Chapter 12 – Factorial Designs: Experiments with More Than One Independent Variable

Factorial Designs: Experiments with More Than One Independent Variable

  • Purpose and scope

    • Builds on earlier chapters that covered one independent variable (IV) and one dependent variable (DV).
    • Now explores experiments with two or more IVs to see how they interact in affecting a DV.
    • Central ideas: main effects, interactions (including how to detect and interpret them), and how factorial designs test theory and generalizability.

  • Key concepts and definitions

    • Variable: something that varies; must have at least two levels.
    • Measured variable: levels are measured (e.g., IQ, depression scores).
    • Manipulated (independent) variable: levels are controlled by the experimenter.
    • Independent variable (IV) vs participant variable: in factorial designs, a IV is manipulated; a participant variable (e.g., age, gender) is typically measured but can be treated as an IV for simplicity in analysis.
    • Main effect: the overall effect of one IV on the DV, averaged over the levels of the other IV(s). Formally, for a 2×2 design with factors A and B:
    • Main effect of A =
      Yˉ<em>A1∙−Yˉ</em>A2∙\bar{Y}<em>{A1\bullet} - \bar{Y}</em>{A2\bullet}
      where ar{Y}{Aiullet} is the marginal mean for level i of A averaged across B.
    • Main effect of B =
      Yˉ<em>∙B1−Yˉ</em>∙B2\bar{Y}<em>{\bullet B1} - \bar{Y}</em>{\bullet B2}
      where ar{Y}{ullet Bj} is the marginal mean for level j of B averaged across A.
    • Interaction: the effect of one IV depends on the level of the other IV. In a 2×2 design, the interaction can be described as a "difference in differences":
      extInteraction=(Yˉ<em>A1B1−Yˉ</em>A2B1)−(Yˉ<em>A1B2−Yˉ</em>A2B2).ext{Interaction} = ( \bar{Y}<em>{A1B1} - \bar{Y}</em>{A2B1} ) - ( \bar{Y}<em>{A1B2} - \bar{Y}</em>{A2B2} ).
      This is the difference between the two simple differences (one for B’s level B1, one for B’s level B2).
    • Difference in differences: another way to express an interaction across two IVs. If you compute the A effect at B1 and subtract the A effect at B2, you get the interaction magnitude.
    • Marginal means: the mean for a level of one IV averaging over the levels of the other IV(s). For a 2×2 design:
    • Marginal mean for A1 across B:
      YˉA1∙\bar{Y}_{A1\bullet}
    • Marginal mean for B1 across A:
      Yˉ∙B1\bar{Y}_{\bullet B1}
    • Crossover interaction: an interaction pattern where lines cross on a line graph (illustrates that the direction or magnitude of the IV effect reverses across levels of the other IV).
    • Spreading interaction: a non-crossing interaction where the effect of one IV depends strongly on the level of the other IV, but lines do not cross.
    • Moderators: a variable that changes the strength or direction of a relation between an IV and the DV; in factorials, moderators are often tested as part of interactions.
    • Three-way interaction: in a 2×2×2 design, an interaction among three IVs; it means the two-way interaction between two IVs depends on the level of the third IV.
    • Three-way interaction general form: the interaction among A, B, and C is significant if the two-way interaction between A and B differs across levels of C. A common algebraic representation (textbook form) is:
      ABC=((Yˉ<em>A1B1C1−Yˉ</em>A1B2C1)−(Yˉ<em>A2B1C1−Yˉ</em>A2B2C1))−((Yˉ<em>A1B1C2−Yˉ</em>A1B2C2)−(Yˉ<em>A2B1C2−Yˉ</em>A2B2C2)).ABC = \big((\bar{Y}<em>{A1B1C1}-\bar{Y}</em>{A1B2C1})-(\bar{Y}<em>{A2B1C1}-\bar{Y}</em>{A2B2C1})\big) - \big((\bar{Y}<em>{A1B1C2}-\bar{Y}</em>{A1B2C2})-(\bar{Y}<em>{A2B1C2}-\bar{Y}</em>{A2B2C2})\big).

    where C is the third IV (levels C1 and C2).

  • Why researchers use factorial designs

    • To test interactions: does the effect of one IV depend on another IV (the core question of causality in more complex systems)?
    • To test generalizability and moderators: do effects hold across different groups (e.g., age groups) or conditions (e.g., traffic levels)?
    • To test theories: many theories predict specific interactions; factorial designs are well-suited to test such predictions.
    • To examine whether effects are robust across levels and contexts (external validity).

  • Examples of classic 2×2 factorial designs in the transcript

    • Strayer & Drews (2004): driving simulator, age (younger vs older) and cell phone use (on vs off) as factors. Design: 2 (Age) × 2 (Cell phone). IVs not manipulated in the real world are treated as manipulated (cell phone use) vs measured (age). They found no interaction: the effect of cell phone use on braking onset time did not differ by age. The simple differences were observed in both age groups:
    • Younger drivers: braking onset time difference between on-phone vs not on-phone ~132 ms.
    • Older drivers: braking onset time difference between on-phone vs not on-phone ~174 ms.
    • Statistical tests showed these differences were not significantly different, i.e., no significant Age × Cell Phone interaction.
    • Bartholow & Heinz (2006): alcohol cues and aggression-related vs neutral words in a memory/word-recognition task. Design: 2 (Photo type: Alcohol vs Plant) × 2 (Word type: Aggressive vs Neutral). DV: reaction time to categorize words. They reported a significant interaction in which alcohol cues sped recognition of aggression-related words more than plant cues did, and the plant cues slowed aggression-related words, demonstrating an interaction between perceptual cues and semantic processing.
    • Chi (1978): memory development in children vs adults using two memory tasks (digits vs chess pieces). Design: 2 (Participant type: Child Experts vs Adult Novices) × 2 (Item type: Digits vs Chess Pieces). DV: number of items recalled. Findings showed an interaction: children excelled at recalling chess pieces (domain knowledge) relative to digits, whereas adults excelled more at digits, illustrating domain-specific memory effects.
    • DeWall, Bushman, Giancola, Webster (2010): alcohol effects on aggression moderated by body weight. Design: 2 (Weight: Light vs Heavy) × 2 (Drinking condition: Placebo vs Drunk). Findings indicated a main effect of drinking (drunk more aggressive) and a main effect of body weight (heavy more aggressive). Importantly, a significant interaction suggested alcohol's effect on aggression was stronger in heavier men than in lighter men.

  • Factorial design notations and why they matter

    • Notation shows the number of independent variables and their levels, e.g.:
    • 2×2 design: two IVs each with two levels.
    • 2×3 design: one IV with two levels, another with three levels.
    • 3×4 design: two IVs, with three and four levels respectively.
    • 2×2×2 design: three IVs, each with two levels.
    • Total cells = product of levels, e.g., 2×2×2 has 8 cells.
    • Mixed, independent-groups, and within-groups factorials
    • Independent-groups factorial (between-subjects): all IVs manipulated between subjects; example: DeWall et al. (2010) with 4 independent groups (placebo/light/heavy/ alcohol across two weight levels).
    • Within-groups factorial (repeated measures): all IVs manipulated within subjects; example: Bartholow & Heinz (2006) where participants saw both photo types and responded to both word types across trials.
    • Mixed factorial: one IV manipulated between-subjects and another within-subjects; example: Strayer & Drews (2004) where age was between-subjects and cell phone condition was within-subjects.
    • Practical implications
    • Within-subjects designs require fewer participants and offer higher statistical power, but demand counterbalancing to control order effects.
    • Independent-groups designs reduce carryover effects but require more participants.

  • Interactions, how to read and detect them

    • Detecting interactions from data
    • From a table: compute the two simple differences and compare them (difference in differences). Example with Bartholow & Heinz (2006):
      • Alcohol photos: aggressive vs neutral difference = 551 − 562 = −11 ms.
      • Plant photos: aggressive vs neutral difference = 559 − 552 = 7 ms.
      • Interaction magnitude = −11 − 7 = −18 ms (significant in the study).
    • From a line graph: nonparallel lines indicate an interaction; a crossover pattern strongly suggests an interaction, but nonparallel lines suffice for a significant interaction.
    • From a bar graph: not always obvious; visual inspection of whether line connections would cross or whether differences between bars change across the x-axis helps infer interactions, which should be confirmed with statistical tests.
    • Interpreting main effects vs interactions
    • In a factorial design with two IVs, there are three main results to inspect: two main effects and one interaction.
    • Interactions are often more informative than main effects when present, because a main effect may mask an interaction (e.g., alcohol effects that are stronger for heavier men).
    • Main effects summarize the overall effect of an IV averaging across levels of the other IV(s) but do not reveal how the effect varies by the other IV(s).
    • Describing an interaction in words
    • Strategy 1: Describe one level of the first IV, then describe how the second IV changes the DV at that level; then move to the next level of the first IV and describe again.
    • Strategy 2: Use evidence-based phrases such as "it depends on" or "especially for" to convey the interaction pattern.
    • Example descriptions from the text:
      • Bartholow & Heinz: “When people saw alcohol photos, they were quicker to recognize aggression words than neutral words, but when people saw plant photos, they were slower to recognize aggression words than neutral words.”
      • DeWall et al.: “The alcohol effect on aggression was stronger for heavier men.”

  • How to analyze and report factorial results

    • Main effects and interactions to report in tables and figures
    • Report marginal means for each level of each IV to illustrate main effects.
    • Report the interaction term and, if significant, provide simple-mean comparisons or simple effects analyses.
    • Three-way factorials (2×2×2)
    • Three main effects (one for each IV), three two-way interactions (AB, AC, BC), and one three-way interaction (ABC).
    • A three-way interaction implies the two-way interactions differ across levels of the third IV (a difference in differences across a third factor).
    • Reading and forecasting outcomes (Summary chart example)
    • A chart can show various possible outcomes for main effects and interactions; one should anticipate which patterns imply interactions and which imply simple main effects.

  • Factorial variations: beyond the basics

    • More levels for IVs (e.g., 2×3, 3×4)
    • With more levels, you still examine main effects and interactions via marginal means; line graphs help detect interactions (nonparallel lines).
    • Three-way and higher order interactions
    • Three-way interactions require examining whether two-way interactions differ by the third factor and are best examined via line graphs across the two levels of the third variable.
    • Increasing the number of independent variables
    • Three-way (2×2×2) shows more complex patterns: three main effects, three two-way interactions, and one three-way interaction.
    • Three-way design interpretation depends on patterns across all three factors and their interactions.

  • How factorial designs are identified in your reading

    • In empirical journal articles:
    • Look for explicit notation like 2×2, 3×2, 2×2×2 in the Methods section.
    • Identify which IVs are between-subjects vs within-subjects (e.g., “Age: Younger vs Older adults” between-subjects; “Task: Single- vs. Dual-task” within-subjects).
    • Check Results for statements about main effects and interactions and whether they are statistically significant (e.g., p-values, F-statistics).
    • In popular media:
    • Look for phrases like “it depends” or “only when” indicating an interaction.
    • Be aware that popular reports may omit design details; you may need to infer a factorial design from context.
    • Participant variables as moderators
    • Journalists may discuss moderator concepts (e.g., self-esteem) that imply interactions without clearly naming a factorial design.

  • Practical implications and takeaways

    • Interactions reveal complex dependencies that are not captured by main effects alone.
    • Factorial designs are powerful for theory testing, especially when theories predict how effects vary across contexts or groups.
    • The layout of the design (independent-groups vs within-groups vs mixed) influences sample size and statistical power.
    • When interpreting results, always consider whether an interaction exists before drawing conclusions about main effects.

  • Quick reference: common notations and definitions

    • Notation for designs:
    • 2×2: two IVs with two levels each
    • 2×3: one IV with two levels, another with three levels
    • 3×4: two IVs with three and four levels respectively
    • 2×2×2: three IVs, each with two levels
    • Cells: product of levels (e.g., 2×2×2 = 8 cells)
    • Notation conventions: sometimes IVs are labeled in parentheses (e.g., Age: Younger vs Older) and Task: Single vs Dual.

  • Check Your Understanding (brief prompts)

    • How would you describe a factorial design as independent-groups vs within-groups vs mixed? Give an example for each.
    • What does a 3×4 design indicate about the number of independent variables and their levels?
    • In a 2×2 design, how do you compute the interaction from a table of cell means?
    • Why are interactions often more informative than main effects in a factorial study?
    • How can you detect a three-way interaction from a 2×2×2 design? What would you look for in a line graph?

  • Summary statement

    • Factorial designs extend causal inference by allowing simultaneous manipulation or measurement of two or more IVs, enabling the detection of main effects, interactions, and higher-order effects. They provide a rigorous framework for examining how effects generalize across people and situations and for testing theories that predict how variables combine to influence outcomes.