Chapter 16: Factorial Analysis of Variance
Student Resources
I use the 4 “P’s” framework to help you learn the material in this chapter: Prepare, Practice, Participate, and Perform. To increase the chances to succeed in this course, I strongly encourage you to complete all four “P’s” for each chapter.
1 Prepare
1.1 Chapter Overview
This chapter introduces factorial ANOVA — the extension of the one-way ANOVA to designs with two or more independent variables (factors) operating simultaneously on a single outcome. Beyond estimating the individual contribution of each factor (the main effects), factorial ANOVA tests whether the effect of one factor depends on the level of another — the interaction effect. You will learn how variance is partitioned in between-subjects factorial, mixed (split-plot), and within-within designs; why the interaction must be interpreted before the main effects; how to decompose a significant interaction using simple effects analysis; and how to select the correct error term for each F-ratio in a mixed ANOVA. Effect size reporting with partial eta-squared and partial omega-squared, along with APA-style write-ups for each design type, complete the chapter.
1.2 Multimedia Resources
The following table provides access to video and slide resources for this chapter. Click the links to open them in an overlay for better viewing on all devices.
| Resource | Description | Link |
|---|---|---|
| Long Video Overview | A detailed video explaining factorial ANOVA, main effects, interactions, mixed designs, simple effects, and APA reporting in movement science research. | 🔗 Watch Video |
| 🆕 Interactive Self-Study Guide | NEW: An interactive, comprehensive module to be completed before attending class. Covers core concepts to prepare you for the lecture. | 🔗 View Guide |
| Slide Deck HTML | Interactive HTML slides for class. During class, the instructor controls the presentation; after class, review at your own pace. | 🔗 Open Slides |
| Slide Deck PDF | PDF version of the slide deck for download and offline viewing. | 🔗 Download PDF |
An alternative interactive study guide is also available for your review: Chapter 16 Slides. Please note that this alternative guide is purely for study purposes and no extra credit points are awarded for completing it.
You can earn extra credit points by completing the Interactive Self-Study Guide before attending the lecture.
Instructions:
- Click the link in the table above to start the guide.
- Complete the activity (you may retake it as many times as you like).
- Once satisfied with your result, take a screenshot of the Final Score Card.
- Submit the screenshot to Canvas.
Important: Your screenshot must clearly show both your score and your full name (ensure you enter your name at the start of the activity). See Canvas for point details.
1.3 Read the Chapter
Read (Weir & Vincent, 2021, p. Ch.14) and (Furtado, 2026, p. Ch.16) to understand factorial ANOVA, interaction effects, mixed designs, and simple effects analysis.
To succeed in this course, you must read the textbook chapters assigned for each topic. This is the only way to learn the material in depth.
Once done, proceed to the next section to practice what you learned.
2 Practice
Practicing what you learned in the chapter is essential to mastering the material. Below are some resources to help you practice the material in this chapter.
2.1 Frequently Asked Questions
A factorial design is one in which the researcher manipulates or observes two or more independent variables — called factors — simultaneously and examines their individual and combined effects on a single outcome variable. Each factor has discrete categories called levels, and every unique combination of factor levels defines a cell. For example, a 2(Sex: Female, Male) × 2(Group: Control, Training) design produces four cells.
Running two separate one-way ANOVAs instead of a factorial ANOVA has three critical problems. First, it inflates the familywise Type I error rate — the more tests performed, the greater the chance of a spurious significant result. Second, and most importantly, separate ANOVAs cannot detect an interaction — the possibility that the effect of training differs between male and female participants simply cannot be evaluated unless both factors are modeled simultaneously. Third, the factorial ANOVA typically uses a smaller error term than separate one-way tests because variance attributable to all factors is removed before estimating error, yielding more powerful tests.
A main effect is the overall effect of one factor, averaging across all levels of all other factors. The main effect of Group represents the Training vs. Control difference collapsed across (ignoring) sex. Main effects describe the simple, unconditional story.
An interaction occurs when the effect of one factor depends on the level of another factor. If the training program improves strength by 12 kg in male participants but only 4 kg in female participants, the Group effect is not the same across sex — there is a Group × Sex interaction. The effect of training is conditional on the participant’s sex.
A quantitative (ordinal) interaction means both groups change in the same direction, but by different amounts. A qualitative (disordinal) interaction means the direction of the effect reverses across levels — one group improves while the other worsens. Qualitative interactions are rarer but more dramatic in movement science data.
When a factorial ANOVA yields a significant interaction, the main effects become incomplete — and potentially misleading — summaries of the data. A significant main effect of Group that averages over very different sex-specific patterns describes neither the male pattern nor the female pattern correctly; it describes only a blend of the two. The cardinal rule of factorial ANOVA is: check the interaction first. If the interaction is significant, decompose it with simple effects analysis, and interpret any main effect statements in the context of that interaction. Only when the interaction is non-significant can the main effects be interpreted straightforwardly as stand-alone findings.
An interaction plot has the levels of one factor on the x-axis, the outcome variable on the y-axis, and separate lines for each level of the second factor. The key diagnostic is whether the lines are parallel:
- Parallel lines → no interaction: the Group effect is the same regardless of Sex
- Diverging lines → quantitative interaction: one group benefits more than the other, but both improve
- Crossing lines → qualitative interaction: the direction of the effect reverses
Always examine the interaction plot before running formal simple effects tests — the plot tells you where the interaction is and which comparisons are theoretically meaningful. Conventional layout places the factor with more levels on the x-axis and uses separate lines for the factor with fewer levels.
In a two-way between-subjects ANOVA, every participant belongs to exactly one cell. Total variance is partitioned into four sources:
\[SS_{\text{total}} = SS_A + SS_B + SS_{A \times B} + SS_{\text{error}}\]
All three effects — Factor A, Factor B, and the A × B interaction — are tested against the same within-cell error term. The F-ratios are:
\[F_A = \frac{MS_A}{MS_{\text{error}}}, \quad F_B = \frac{MS_B}{MS_{\text{error}}}, \quad F_{A \times B} = \frac{MS_{A \times B}}{MS_{\text{error}}}\]
A mixed factorial ANOVA (also called a split-plot design) combines at least one between-subjects factor (such as Group: Control vs. Training) with at least one within-subjects factor (such as Time: Pre, Mid, Post). It is the most common factorial design in training and rehabilitation research because it directly answers: do the groups follow different trajectories of change over time? A significant Group × Time interaction confirms that the groups’ strength curves diverge — which is the scientific core of most intervention studies.
The mixed design has fundamentally different sources of variability for between-subjects and within-subjects effects, so they require different denominators for their F-ratios:
| Effect | F-ratio | Error term |
|---|---|---|
| Group (between-subjects) | \(MS_{\text{Group}} / MS_{\text{Subjects/Group}}\) | Large (includes stable individual differences) |
| Time (within-subjects) | \(MS_{\text{Time}} / MS_{\text{Time × Subjects/Group}}\) | Small (only within-person inconsistency) |
| Group × Time interaction | \(MS_{\text{Group × Time}} / MS_{\text{Time × Subjects/Group}}\) | Small (same as Time) |
Because the within-subjects error is typically much smaller than the between-subjects error, the Time and interaction F-ratios tend to be much larger — and better powered — than the Group F-ratio. This asymmetry is why you should never compare F values across these sources to judge relative importance; always use η²_p or ω²_p.
Yes. The sphericity assumption — that the variances of all pairwise difference scores among the within-subjects factor levels are approximately equal — applies to both the within-subjects main effect (Time) and the interaction term (Group × Time) in a mixed ANOVA, because both use the same within-subjects error. Before reading the Time and interaction F-values, check Mauchly’s test:
- Mauchly’s p > .05: Use “Sphericity Assumed” rows
- Mauchly’s p < .05, ε_GG < .75: Apply the Greenhouse-Geisser correction
- Mauchly’s p < .05, ε_GG ≥ .75: Apply the Huynh-Feldt correction
The between-subjects Group effect is not affected by the sphericity assumption.
Simple effects analysis is the follow-up procedure used when a factorial interaction is significant. It tests the effect of one factor separately at each level of the other factor. For a significant Group × Time interaction, two complementary questions can be asked:
- Does strength change across Time within the Control group?
- Does strength change across Time within the Training group?
Or, flipped: Do the groups differ at Pre? At Mid? At Post?
In SPSS, simple effects are accessed via the General Linear Model → Repeated Measures dialog (Options → Compare main effects) or by running separate one-way ANOVAs after splitting the file by one factor. Simple effects tests should not be run after a non-significant interaction — that would capitalize on chance and inflate Type I error.
Partial eta-squared (η²_p) is computed separately for each effect using only that effect’s SS and the relevant error SS:
\[\eta^2_p = \frac{SS_{\text{effect}}}{SS_{\text{effect}} + SS_{\text{error}}}\]
Note that the “error” in this formula differs by effect type in a mixed ANOVA: the between-subjects error (\(SS_{\text{Subjects/Group}}\)) is used for the Group main effect, while the within-subjects error is used for Time and the interaction. Cohen’s (1988) benchmarks: small ≈ .01, medium ≈ .06, large ≥ .14.
Partial omega-squared (ω²_p) corrects for positive bias — η²_p tends to overestimate the population effect, especially with small samples:
\[\omega^2_p = \frac{SS_{\text{effect}} - df_{\text{effect}} \cdot MS_{\text{error}}}{SS_{\text{effect}} + (N \cdot p - df_{\text{effect}}) \cdot MS_{\text{error}}}\]
where \(p\) is the number of within-subjects levels (or 1 for between-subjects effects) and \(N\) is the total sample size. SPSS does not compute ω²_p automatically; calculate it from the ANOVA source table. Always report η²_p (SPSS default) and ideally ω²_p for every source of variance.
A complete report must include:
- Design statement — identify both factors, their type (between vs. within), and the outcome variable
- Sphericity check — Mauchly’s W, df, and p; correction applied (if any) and epsilon value
- Interaction — F(df1, df2), p, η²_p, ω²_p — always report this first
- Simple effects (if interaction is significant) — which groups/time points differ
- Main effects — F, df, p, η²_p, ω²_p for each factor
- Descriptive statistics — M and SD for all cells
Example (mixed ANOVA, significant interaction):
“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. Simple effects analysis revealed a significant effect of Time within the training group, with progressive gains from pre (M = 79.67 kg) to post (M = 85.06 kg), while the control group remained essentially unchanged (pre: 76.34 kg, post: 77.14 kg). The main effect of Time was significant, F(2, 116) = 108.55, p < .001, η²_p = .65, and the main effect of Group was not significant, F(1, 58) = 2.52, p = .117, η²_p = .04.”
Interpreting main effects without checking the interaction: When the interaction is significant, main effects are incomplete summaries. Always test the interaction first and decompose it before making statements about individual factors.
Concluding “no interaction” from a non-significant p-value: A non-significant interaction F tells you the data are consistent with no interaction, not that no interaction exists. In an underpowered study, even a moderate interaction may be missed. Always report η²_p for the interaction regardless of significance.
Using the wrong error term in a mixed ANOVA: The Group F-ratio uses the between-subjects error; the Time and interaction F-ratios use the within-subjects error. These are fundamentally different denominators and F-values across these sources cannot be meaningfully compared by magnitude.
Failing to report effect sizes for all effects: Report η²_p and ω²_p for every source — Group, Time, and the interaction — not only for the “significant” effects. Readers need all values to evaluate power and practical significance.
2.2 Test your Knowledge
Take this low-stakes quiz to test your knowledge of the material in this chapter. This quiz is for practice only and will help you identify areas where you may need additional review.
3 Participate
This section includes activities and discussions that will be completed during class time. Your active participation is essential for deepening your understanding of the material.
During class, we will: - Identify whether a research scenario calls for a between-subjects, mixed, or within-within factorial ANOVA - Sketch interaction plots and decide whether lines are parallel or diverging - Determine whether an interaction is qualitative (disordinal) or quantitative (ordinal) from a graph - Select the correct error term for each F-ratio in a mixed ANOVA source table - Interpret Mauchly’s test and apply the appropriate sphericity correction to within-subjects effects - Conduct simple effects analysis to decompose a significant Group × Time interaction - Calculate partial eta-squared and partial omega-squared for each source in a factorial ANOVA - Practice writing a complete APA-style mixed factorial ANOVA report
4 Perform
4.1 Apply Your Learning
Now that you’ve prepared, practiced, and participated, it’s time to demonstrate your mastery of the material through assignments and assessments.
I strongly encourage you to complete the previous “Ps” (Prepare, Practice, Participate) before attempting any assignments or assessments associated with this chapter.