Types of Factorial Design: Within-Subject, Between-Subject, and Mixed Designs
Introduction
Once a researcher decides to use a factorial design, the next critical question is: Should participants experience all conditions or only some? This choice determines whether the design is a within-subject, between-subject, or mixed factorial design. Each has different implications for statistical power, participant burden, carry-over effects, and practical feasibility.
This unit explores all three types with classic psychological experiments, layout tables, and practical guidance for choosing the right design.
📄 Block-3/Unit-2.pdf — Pages 23–26
📚 MPC-005 Research Methods
2.4.1 Layout of Factorial Design: A Visual Foundation
Before exploring the types, understanding the standard layout is essential.
2×2 Factorial Design Layout
The simplest factorial design: 2 factors, each with 2 levels = 4 cells.
| Factor A: Level A₁ | Factor A: Level A₂ | |
|---|---|---|
| Factor B: Level B₁ | Cell A₁B₁ | Cell A₂B₁ |
| Factor B: Level B₂ | Cell A₁B₂ | Cell A₂B₂ |
Each cell represents one experimental condition. The cell means are the foundation for calculating both main effects and interaction effects.
2×3 Factorial Design Layout
2 levels of Factor B, 3 levels of Factor A = 6 cells:
| A₁ | A₂ | A₃ | Row Mean | |
|---|---|---|---|---|
| B₁ | A₁B₁ | A₂B₁ | A₃B₁ | Main Effect B₁ |
| B₂ | A₁B₂ | A₂B₂ | A₃B₂ | Main Effect B₂ |
| Column Mean | Main Effect A₁ | Main Effect A₂ | Main Effect A₃ | Grand Mean |
- Column means → Main effect of Factor A (ignoring B)
- Row means → Main effect of Factor B (ignoring A)
- Cell pattern → Reveals interaction
2.5.1 Within-Subject Factorial Design
Definition
In a within-subject factorial design, every participant experiences every condition — all combinations of all factor levels. This is also called a repeated-measures factorial design.
Classic Study: Godden & Baddeley (1975)
This landmark study on context-dependent memory used a perfect within-subject factorial design.
Research question: Does memory performance improve when the context at recall matches the context during learning?
Participants: Deep-sea divers (a highly specialized, motivated sample)
Factors:
- Factor A (Learning Place): 2 levels — On the Beach (A₁) vs. Under Sea (A₂)
- Factor B (Testing Place): 2 levels — On the Beach (B₁) vs. Under Sea (B₂)
Conditions created (2×2 = 4):
- A₁B₁: Learned on beach → Recalled on beach (Matching context)
- A₁B₂: Learned on beach → Recalled under sea (Mismatching context)
- A₂B₁: Learned under sea → Recalled on beach (Mismatching context)
- A₂B₂: Learned under sea → Recalled under sea (Matching context)
Key design feature: Every diver participated in all four conditions. The word lists were randomly assigned to conditions and the order of testing was randomized to control for carry-over effects.
Layout Table: Within-Subject Design
| A₁ (Learning: Beach) | A₂ (Learning: Under Sea) | |
|---|---|---|
| B₁ (Testing: Beach) | S₁, S₂, S₃, S₄, S₅, S₆ | S₁, S₂, S₃, S₄, S₅, S₆ |
| B₂ (Testing: Under Sea) | S₁, S₂, S₃, S₄, S₅, S₆ | S₁, S₂, S₃, S₄, S₅, S₆ |
Notice: The same participants appear in every cell — they experience all four conditions.
Result of Godden & Baddeley (1975): Divers recalled significantly more words when tested in the same environment where they had studied — supporting the Encoding Specificity Principle.
Advantages of Within-Subject Factorial Design
| Advantage | Explanation |
|---|---|
| Statistical power | Individual differences are controlled — same person is their own comparison |
| Fewer participants needed | Each participant contributes data to every cell |
| Controls individual differences | Person-level variables (IQ, anxiety) don't confound comparisons |
| More observations per cell | Richer data per condition |
Disadvantages
| Disadvantage | Explanation |
|---|---|
| Carry-over effects | Experiences in one condition may affect subsequent conditions |
| Order effects | Earlier conditions may prime or fatigue participants for later ones |
| Practice/fatigue effects | Learning or exhaustion across conditions |
| Not always feasible | Some manipulations are irreversible (e.g., learning a fact cannot be un-learned) |
Counteracting carry-over effects: Randomizing or counterbalancing the order of conditions across participants.
2.5.2 Between-Subject Factorial Design
Definition
In a between-subject factorial design, each participant experiences only one condition — one specific combination of factor levels. Different groups of participants are assigned to different cells.
Layout Table: Between-Subject 2×2 Design
With 6 participants per cell in a 2×2 design, 24 total participants are needed:
| Factor A: A₁ | Factor A: A₂ | |
|---|---|---|
| Factor B: B₁ | S₁, S₂, S₃, S₄, S₅, S₆ | S₁₃, S₁₄, S₁₅, S₁₆, S₁₇, S₁₈ |
| Factor B: B₂ | S₇, S₈, S₉, S₁₀, S₁₁, S₁₂ | S₁₉, S₂₀, S₂₁, S₂₂, S₂₃, S₂₄ |
Each participant appears in only one cell.
Example: Drug Treatment Study
A study examines two drugs (Drug A, Drug B) and two dosage levels (Low, High) on pain reduction.
- Group 1: Drug A, Low dose
- Group 2: Drug A, High dose
- Group 3: Drug B, Low dose
- Group 4: Drug B, High dose
Each patient receives only one treatment. This is necessary because giving all drugs to the same patient would be medically inappropriate and would confound results.
Advantages of Between-Subject Design
| Advantage | Explanation |
|---|---|
| No carry-over effects | Conditions don't contaminate each other |
| Works for irreversible manipulations | Participant can receive treatment only once |
| Simpler participant protocol | Each person has a shorter, simpler experience |
| Appropriate when conditions are mutually exclusive | E.g., gender, diagnosis, school type |
Disadvantages
| Disadvantage | Explanation |
|---|---|
| Requires more participants | N × number of conditions participants needed |
| Less statistical power | Individual differences become error variance |
| Random assignment challenges | Must ensure groups are equivalent at baseline |
2.5.3 Mixed Factorial Design
Definition
A mixed factorial design contains at least one within-subject factor and at least one between-subject factor. It is sometimes called a split-plot design in the statistical literature.
This design is used when:
- Some variables logically must be between-subject (e.g., gender, diagnosis, culture)
- Other variables can be repeated within participants (e.g., time, drug dose, condition)
Classic Example: Gender × Alcohol × Risk-Taking
Research question: How do gender and alcohol level interact to affect risk-taking behavior while driving?
Factors:
- Factor A (Gender): 2 levels — Male (A₁) vs. Female (A₂) → BETWEEN-subject (a participant can only be one gender)
- Factor B (Alcohol Level): 4 levels — B₁, B₂, B₃, B₄ → WITHIN-subject (each participant tested at all 4 alcohol concentrations, in randomized order)
Participants: 10 total — 5 males, 5 females
Mixed Factorial Layout
| B₁ (Alcohol Level 1) | B₂ | B₃ | B₄ | |
|---|---|---|---|---|
| A₁ (Male) | S₁, S₂, S₃, S₄, S₅ | S₁, S₂, S₃, S₄, S₅ | S₁, S₂, S₃, S₄, S₅ | S₁, S₂, S₃, S₄, S₅ |
| A₂ (Female) | S₆, S₇, S₈, S₉, S₁₀ | S₆, S₇, S₈, S₉, S₁₀ | S₆, S₇, S₈, S₉, S₁₀ | S₆, S₇, S₈, S₉, S₁₀ |
- Males and females are separate groups (between-subject for gender)
- All participants experience all 4 alcohol levels (within-subject for alcohol)
Possible findings: Males may show a different pattern of increasing risk-taking with alcohol levels compared to females — this gender × alcohol interaction could only be detected with this mixed design.
2.4.3 Graphical Representation of Interaction
The graphical method is the most intuitive way to detect and communicate interaction effects.
Reading Interaction Graphs
Rule: If lines in the graph are not parallel, an interaction is present.
Three scenarios from the text (Task Complexity × Arousal Study):
| Study | Data Pattern | Interpretation |
|---|---|---|
| Study I (Task complexity only) | High arousal → Simple: 7.6s, Complex: 13.1s | Main effect of task complexity only |
| Study II (Arousal only) | Low arousal: 10.3s, High arousal: 10.4s | No effect of arousal alone |
| Study III (Both factors) | Simple task: Low 9.0s, High 6.2s; Complex task: Low 11.8s, High 14.4s | Interaction! High arousal helps simple tasks but hurts complex tasks |
Why Study III shows interaction: For simple tasks, high arousal improves performance. For complex tasks, high arousal hurts performance. The direction of arousal's effect reverses based on task complexity — the hallmark of interaction.
This pattern follows the Yerkes-Dodson Law: an inverted-U relationship between arousal and performance, with the optimal level depending on task complexity.
Graphical Indicators of Interaction
No Interaction (Parallel lines):
Performance
| —————————— Complex
| —————————— Simple
|________________
Low High
Arousal
Interaction (Non-parallel lines, diverging):
Performance
| /Complex
| /
| ————————/Simple (flat)
|______________
Low High
Arousal
Interaction (Crossing lines):
Performance
| Simple\ /Complex
| \/
| /\
|_____________
Low High
Arousal
2.4.4 Importance of Interaction Effects
Understanding interactions is arguably more important than understanding main effects, for several reasons:
-
Main effects can be misleading when interactions are present. A main effect that looks impressive may completely disappear — or even reverse — at certain levels of another variable.
-
Interactions reveal practical nuance. "Method A is better than Method B" is far less informative than "Method A is better than Method B — but only for students with high prior knowledge."
-
Interactions are the heart of moderation analysis. When researchers ask "does the effect of X depend on Y?", they're asking about interactions.
-
Simple effects analysis. When interaction is significant, researchers analyze simple effects — the effect of one factor at each specific level of the other factor. Example: In a 2×2 design, find the effect of Factor A separately for B₁ and separately for B₂.
Practical example: A study comparing two teaching methods for students of low vs. high intelligence. The researcher already knows high-IQ students learn faster — this main effect is not interesting. What's interesting: Does method effectiveness interact with intelligence? Perhaps Method 1 works better for slower learners but Method 2 works better for faster learners — a crucial interaction with practical educational implications.
Comparison: Three Types of Factorial Designs
| Feature | Within-Subject | Between-Subject | Mixed |
|---|---|---|---|
| Participants per condition | Same participants | Different participants | Depends on factor |
| Carry-over effects | High risk | No risk | Partial risk |
| Statistical power | Highest | Lowest | Intermediate |
| Participants needed | Fewest | Most | Intermediate |
| Appropriate when | Reversible manipulations | Irreversible or impossible to repeat | Some factors fixed, others can repeat |
| Controls individual differences | Best | Weakest | Partial |
| Example variable | Task condition | Gender | Gender × Drug dose |
Memory Aid: W-B-M Rule
Within = Whole sample in every condition
Between = Big samples needed (separate groups)
Mixed = Marriage of both — one factor each way
Indian Psychology Context
Mixed factorial designs are especially common in Indian psychological research:
- NEET preparation studies: Coaching method (between: different coaching institutes) × Study hours (within: across months) → Exam performance
- Clinical psychology: Diagnosis group (between: anxiety, depression, controls) × Therapy session (within: sessions 1–8) → Symptom reduction
- Gender studies: Gender (between) × Task complexity (within: easy/medium/hard) → Cognitive performance, addressing debates about gender differences in Indian educational contexts
External Resources
Wikipedia
- 🌐 Repeated Measures Design — Wikipedia
- 🌐 Mixed Design Analysis of Variance — Wikipedia
- 🌐 Between-Subjects Design — Wikipedia
Educational Videos
Research Papers
- 📄 Godden, D. R., & Baddeley, A. D. (1975). Context-dependent memory in two natural environments: On land and underwater. British Journal of Psychology, 66(3), 325–331.
- 📄 Yerkes, R. M., & Dodson, J. D. (1908). The relation of strength of stimulus to rapidity of habit-formation. Journal of Comparative Neurology and Psychology, 18(5), 459–482.
- 📄 Field, A. P., & Hole, G. (2002). How to Design and Report Experiments. Sage Publications.
Online Resources
- 🔗 Mixed ANOVA — Laerd Statistics
- 🔗 Within-Subject Design — Simply Psychology
- 🔗 Between-Subject Design — Simply Psychology
- 🔗 Interaction Effects Visualization — Psychonomic Bulletin
Self-Assessment Questions
Level 1 — Recall
- In a within-subject factorial design, how many groups of participants are there?
- What is the defining feature of a mixed factorial design?
Level 2 — Understanding 3. Why would a researcher choose a between-subject design over a within-subject design, even though it requires more participants? 4. Explain with a graph why parallel lines indicate no interaction, while non-parallel lines indicate interaction.
Level 3 — Application 5. A researcher wants to study the effects of (a) type of therapy (CBT vs. psychodynamic) and (b) number of sessions (4 vs. 8 vs. 12) on depression scores. Which type of factorial design would be most appropriate — within-subject, between-subject, or mixed? Justify your answer.
Design Identification Practice:
| Study | Design Type |
|---|---|
| Gender × Study method on exam scores | Mixed (gender=between, method=within) |
| Same participants under 3 noise conditions × 2 task types | Within-subject |
| 4 different therapy groups each tested at 3 time points | Mixed |
| Different drug groups (A, B, placebo) tested once | Between-subject |
Previous: ← Factorial Design Basics
Next: Advantages, Limitations & Applications of Factorial Design →