three way anova

Introduction to Three-Way ANOVA

  • Definition: Analysis of Variance (ANOVA) used when there are 3 independent variables (IVs) that potentially interact with each other. It allows researchers to understand the combined influence of three categorical predictors on a single continuous dependent variable (DV).

  • Structure: Can be defined with various independent variables and their levels, forming a factorial design.

  • Example: A common design is 2 \times 2 \times 2, involving three factors where each factor has two levels. This results in 8 distinct experimental conditions or treatment cells.

  • Interactions and Effects:

    • Interactions: There are four possible interaction terms to evaluate:

    • One 3-way interaction: Exploring how the interaction of two IVs varies across levels of the third IV. A significant 3-way interaction indicates that the moderation effect observed in a 2-way interaction is itself moderated by a third variable.

    • Three 2-way interactions: Examining the effect of one IV at different levels of a second IV, while ignoring (collapsing across) the third variable (e.g., IV1 \times IV2, IV1 \times IV3, and IV2 \times IV3).

    • Main Effects:

    • Each IV has one main effect, making three total main effects (IV1, IV2, and IV_3). A main effect represents the individual impact of one predictor on the DV.

Example of Three-Way ANOVA Study

  • Research Scenario: Assessing how age, sex, and modality affect memory recall performance.

    • Age groups: Young adults vs. Older adults.

    • Sex groups: Male vs. Female.

    • Modality: Visual (images) vs. Verbal (words).

Three-Way Interactions

  • Study Breakdown:

    • Investigating if there are differences in memory performance across combinations of Age, Sex, and Modality.

    • The analysis attempts to determine if the way Sex affects Modality-based recall differs between the Young and the Old.

  • Key Questions:

    • Is there a significant main effect of Age?: Do younger participants consistently recall more than older ones regardless of other factors?

    • How does the effect of Sex vary across Age groups?: (Age x Sex interaction).

    • Is the effect of Modality different for each Age and Sex group?: (3-way interaction).

Detailed Interaction Analysis

Three Main Effects

  1. Age: Differences in memory baseline attributable strictly to age category.

  2. Sex: Differences in memory baseline attributable strictly to gender.

  3. Modality: Differences in memory baseline depending on whether the format was visual or verbal.

Two-Way Interactions

  1. Age x Sex: Investigates if the sex difference in memory is consistent across both age groups or if, for example, the gap widens in older age.

  2. Age x Modality: Do younger people show a visual advantage while older people show a verbal advantage?

  3. Sex x Modality: Is the memory effect of Modality different for Males vs. Females?

Three-Way Interaction

  • Core Concept: Identifies if the interaction between two IVs changes in nature or magnitude depending on the level of the third IV.

  • Visualization: If lines in an interaction plot are parallel in one group (e.g., Males) but cross in another group (e.g., Females), a 3-way interaction is likely present.

  • Statistical Follow-up: When significant, researchers "unpack" the interaction by running Simple Interaction Effects (e.g., examining the 2 \times 2 interaction separately for each age group).

Example Study Design: Sequence Recall

Hypothesis

A study on participants’ ability to recall sequences of actions involving different instructional methods:

  • Conditions (Instruction Type):

    • Verbal-only instructions.

    • Verbal + enactment (doing the action).

    • Verbal + demonstration (watching someone else do the action).

  • Age Groups: Children vs. Adults.

  • Sequence Complexity: Simple sequences vs. Complex sequences.

Experimental Design

  • Sample Size: 20 Participants assigned to each age group.

  • Statistical Method: Analyze using a mixed-model 2 \times 2 \times 3 ANOVA.

    • Between-subjects: Age group (Children vs. Adults).

    • Within-subjects: Complexity (Simple/Complex) and Instruction (Verbal/Enactment/Demonstration).

Statistical Procedures

Running the ANOVA in SPSS

  1. Define Factors: Use the 'Repeated Measures' tool for within-subject components.

  2. Input Variables: Map data columns to the corresponding factor levels in the model builder.

  3. Modeling:

    • Contrasts: Set up specific comparisons (e.g., comparing enactment to the verbal-only control group).

    • Plots: Request profile plots to visually inspect potential 3-way and 2-way interactions.

Analysis of Results

  • Assumption Testing:

    • Sphericity: Checked via Mauchly’s test. If significant (p < .05), apply Greenhouse-Geisser or Huynh-Feldt corrections.

    • Normality and Homogeneity: Ensure the data distribution and variances (Levene's test) meet ANOVA standards.

  • Interpreting Output: Generally, one should interpret the 3-Way Interaction fix first. If it is significant, the interpretation of lower-order effects (2-way and main effects) remains conditional on the 3-way interaction.

Limitations of ANOVA

  • Main Points:

    • Rigidity: ANOVA is designed for categorical predictors and assumes balanced, controlled experimental designs.

    • Assumption Violations: Real-world data often contain outliers or violate normality and homogeneity of variance, which can inflate Type I or Type II error rates.

    • Sensitivity: Traditional ANOVA is highly sensitive to extreme outliers which can distort the mean.

  • Alternatives:

    • Non-parametric tests: (e.g., Kruskal-Wallis) lack the ability to easily test multi-way interactions.

    • Robust Methods: Utilize techniques like trimmed means or bootstrapping (resampling thousands of times) to calculate more reliable statistics and p-values when traditional assumptions fail.

Understanding Bayesian ANOVA

  • Concept: Bayesian inference serves as an alternative to Null Hypothesis Significance Testing (NHST). Instead of focusing on p-values, it focuses on the relative likelihood of competing models.

  • Key Metric: Bayes Factor (BF): A ratio quantify the support for the alternative hypothesis (H1) relative to the null (H0).

    • BF{10}: If BF{10} = 200, the data are 200 times more likely under H1 than under H0. This is considered decisive evidence favoring the experimental effect.

    • Interpretation Guide:

    • 1 - 3: Anecdotal evidence.

    • 3 - 10: Moderate evidence.

    • 10 - 30: Strong evidence.

    • > 100: Decisive evidence.

Practical Applications of Bayesian Analysis

  • Example: Life Satisfaction:

    • Investigating whether life satisfaction scores vary significantly by age group.

    • Unlike NHST, Bayesian ANOVA can provide evidence in favor of the null hypothesis (e.g., confirming that age has no effect on life satisfaction if BF_{01} is large).

  • Output interpretation: Provides a clearer understanding of effect uncertainty and allows for the cumulative updating of scientific knowledge as new data is collected.

Review & Test Yourself Questions

  1. Describe the difference between a main effect and an interaction in a factorial design.

  2. In a 2 \times 2 \times 2 design, how many simple 2-way interactions must be checked if the 3-way interaction is significant?

  3. What is the Greenhouse-Geisser adjustment, and when should a researcher use it?

  4. Explain how a robust ANOVA handles data that is not normally distributed.

  5. If a researcher finds BF_{10} = 0.2, what does this tell them about their hypothesis?

  6. Outline the steps to "unpack" a non-significant 3-way interaction compared to a significant one.

  7. Compare the goals of NHST (based on frequencies) versus Bayesian inference (based on probability distributions).