, and the interaction — including non-significant effects. Readers need all values to evaluate power.

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🎯 Knowledge Check

A Group × Time interaction is significant. Simple effects show the Group effect at Time 1 is F(1, 46) = 0.23, p = .63. What does this finding mean?

C is correct. A non-significant simple effect at one level is entirely consistent with a significant interaction — it tells you where the interaction does not manifest. The group difference simply emerged at a later time point. This is a substantive finding to report, not evidence against the interaction or grounds for excluding data.

KC · 9 / 10

🎯 Knowledge Check

The Group × Time interaction yields η²_p = .42, p < .001. A colleague concludes: "The training program caused 42% of all outcome variance." What is wrong with this statement?

C is correct. η²_p = SS_effect / (SS_effect + SS_error) — variance from other effects is removed from the denominator, so partial η² is always ≥ total η² and cannot be interpreted as a proportion of total variance. More critically, no ANOVA statistic implies causation — that inference requires experimental design logic, not the size of F.

KC · 4 / 10

Power Planning

Cohen's f for Power Calculations

G*Power requires Cohen's f — not η²_p — as its effect size input for factorial ANOVA power analyses.

f = √(η²_p / (1 − η²_p))

Conversions from our example

Effect	η²_p	f	Interpretation
Group × Time	.513	1.03	Very large
Time	.652	1.37	Very large
Group	.042	0.21	Small-medium

Power asymmetry in mixed designs

The interaction (f = 1.03) and Time (f = 1.37) have enormous effect sizes because they're tested against the tiny within-subjects error. Group (f = 0.21) uses the noisy between-subjects error — it requires a much larger N to achieve equivalent power.

G*Power settings for mixed ANOVA

Use: F-test → ANOVA: Repeated Measures, between-within interaction. Key extra input: expected correlation among repeated measures. In training studies with strength, pre-mid-post correlations are typically r = .96–.98, explaining the very high power of within-subjects effects.

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Reporting

APA Reporting Template

Required order — always follow this sequence:

1State the design: "A 2(Group: control, training) × 3(Time: pre, mid, post) mixed ANOVA was conducted with [DV] as the dependent variable."

2Report Mauchly's test: W(df) = value, p = value. State whether sphericity was violated and which correction was applied.

3Interaction first: F(df1, df2) = value, p = value, η²_p = value, ω²_p = value. State what the interaction means substantively.

4If interaction significant: report simple effects findings with descriptive statistics (M, SD) for key comparisons.

5Main effects: report F, p, η²_p, ω²_p for each. Interpret in context of the interaction.

Never omit effect sizes for non-significant effects

Report η²_p and ω²_p for every source. A non-significant Group effect with η²_p = .04 tells a different story than one with η²_p = .30 that a study was underpowered to detect.

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Full Write-Up

APA Example — Group × Time Mixed ANOVA

"A 2 (Group: control, training) × 3 (Time: pre, mid, post) mixed ANOVA was conducted with strength (kg) as the dependent variable. Mauchly's test indicated that the sphericity assumption was not violated, W(2) = .93, p = .059.

The Group × Time interaction was statistically significant, F(2, 116) = 61.00, p < .001, η²_p = .51, ω²_p = .40, indicating that the two groups followed different strength trajectories over the 12-week program. Simple effects analysis revealed a significant effect of Time within the training group, with progressive gains from pre-test (M = 79.67 kg, SD = 12.26) to post-test (M = 85.06 kg, SD = 12.48). The effect of Time was not significant within the control group, which remained essentially unchanged (76.34, 76.85, 77.14 kg).

The main effect of Time was significant, F(2, 116) = 108.55, p < .001, η²_p = .65, ω²_p = .54. The main effect of Group was not significant, F(1, 58) = 2.52, p = .117, η²_p = .04, ω²_p = .008."

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Avoid These Mistakes

Common Pitfalls

Interpreting main effects when the interaction is significant. A main effect of Group averaged over two different sex-specific patterns is a statistical fiction — it describes neither pattern accurately. Interaction first, always.
Concluding "no interaction" from a non-significant p-value. An underpowered study cannot distinguish a small interaction from no interaction. Report η²_p for the interaction regardless of significance.
Using the wrong error term in a mixed ANOVA. Group uses the between-subjects error; Time and the interaction use the within-subjects error. Never compare those F-values directly to judge importance.
Reporting effect sizes only for significant effects. Every source needs η²_p and ω²_p. Non-significant effects with large η²_p signal underpowered studies, not trivial effects.
Running simple effects without a significant interaction. Simple effects tests are only justified after a significant omnibus interaction. Running them otherwise capitalizes on chance and inflates Type I error.

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Chapter 16 Summary

Key Takeaways

Factorial ANOVA simultaneously tests main effects and the interaction — the interaction is its exclusive contribution over separate one-way tests
Between-subjects factorial: all factors between-groups; all effects share one within-cell error term
Mixed factorial: between-subjects Group + within-subjects Time; two error terms — Group uses between-subjects error, Time and interaction use within-subjects error
The interaction must be interpreted first — when significant, main effects alone are incomplete and can mislead

Simple effects analysis decomposes a significant interaction — test one factor at each level of the other; requires significant omnibus interaction
Sphericity (Mauchly's test, GG/HF corrections) applies only to within-subjects effects — not to the between-subjects Group main effect
Report η²_p and ω²_p for every source — Group, Time, and the interaction — including non-significant effects
APA sequence: design → Mauchly's → interaction first → simple effects → main effects → descriptives

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🎯 Knowledge Check

Which of the following correctly reports a mixed ANOVA interaction effect in APA format?

D is correct. APA format requires: narrative statement first, then F-ratio with both degrees of freedom, exact p-value, and a numeric effect size — all in that order. Option A omits the effect size. Option B gives no F-ratio or numeric effect size. Option C puts the statistics before the narrative, which is non-standard.

KC · 10 / 10

Reference

Key Terms

Design

Factorial design
Factor / Level / Cell
Between-subjects factorial
Mixed factorial (split-plot)
Within-within factorial
Balanced / Unbalanced
Type III SS

Effects

Main effect
Interaction effect
Quantitative interaction
Qualitative interaction
Simple effects analysis
Interaction plot
Between- vs. within-subjects error

Measurement

Partial eta-squared (η²_p)
Partial omega-squared (ω²_p)
Cohen's f
Sphericity
Mauchly's test
Greenhouse-Geisser (ε_GG)
Huynh-Feldt correction

Navigation

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Ch. 16 · Achievement

Badge Collection

Earn badges by answering Knowledge Check questions correctly

Save as PDF opens your browser's print dialog — choose "Save as PDF." | Try Again clears badges and returns to slide 1.

Factorial Analysisof Variance

How to Use This Deck

🎓 Identify yourself

🏅 How badges work

Chapter Roadmap

Three design types covered

Learning Objectives

Workflow for Factorial ANOVA

Factors, Levels, and Cells

Why Not Run Separate One-Way ANOVAs?

Three Fatal Flaws

Main Effects

Collapsing to a Main Effect

Interactions: Quantitative vs. Qualitative

Quantitative (Ordinal) Interaction

Qualitative (Disordinal) Interaction

Reading Interaction Plots

Variance Partitioning

2(Sex) × 2(Group) Between-Subjects ANOVA

2 × 2 Results and Interpretation

The Workhorse of Training Research

Why two separate error terms?

Two Error Terms — One Model

Three F-Ratios

Variance Partitioning — 2(Group) × 3(Time)

Between-Subjects Partition

Within-Subjects Partition

Why within-subjects error is smaller

Sphericity in the Mixed ANOVA

2(Group) × 3(Time) Mixed ANOVA — Setup

Sphericity Check

2 × 3 Mixed ANOVA Results

Group × Time Interaction — Diverging Trajectories

Key Observations

Simple Effects Analysis

Perspective 1: Time within each Group

Perspective 2: Group at each Time point

Effect Sizes: η²p and ω²p

Partial Eta-Squared (η²p)

Partial Omega-Squared (ω²p)

Cohen's f for Power Calculations

Conversions from our example

APA Reporting Template

APA Example — Group × Time Mixed ANOVA

Common Pitfalls

Key Takeaways

Key Terms

Design

Effects

Measurement

Badge Collection

Factorial Analysis
of Variance

Effect Sizes: η²_p and ω²_p

Partial Eta-Squared (η²_p)

Partial Omega-Squared (ω²_p)