Definition: A one-way repeated measures ANOVA is a statistical method used when a single group is tested under different conditions or at multiple time points. It is the within-subjects equivalent of a one-way independent ANOVA.
Structure:
Independent Variable (IV): Categorical (e.g., type of condition). All participants experience every level of the IV.
Dependent Variable (DV): Continuous (e.g., reaction time, score).
F-ratio: This ratio assesses how well the model fits the data compared to a null model. In RM ANOVA, the error term (denominator) is reduced because the variance due to individual differences (between-person variance) is partitioned out, making the test generally more powerful than independent designs.
F = \frac{MS{model}}{MS{error}}
Assumptions of One-way RM ANOVA
Normal Distribution: The dependent variable should be normally distributed across all levels of the IV. This is often checked using Shapiro-Wilk tests or Q-Q plots.
Continuous Scale: The dependent variable should be measured on an interval or ratio scale.
Sphericity: This is the within-subjects equivalent of the homogeneity of variance assumption. It requires that the variances of the differences between all pairs of related groups are equal.
Mauchly's Test: Used if there are three or more levels. If significant (p < .05), the assumption is violated.
Violations: If sphericity is violated, corrections must be applied to the degrees of freedom to prevent Type I errors. Common corrections include:
Greenhouse-Geisser (GG): Conservative, used when epsilon (\epsilon) is < .75.
Huynh-Feldt (HF): Less conservative, used when epsilon (\epsilon) is > .75.
Follow-up Tests
Post Hoc Comparisons: If the main effect is significant, pairwise comparisons are used to identify which specific levels differ.
Adjustments: Since multiple comparisons increase the risk of Type I errors, adjustments like Bonferroni (dividing the alpha level by the number of tests) are commonly applied.
Example of One-way RM ANOVA
Experimental Design: Examining the Stroop Effect.
Hypothesis: Participant reaction times (RT) will be faster in the 'Congruent' condition (e.g., the word 'RED' printed in red ink) than in the 'Incongruent' condition (e.g., 'RED' printed in blue ink).
Data Representation
Data Layout: In SPSS/JASP, each level of the repeated measures factor must be a separate column (wide format).
Analysis Path: Select "Repeated Measures" option in the General Linear Model menu.
Naming Factors and Contrasts
Factor Naming: It is essential to name the within-subject factor (e.g., "Congruency") and define the levels (e.g., Congruent, Incongruent, Neutral).
Contrasts Menu: Used for planned comparisons to test specific hypotheses:
Simple: Compares each level to a reference level (either the first or the last).
Repeated: Compares each level (except the first) to the previous level.
Helmert: Compares each level to the mean of all subsequent levels.
Profile Plots and Effect Size
Profile Plots: Line graphs that show the mean scores across different conditions. They are helpful for visualizing the direction and magnitude of effects.
Effect Size (Partial Eta Squared - \eta^2_p): Indicates the proportion of variance accounted for by the IV, excluding other factors.
Benchmarks:
Small: \approx .01
Medium: \approx .06
Large: \approx .14
Reporting: Include the F-statistic, degrees of freedom (effect and error), p-value, and effect size. Example: F(2, 48) = 15.40, p < .001, \eta^2_p = .39.
Factorial ANOVAs
Definition and Structure
Definition: Factorial ANOVAs involve two or more independent variables (IVs). These allow researchers to look at the effect of each IV individually (Main Effects) and how they work together (Interaction Effects).
Design Types:
Between-Subjects: All IVs are independent groups.
Within-Subjects: All IVs are repeated measures.
Mixed Design: Includes at least one between-subjects IV and one within-subjects IV.
Introduction to 2x2 ANOVA Design
Structure: The simplest factorial design. Two IVs, each with two levels, creating 4 distinct conditions.
Example: IV1 (Time: Pre-test vs Post-test) and IV2 (Group: Control vs Intervention).
Interactions in 2x2 ANOVA
Definition: An interaction occurs when the effect of one IV on the DV changes depending on the level of the second IV.
Visualizing: Interaction is present if the lines on a profile plot are not parallel (e.g., they cross, converge, or diverge).
Follow-up: If an interaction is significant, researchers typically perform Simple Effects Analysis to examine the effect of one IV at each individual level of the other IV.
Procedure for Mixed ANOVA Examples
Example: The influence of Altruism (Between-subjects factor: Charitable vs. Uncharitable) on performance in a Monetary Stroop task (Within-subjects factor: Congruent vs. Incongruent).
Hypothesis: There is a moderation effect; charitable participants may show higher control (smaller Stroop effect) in a context involving social/monetary value.
Marginal Means: The average of all levels of one factor, collapsed across the levels of another factor, used to identify Main Effects.
Interpretation and Reporting of ANOVA Results
Main Effect of IV1: Is there a difference regardless of IV2?
Main Effect of IV2: Is there a difference regardless of IV1?
Interaction Effect (IV1 x IV2): Does the 'Stroop effect' (Incongruent - Congruent) differ significantly between ‘Charitable’ and ‘Uncharitable’ groups?
Reporting: Always report the interaction first. If the interaction is significant, the main effects must be interpreted with caution, as the interaction explains the relationship more specifically.
Reference Material and Further Learning
Field, A. (2017):Discovering Statistics Using SPSS (5th Edition).