Module S05

Two-Way ANOVA

Analyzing two factors and their interaction

Factorial Designs

  • 01

    Two Factors

    Testing two IVs simultaneously

  • 02

    Main Effects

    Effect of each factor separately

  • 03

    Interaction

    Do factors combine non-additively?

  • 04

    Interpretation

    Reading interaction plots

01

Why Two-Way ANOVA?

Multiple Factors

One-Way ANOVA
One factor (IV)
e.g., Drug type (A, B, C)
Question: Does drug type affect outcome?
Two-Way ANOVA
Two factors (IVs)
e.g., Drug type × Dosage
Questions: Drug effect? Dosage effect? Do they interact?

Two-way ANOVA extends one-way by testing two independent variables simultaneously. This lets us see not just if each factor matters, but whether they interact - whether the effect of one depends on the level of the other.

Factors
2 independent variables
Tests
2 main effects + 1 interaction

Factorial Design

Example: Testing drug effectiveness by type and dosage
Low DoseMedium DoseHigh Dose
Drug ACell 1Cell 2Cell 3
Drug BCell 4Cell 5Cell 6
Drug CCell 7Cell 8Cell 9
3 × 3 = 9 cells, each with multiple observations

A 3×3 factorial design has 9 'cells' - every combination of factor levels. Each cell contains multiple participants. Two-way ANOVA partitions variance into: Factor A, Factor B, A×B interaction, and error.

Factors
2 independent variables
Tests
2 main effects + 1 interaction

What Two-Way Tests

1. Main Effect of Factor A
Ignoring B, does A affect the outcome?
e.g., Does drug type matter (averaging across dosages)?
2. Main Effect of Factor B
Ignoring A, does B affect the outcome?
e.g., Does dosage matter (averaging across drugs)?
3. A × B Interaction
Does the effect of A depend on the level of B?
e.g., Does drug type's effect vary by dosage?

Two-way ANOVA produces three F-tests: one for each main effect and one for the interaction. The interaction is often the most interesting - it reveals whether factors combine non-additively.

Factors
2 independent variables
Tests
2 main effects + 1 interaction
02

Main Effects

What Are Main Effects?

LowMedHighMain Effect of Dosage
Averaging across drug types, does dosage affect outcome?

A main effect is the overall effect of one factor, averaging (or collapsing) across levels of the other factor. It answers: 'Ignoring the other factor, does this factor matter?'

Main Effect
Marginal means differ?
Collapse
Average across other factor

Calculating Main Effects

LowHighRow Mean
Drug A204030
Drug B406050
Col Mean305040
Main effect of Drug: B (50) > A (30) → 20-point difference
Main effect of Dose: High (50) > Low (30) → 20-point difference

Row means test Factor A (Drug), column means test Factor B (Dose). Here, both factors have 20-point main effects. But this doesn't tell us about interaction - whether the drug effect differs by dose.

Main Effect
Marginal means differ?
Collapse
Average across other factor
03

Interaction Effects

The Key Concept

Interaction = Effect of A depends on level of B
NO Interaction
Parallel lines
YES Interaction
Non-parallel lines

Interaction means the effect of one factor changes depending on the level of the other. In plots, parallel lines = no interaction; crossing or diverging lines = interaction present.

Interaction
Effect depends on other factor
Visual
Non-parallel lines

Real Interactions

Crossover Interaction
Drug A works better at low dose, Drug B works better at high dose. Lines cross - the 'best' drug depends on dosage.
Ordinal Interaction
Drug A is always better than B, but the difference is larger at high dose. Lines spread apart but don't cross.
No Interaction (Additive)
Drug A is always 20 points better than B, regardless of dose. Lines are parallel. Effects simply add up.

Crossover interactions are most dramatic - they mean qualitative differences (which is 'best' changes). Ordinal interactions are quantitative - the ordering stays the same but magnitude differs.

Interaction
Effect depends on other factor
Visual
Non-parallel lines

What to Report

If Interaction is Significant:
Main effects may be misleading! Don't just say 'Drug A is better' if it depends on dosage. Focus on the interaction.
If NO Interaction:
Main effects are interpretable. Can say 'Drug A is better than B' and 'Higher dose is better' without qualification.

When there's a significant interaction, main effects can be misleading or even meaningless. Always check interaction first. If significant, describe the pattern rather than averaging across factors.

Interaction
Effect depends on other factor
Visual
Non-parallel lines
04

Reading ANOVA Output

The Output

SourceSSdfMSFp
Drug (A)20021005.0.012*
Dose (B)400220010.0.001*
A × B804201.0.42
Error7203620--
* p < .05

Two-way ANOVA output shows three F-tests. Here: significant main effects of Drug (p=.012) and Dose (p=.001), but NO significant interaction (p=.42). Main effects are interpretable.

Effect Size
η² (eta squared)
Key
Check interaction first

How to Report

APA Style Example:

"A 3 (Drug: A, B, C) × 3 (Dose: Low, Medium, High) between-subjects ANOVA revealed significant main effects of drug type, F(2, 36) = 5.0, p = .012, η² = .14, and dosage, F(2, 36) = 10.0, p = .001, η² = .28. The Drug × Dose interaction was not significant, F(4, 36) = 1.0, p = .42."

Always report: the design (factors and levels), degrees of freedom, F-value, p-value, and effect size (η²). Report the interaction test even if not significant - absence of interaction is informative.

Effect Size
η² (eta squared)
Key
Check interaction first
★

Real-World Examples

Apply your knowledge to practical scenarios

01

Therapy × Medication Study

A study compares depression treatments: 2 therapy types (CBT, Psychodynamic) × 2 medication conditions (Drug, Placebo). Outcome is depression score reduction (higher = better).

Q What would a significant main effect of therapy mean? +
A Averaging across medication conditions, one therapy type produces better outcomes than the other. e.g., CBT > Psychodynamic regardless of whether patients got drug or placebo.
Q What would a significant interaction mean? +
A The effectiveness of therapy depends on medication status. e.g., CBT works equally well with/without drugs, but Psychodynamic only works well when combined with medication.
Q Results: Main effect of therapy (p=.03), no main effect of medication (p=.25), significant interaction (p=.01). How to interpret? +
A The interaction qualifies the therapy effect. Can't simply say 'therapy works' - must specify: therapy type matters, but the pattern depends on medication status. Need to look at cell means and interaction plot.
Key Insights
  • Significant interaction means main effects need qualification
  • Plot the cell means to understand the pattern
  • Clinical recommendations depend on the interaction pattern
?

Two-Way ANOVA Quiz

Test your understanding • Passing score: 70%

0 of 5 answered
Q1 multiple-choice

Two-way ANOVA tests:

Explanation: Two-way ANOVA examines two independent variables (factors) and their interaction - whether the effect of one factor depends on the level of the other.
Q2 multiple-choice

In an interaction plot, parallel lines indicate:

Explanation: Parallel lines mean the effect of one factor is the same at all levels of the other - no interaction. Non-parallel (crossing or diverging) lines indicate interaction.
Q3 multiple-choice

A significant interaction means:

Explanation: Interaction means the effect of Factor A changes depending on the level of Factor B. The factors combine non-additively.
Q4 multiple-choice

If there's a significant interaction, you should:

Explanation: Report main effects but interpret them in context of the interaction. A main effect that says 'A is better than B overall' may be misleading if A is only better at certain levels of the other factor.
Q5 multiple-choice

A 2×3 factorial design has how many cells?

Explanation: A 2×3 design has 2 levels of Factor A × 3 levels of Factor B = 6 cells, each containing different combinations of factor levels.

Quiz Results

0 %

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