The Psych Diaries

What is factorial ANOVA?

• Extension of one-way ANOVA to multiple independent variables (IVs)
• Purpose: examine
  1. Main effects of each IV
  2. Interaction effect between IVs
• Design: between-subjects (independent groups)
• DV: numerical (interval/ratio)
• IVs: categorical (factors) with ≥ 2 levels each

Key Terminology

• Factor = independent variable (IV)
• Levels = categories within a factor
• Cell = combination of levels across factors
• Cell mean = mean DV for each combination
• Design notation:
  • 2 × 2 = two IVs, each with 2 levels
  • 3 × 2 = IV1 has 3 levels, IV2 has 2 levels
• k-way design = number of IVs (e.g., 2-way, 3-way)

What Factorial ANOVA Tests

For a 2-way ANOVA, there are three effects:

1. Main Effect of A

• Effect of IV1 on DV, ignoring IV2
• Equivalent to a one-way ANOVA across levels of A

2. Main Effect of B

• Effect of IV2 on DV, ignoring IV1

3. Interaction Effect (A × B)

• Tests whether:

the effect of one IV on the DV differs across levels of the other IV

Understanding Interaction

Definition

An interaction occurs when:

the effect of one IV depends on the level of another IV

• Compare differences between group means
• Example:
  • Difference between male vs female at Year 1
  • compared to difference at Year 2, Year 3, etc.

Types of Interaction

1. Directional interaction

• Effect reverses depending on the other IV
• Graph: lines cross

2. Magnitude interaction

• Effect exists but differs in strength
• Graph: lines not parallel

No interaction:

• Effect of one IV is consistent across levels of the other IV
• Graph: lines parallel

Why Not Run Multiple One-Way ANOVAs?

• Would miss moderating effects (interactions)
• Factorial ANOVA tests:
  • IV1 effect
  • IV2 effect
  • interaction
    simultaneously

Analysis The Output

Step 1: Test interaction

• If significant (p < .05):
  • Interpret interaction
  • Conduct simple effects analysis
  • Do not rely on main effects alone

Step 2: If interaction is not significant

• Interpret main effects

Step 3: If IV has >2 levels

• Conduct post-hoc comparisons (e.g., Tukey)

Graph Interpretation

• Interaction:
  • Lines not parallel
• No interaction:
  • Lines parallel

Tips:

• Factor with more levels usually on x-axis
• Lines represent second factor

Stata Output

Command:

anova DV IV1##IV2

Output includes:

• Main effect of IV1
• Main effect of IV2
• Interaction (IV1#IV2)

Each has:

• F-value
• p-value

Variance Partitioning

Total variance is divided into:$$SS_{Total} = SS_A + SS_B + SS_{AB} + SS_{Error}$$SSTotal​=SSA​+SSB​+SSAB​+SSError​

Where:

• $SS_A$SSA​ = variance due to IV1
• $SS_B$SSB​ = variance due to IV2
• $SS_{AB}$SSAB​ = interaction variance
• $SS_{Error}$SSError​ = unexplained variance

Effect Size

Use partial eta squared (ηp²):$$\eta_p^2 = \frac{SS_{\text{effect}}}{SS_{\text{effect}} + SS_{\text{residual}}}$$ηp2​=SSeffect​+SSresidual​SSeffect​​

Interpretation guidelines:

• .01 = small
• .06 = medium
• .14 = large

Each effect has its own ηp²:

• IV1
• IV2
• Interaction

Assumptions

• DV is numerical
• Independence of observations
• Homogeneity of variance
• Approximate normality

Summary

• Factorial ANOVA tests multiple IVs simultaneously
• Produces:
  • main effect A
  • main effect B
  • interaction
• Interaction is the key new concept
• Always:
  1. Check interaction first
  2. Then interpret main effects if appropriate
• Effect size reported as partial eta squared

Example Write-Ups

(Interaction Significant)

A two-way between-subjects ANOVA was conducted to examine the effects of sleep quality and caffeine on memory performance.
There was a significant interaction between sleep quality and caffeine, F(2, 60) = 5.12, p = .009, ηp² = .15. This indicates that the effect of sleep quality on memory performance differed depending on caffeine intake.
Simple effects analyses showed that for participants who consumed caffeine, memory performance did not differ significantly across sleep conditions. However, for participants who did not consume caffeine, those with good sleep performed significantly better than those with poor sleep.

(No Interaction)

A two-way between-subjects ANOVA was conducted to examine the effects of sleep quality and caffeine on memory performance.
The interaction between sleep quality and caffeine was not significant, F(2, 60) = 1.23, p = .30.
There was a significant main effect of sleep quality, F(2, 60) = 8.45, p < .001, ηp² = .22, with participants who reported good sleep demonstrating higher memory performance than those with poor sleep.
The main effect of caffeine was also significant, F(1, 60) = 4.10, p = .047, ηp² = .06, with caffeine consumption associated with higher memory scores.