Factorial Analysis of Variance: In-Depth Notes for Psychology Statistics Exam

Basic Logic of Factorial Designs and Interaction Effects

  • Importance of Factorial Designs in Psychology
    • Most behaviors are influenced by multiple variables.
    • Factorial research design:
    • Examines the effect of two or more variables simultaneously.
    • More efficient than single-variable designs.
    • Effects to Consider:
    • Additive effects: One variable has an impact when considered alone or with another variable.
    • Interaction effects:
      • Occur when the combination of variables produces a unique effect not seen when examining variables independently.
      • Variables that change the relationships among others are called moderators.

Relationship Between ANOVA Types

  • One-way ANOVA:
    • Evaluates the impact of one grouping variable.
  • Two-way ANOVA:
    • Examines two grouping variables.
    • Each grouping variable is considered a "way" in the analysis.

Main Effects

  • Definition:
    • A main effect exists when the average result for a grouping variable is significant, regardless of other variables.
  • Key terms:
    • Cell: Each unique combination of conditions in a factorial design.
    • Example: In a $2 imes 2$ factorial design, there are four conditions (cells).
    • Cell Mean: Average score of a particular condition.
    • Marginal Means: Averages calculated for one grouping variable across all levels of another.

Recognizing and Interpreting Interaction Effects

  • Characteristics of Interaction Effects:
    • The effect of one variable is influenced by the level of another.
    • To identify interaction effects, observe the patterns of cell means.
    • An interaction effect is indicated when differences in means vary across different levels of the other variable.

Two-Way ANOVA: Understanding F Ratios

  • Main Effects:
    • Column Main Effect:
    • Numerator: Between-groups variance (variation between column marginal means).
    • Denominator: Within-groups variance (average variance from each cell).
    • Row Main Effect:
    • Similar structure as column main effect but based on row marginal means.
    • Interaction Effect:
    • Uses means from diagonals in the design for its estimation.

Relation of Interaction and Main Effects

  • Complex Relationships:
    • A study can show main effects without interactions and vice versa, making interpretation more complex when interactions exist.

Assumptions of Two-Way ANOVA

  • The populations should:
    • Follow a normal distribution.
    • Exhibit equal variances.
    • These assumptions hold true for the populations in each cell of the analysis.

Controversies and Limitations

  • Dichotomizing Numeric Variables:
    • Use of median splits can lead to loss of information and is often controversial.