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TipStudy instructions
Work through each section in order — every concept builds on the last.
Complete every Knowledge Check before moving on. Your running score appears in the bottom-right corner of the screen.
Fully expand and interact with the Decision Trees and Flashcards when you encounter them.
Use the APA Builder to practice writing your results.
Click Reveal My Score at the end to see your final result and targeted study advice.
Start here: complete the pre-check, then work through each section.
📋 Pre-Check
A sport scientist wants to know whether a training program improved strength AND whether the improvement was the same for male and female athletes. Why is a one-way ANOVA insufficient for this question?
1. From One Factor to Two: Why Factorial ANOVA?
Factorial ANOVA extends the one-way framework to designs where two or more factors operate simultaneously on a single outcome. The key contribution is the ability to test interactions — whether the effect of one factor depends on the level of another. This is impossible with separate one-way ANOVAs.
Problems with running separate one-way ANOVAs
Problem
Separate one-way ANOVAs
Factorial ANOVA
Type I error control
Inflated — each test adds error
Single family-wise α maintained
Interactions
Cannot be detected
Explicitly tested
Statistical power
Lower (larger error term)
Higher (variance from other factors removed from error)
What is answered
Each factor in isolation
Each factor AND their combined conditional effect
Vocabulary
Factor — an independent variable (e.g., Sex, Group, Time)
Level — a category of a factor (e.g., Female / Male)
Cell — a unique combination of factor levels (e.g., Female-Training)
What is the most critical scientific advantage of using a factorial ANOVA design rather than running separate one-way ANOVAs for each independent variable?
2. Main Effects and Interactions
Every factorial ANOVA produces at least three results: the main effect of Factor A, the main effect of Factor B, and the A × B interaction. The interaction must always be interpreted first.
Main effects
A main effect represents the overall effect of one factor, averaged across all levels of the other factor(s).
Main effect of Group: The overall impact of Training vs. Control, averaged across all time points.
Main effect of Time: The overall pattern across Pre, Mid, and Post, averaged across both groups.
Main effects give the “marginal means” story — what happens to the outcome when you ignore everything else.
Interactions
An interaction occurs when the effect of one factor changes depending on the level of another factor.
Type
Description
Lines on interaction plot
No interaction
The effect of one factor does not depend on the other (e.g., Groups change uniformly over Time)
Nearly parallel lines
Quantitative (ordinal)
Both groups change in the same direction, but the magnitude differs between groups
Diverging lines (do not cross)
Qualitative (disordinal)
The direction of the effect reverses compared to the other group
Lines cross each other
ImportantCardinal rule of factorial ANOVA
Always interpret the interaction before the main effects. When the interaction is significant, the main effects cannot be correctly interpreted in isolation — they describe an averaged pattern that may not represent any real group.
Knowledge Check 2
A 2(Sex) × 2(Group) ANOVA shows: training increases strength by 10 kg in males but by 3 kg in females. Which term BEST describes this result?
3. Between, Mixed, and Within-Within Designs
The design type determines how variance is partitioned and which error terms are used.
Between-subjects factorial ANOVA
Every participant belongs to exactly one cell. All effects share the same within-cell error. Because it relies purely on between-subjects variance, this design requires the largest sample size:
Combines a between-subjects factor (e.g., Group) with a within-subjects factor (e.g., Time). Uses two different error terms:
Effect
Error term
Why?
Group (between)
MSSubjects/Group
Stable individual differences — large denominator
Time (within)
MSTime × Subjects/Group
Within-person inconsistency only — small denominator
Group × Time
MSTime × Subjects/Group
Same within-subjects error as Time
Consequence: Time and interaction F-values are typically much larger than the Group F — because they are divided by a much smaller denominator. Do not compare F values across these effects to judge practical importance.
Within-within factorial ANOVA
Both factors are within-subjects. For example, measuring muscle activation across two different Speeds (Fast, Slow) and two different Loads (Heavy, Light) in the same participants.
This design is incredibly efficient for statistical power, but introduces massive complexity: It utilizes three separate error terms (one for Factor A, one for Factor B, and one for the A×B interaction). Consequently, you must check sphericity strictly for each of these three terms independently!
Knowledge Check - Design Types
When transitioning your study design from a Mixed Factorial ANOVA (where one factor is between-groups and one is within-groups) to a Within-Within Factorial ANOVA (where both factors are repeated measures for all participants), what structural change must occur regarding your error terms?
Knowledge Check 3
In a 2(Group) × 3(Time) mixed ANOVA, the Group F(1, 58) = 2.52, p = .117 while Time F(2, 116) = 108.55, p < .001. A student concludes "Time is 43 times more important than Group because its F-value is 43 times larger." What is wrong?
4. Interpreting the Interaction: A Decision Tree
When you get your ANOVA results, follow a systematic sequence. This interactive decision tree walks you through the key steps.
Step 1: Check the interaction F-test
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Is the Group × Time interaction statistically significant (p < .05)?
↓
YES
NO
Interaction IS significant
↓
The main effects cannot be interpreted in isolation. What do you do?
↓
✅ Examine the interaction plot to identify the pattern (e.g., qualitative or quantitative)
✅ Run simple effects analysis to examine the effect of one factor separately within each level of the other (e.g., does Time produce significant change within each Group individually?)
✅ Report the interaction F, η²_p, and ω²_p FIRST
✅ Report main effects with the caveat that they must be understood in context of the interaction
⚠️ Do NOT interpret the Group main effect as if the groups behaved uniformly across time
Interaction is NOT significant
↓
The groups follow parallel trajectories. What do you do?
↓
✅ Report the non-significant interaction (including η²_p — never omit effect sizes)
✅ Interpret main effects directly as stand-alone findings
✅ If Time has ≥ 3 levels and the main effect is significant, run Bonferroni post hoc comparisons
✅ If Group has only 2 levels and the main effect is significant, no further test is needed
⚠️ Do NOT run simple effects after a non-significant interaction
Knowledge Check 4
A 2(Group) × 3(Time) mixed ANOVA yields Group × Time F(2,116) = 61.00, p < .001. A student then reports "the main effect of Group was not significant, F(1,58) = 2.52, p = .117, so training did not improve strength." What is the error?
5. Sphericity in the Mixed ANOVA
The sphericity assumption applies to within-subjects effects — both the Time main effect and the Group × Time interaction. Always check Mauchly's test before reporting these F-values.
Which effects require a sphericity check?
Effect
Requires sphericity check?
Group (between-subjects)
No — uses between-subjects error
Time (within-subjects)
Yes — uses within-subjects error
Group × Time (within-subjects)
Yes — uses same within-subjects error
Decision rule (same as Chapter 15)
Mauchly’s result
Action
p > .05
Use “Sphericity Assumed” rows for Time and Group × Time
p < .05, ε_GG < .75
Apply Greenhouse-Geisser correction
p < .05, ε_GG ≥ .75
Apply Huynh-Feldt correction
Knowledge Check 5
In a mixed factorial ANOVA (e.g., examining Group over Time), to which elements does the sphericity assumption strictly apply, and why?
6. Power Analysis for Factorial Designs
Factorial ANOVA allows us to calculate power separately for each hypothesis being tested: the main effect of Factor A, the main effect of Factor B, and the interaction between A and B.
When planning sample sizes in **G*Power**, you generally must compute power for the highest-order interaction (the A × B interaction). Since interactions isolate variance across very specific conditional combinations rather than collapsing them like main effects do, they often require significantly more participants to achieve the exact same statistical power.
Instead of partial eta-squared (\(\eta^2_p\)), GPower requires you to input Cohen’s f as the effect size metric for F-tests. You can algebraically convert between the two, but conceptually remember that Cohen’s f* scales differently than \(\eta^2_p\).
Knowledge Check - Power Analysis
When conducting an a priori sample size estimation in G*Power for a 2 × 2 Factorial ANOVA, which of the following effects generally demands the largest sample size to achieve adequate statistical power?
7. Effect Sizes: η²_p and ω²_p
Report partial eta-squared (η²_p, SPSS default) and partial omega-squared (ω²_p, less biased) for every source of variance — Group, Time, and the interaction.
Interpretation over Calculation
In SPSS, partial eta-squared (\(\eta^2_p\)) is computed automatically for every effect. You do not need to memorize its formula or specific cut-off standards. What matters is understanding that it represents the proportion of variance explained by a specific effect, independently of other effects in the model.
Because within-subjects factors (like Time) have the stable between-person differences removed from their error terms, their \(\eta^2_p\) values are often substantially larger than between-subjects effects. In well-controlled lab training studies, finding \(\eta^2_p > .50\) for within-subjects effects is common.
While \(\eta^2_p\) is standard, it tends to overestimate the true effect in the population slightly, especially with small samples. Partial omega-squared (\(\omega^2_p\)) offers a less-biased alternative and is often preferred for more rigorous reporting.
Knowledge Check 6
A mixed ANOVA output shows a Time effect with an unusually large $\eta^2_p$. Why is the within-subjects effect size (Time) often so much larger than the between-subjects effect size (Group) in these designs?
8. Flashcards: Key Terms
Click each card to flip it.
Main effect
The overall effect of one factor, averaged across all levels of the other factor(s). Interpretable directly only when the interaction is NOT significant.
Interaction effect
Occurs when the effect of one factor DEPENDS on the level of another. Detected only in factorial designs — not in separate one-way ANOVAs.
Qualitative interaction
The direction of the effect REVERSES across groups — one group improves while the other worsens. Lines CROSS on the interaction plot.
Quantitative interaction
Both groups change in the SAME direction, but by different amounts. Lines DIVERGE (but do not cross) on the interaction plot.
Mixed ANOVA
Combines ≥1 between-subjects factor (e.g., Group) with ≥1 within-subjects factor (e.g., Time). Uses TWO different error terms.
Between-subjects error
The error term used to test the Group effect in a mixed ANOVA. Large, because it includes stable individual differences in baseline scores.
Within-subjects error
The error term used to test Time and Interaction effects. Small, because individual baselines have been removed.
Simple effects analysis
Tests the effect of one factor SEPARATELY at each level of the other. Used ONLY after a statistically significant interaction to decompose it.
η²_p (partial eta-squared)
Represents the proportion of variance explained by a specific effect. It is the SPSS default but tends to overestimate population effect in small samples.
ω²_p (partial omega-squared)
A closely related metric to η²_p that provides a less-biased estimate of the population effect size. Much preferred for small-to-moderate samples.
Cohen's *f*
An effect size metric. Unlike partial eta-squared variance proportions, Cohen's *f* is required as the default F-Test input parameter when calculating power requirements in G*Power.
9. APA Reporting Builder
Practice writing a complete APA-style report for a mixed factorial ANOVA. Fill in the values below and click Generate Report.
Design information
Mauchly's test
Interaction (A × B)
10. Common Pitfalls: Final Knowledge Checks
Knowledge Check 7
A researcher reports a significant main effect of Group in a factorial ANOVA without first checking the interaction. The interaction turns out to be significant (η²_p = .42). What should the researcher do?
Knowledge Check 8
Which of the following is a complete APA-style statement for an interaction result in a mixed ANOVA?
Reveal My Score
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References
1. Furtado, O., Jr. (2026). Statistics for movement science: A hands-on guide with SPSS (1st ed.). https://drfurtado.github.io/sms/
