Factorial ANOVA: Main Effects and Interactions

Overview

• The lecture introduces factorial ANOVA (chapter 13), focusing on the logic of multiple independent variables, main effects, and interactions.
• It builds on earlier chapters (t‑tests and one‑way ANOVA) to show when and why to use factorial designs.

Review of Earlier Chapters and Test Selection

• Chapter 9: One‑sample t‑test
  • Compare a sample mean to a known or claimed population mean.
• Chapter 10: Two‑sample t‑tests
  • Independent samples t‑test: two unrelated groups.
  • Paired (related) samples t‑test: repeated measures or matched pairs.
• Chapter 11: One‑way ANOVA (independent samples)
  • One independent variable (IV) with 3+ levels (3+ group means).
  • Use an overall F‑test to decide on the null; then do follow‑up pairwise comparisons (e.g., Tukey).
• Chapter 12 (mentioned but skipped)
  • Same logic as chapter 11, but for repeated‑measures ANOVA (one IV, related samples, 3+ levels).

Choosing the Appropriate Statistical Test

• First identify: independent samples vs related samples.
• Second identify: number of independent variables (factors) and their levels.

• Goal: given a research design, be able to select the proper test using independence/relatedness and number of IVs/levels.

From One‑Way ANOVA to Factorial ANOVA

• Chapter 11/12:
  • One independent variable (one factor), possibly with 3+ levels.
  • Example: Drug dosage with four levels (0, 5, 10, 20 mg) → four means, one F‑test.
• Chapter 13: Factorial designs (factorial ANOVA)
  • Two or more independent variables (factors).
  • Terminology: “factor” = “independent variable.”
  • We now consider situations with multiple IVs simultaneously.

Visual Organization of Factorial Designs

• Example: 2 × 3 factorial design
  • Two independent variables: Factor A (3 levels), Factor B (2 levels).
  • Factor A levels might be alcohol dose:
    • A1 = placebo/no alcohol
    • A2 = some alcohol
    • A3 = a lot of alcohol
  • Factor B levels might be gender:
    • B1 = male
    • B2 = female
  • Creates 6 cells: A1B1, A2B1, A3B1, A1B2, A2B2, A3B2 (each with its own mean).

• Each cell has test scores (DV), e.g., exam performance after drinking.
• Marginal means:
  • Row/column averages that collapse across the other factor.
  • Example: mean of all males (B1), mean of all females (B2).
  • Example: mean of A1, A2, A3 (alcohol levels), collapsing across gender.

Links Back to Earlier Analyses

• Marginal means for gender (B1 vs B2) could be compared with a t‑test.
• Marginal means for alcohol levels (A1, A2, A3) could be compared with a one‑way ANOVA.
• New in Chapter 13: considering all cells jointly to test for interactions and multiple main effects.

Main Effects and Interactions

Main Effect

• Concept: effect of one independent variable by itself, averaging over levels of the other IV(s).
• In the alcohol–gender example:
  • Main effect of gender:
    • Are overall test scores different for males vs females, ignoring alcohol level?
  • Main effect of alcohol:
    • Are overall test scores different across no/some/a‑lot alcohol, ignoring gender?
• A main effect is about whether marginal means differ.

Interaction

• Interaction: the effect of one independent variable depends on the level of another independent variable.
• Informal definition: “The IVs work together in such a way that one IV’s effect changes across the levels of the other IV.”
• In practice:
  • You expect or hypothesize that there is such a combined effect.
  • The interaction is usually the primary focus in factorial ANOVA.

Example: Drug Effect for Males vs Females

• Design: 2 × 2 factorial
  • Factor 1: Gender (Male, Female).
  • Factor 2: Drug (No drug, Drug).
• If you ignore gender and only compare No drug vs Drug with a t‑test, you might find no overall difference.
  • Example pattern:
    • Drug helps males a lot (big increase).
    • Drug does nothing for females (no change).
  • When averaged together (50 males + 50 females), the overall effect might “wash out.”
• When analyzed as factorial ANOVA with cells (Male/No drug, Male/Drug, Female/No drug, Female/Drug):
  • You see that the drug works for males but not for females.
  • This pattern is the interaction between Gender and Drug on the DV.

Party Satisfaction Examples (From Textbook)

Example 1: Shy vs Outgoing × Activity (Talk vs Games)

• DV: Party rating (0 = awful time, 10 = great time).
• IV1: Personality
  • Shy
  • Outgoing
• IV2: Activity at party
  • Talk
  • Games

• Marginal means for activity:
  • Talk: (2 + 7) ÷ 2 = 4.5
  • Games: (3 + 8) ÷ 2 = 5.5

Interpretation:

• Main effect of personality (shy vs outgoing):
  • Outgoing people report much higher party ratings than shy people.
  • 2.5 vs 7.5 suggests a likely significant main effect of personality.
• Main effect of activity (talk vs games):
  • 4.5 vs 5.5 is a small difference; likely not significant.
• Interaction (Personality × Activity):
  • Lines graphed for shy vs outgoing are parallel.
  • Pattern: shy are always low, outgoing are always high; the gap is similar for talk and games.
  • Conclusion: no meaningful interaction; personality is the key predictor.

Example 2: Companions × Activity (Talk vs Games)

• DV: Party rating (0–10).

• IV1: Companions

  • One (or by yourself)
  • Several (you brought several people)

• IV2: Activity

  • Talk
  • Games

• Given means (conceptual pattern):

  • For “One” companions:
    • Talking → relatively high rating.
    • Games → relatively low rating.
  • For “Several” companions:
    • Talking → relatively low rating.
    • Games → relatively high rating.

• Marginal means:

  • Average rating for “One” vs “Several” are both about 5.5.
  • Average rating for “Talk” vs “Games” are also both about 5.5.

Interpretation:

• No main effect of companions alone (One vs Several → same overall mean).
• No main effect of activity alone (Talk vs Games → same overall mean).
• Clear interaction between companions and activity:
  • If you come with one person/by yourself, you have more fun talking than playing games.
  • If you come with several people, you have more fun playing games than just talking.
• Conclusion: Your rating of the party depends on both what you do (activity) and how many people you come with (companions).
  • Neither IV alone predicts enjoyment well; the combination does.

Graphing Factorial ANOVA and Interpreting Interactions

• DV is plotted on the y‑axis; IV on the x‑axis; the other IV is shown via separate lines (or bar groups).

Parallel vs Non‑Parallel Lines

• Parallel lines (or nearly parallel):
  • Suggest no interaction.
  • Example: Shy vs Outgoing across Talk vs Games → roughly parallel.
• Non‑parallel lines (especially crossing lines):
  • Suggest an interaction.
  • Example: Companions (One vs Several) × Activity (Talk vs Games) → clearly crossing lines.

Why Graphing Matters

• Interactions are easiest to understand from a graph rather than from a numeric ANOVA summary table.
• To interpret an interaction:
  • Plot means for each cell.
  • Describe, in plain English, how the effect of one IV changes across levels of the other IV.

How to Write an Interpretation (When Interaction Is Significant)

• Step 1: State that there is a significant interaction.
• Step 2: Explain the pattern using “both” or “depends on” language.
  • Example for Companions × Activity:
    • “How much fun people have at the party depends on both the number of companions and the activity.
      When people come by themselves, they enjoy talking more than games.
      When people come with several others, they enjoy games more than talking.”
• Step 3: You do not need to separately detail every simple comparison if you clearly describe the overall interaction pattern.
• Important:
  • If there is a significant interaction, you cannot interpret main effects alone as if they were simple, standalone effects.
  • Your focus shifts to the combined pattern.

Assumptions for Factorial ANOVA

• In addition to one‑way ANOVA assumptions (normality, homogeneity of variance, independence of observations, etc.), factorial ANOVA adds conditions.

Key Assumptions (Factorial ANOVA)

• Equal number of scores in each cell
  • Each cell (e.g., A1B1, A2B1, etc.) should have the same sample size.
  • Example: 25 participants in every alcohol × gender cell.
  • Cannot have extremely unbalanced cells (e.g., 70 in one cell, 2 in another).
• Independence of cells
  • Membership in one cell is independent of membership in another.
  • No repeated measures or pairing across cells; no systematic self‑selection linking particular types into particular cells.
• Factors chosen by the experimenter
  • Factor levels (e.g., dosage levels, activity types) are typically set by the researcher.
  • If factors are randomized in a different way (e.g., random factors), different statistical models and equations may be needed.

Key Terms & Definitions

Factor: another word for independent variable (IV) in ANOVA.

• Level: a specific category or condition within a factor (e.g., no alcohol, some alcohol, high alcohol).
• Factorial design: a design with two or more independent variables (factors), each with at least two levels.
• 2 × 3 design: two IVs, first with 2 levels, second with 3 levels, yielding 6 cells.
• Cell: specific combination of factor levels (e.g., A2B1 = some alcohol, male).
• Marginal mean: mean for one level of a factor, averaging across levels of other factor(s).
• Main effect: effect of one factor on the DV, collapsing across other factors.
• Interaction: situation where the effect of one factor on the DV depends on the level of another factor.
• Grand mean: overall mean across all participants and all cells (all levels of all factors).

Action Items / Next Steps

• Practice identifying:
  • Whether a design uses independent or related samples.
  • How many IVs and levels there are, and which test fits.
• Practice drawing factorial ANOVA graphs:
  • Put DV on y‑axis, a factor on x‑axis, and use separate lines for the other factor.
  • Check visually for parallel vs crossing lines to spot interactions.
• When given a factorial ANOVA output:
  • First check for an interaction, graph it, and interpret in plain English if significant.
  • Then consider main effects, being cautious when an interaction is present.
• Review and memorize assumptions for factorial ANOVA, including equal cell sizes and independence of cells.