9.3 Analysis of repeated measures data

Repeated measures designs allow researchers to analyze data from the same participants across multiple conditions or time points. This approach reduces variability and increases statistical power compared to between-subjects designs, making it a valuable tool in experimental research.

Repeated measures ANOVA and MANOVA are key statistical techniques for analyzing within-subjects data. These methods help researchers uncover significant differences across conditions while accounting for the correlated nature of repeated measurements and multiple dependent variables.

Repeated Measures ANOVA and MANOVA

Repeated Measures ANOVA for Within-Subjects Designs

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• Analyzes differences between related means from the same participants measured at different times or under different conditions
• Reduces unsystematic variability by allowing each participant to serve as their own control
• Requires fewer participants than a between-subjects design to achieve the same statistical power
• Effect size quantifies the magnitude of the difference between groups or the relationship between variables ($\eta^2$, Cohen's $d$)
  • Partial eta squared ($\eta_p^2$) commonly used effect size measure in repeated measures ANOVA
    • Proportion of variance accounted for by the independent variable, while controlling for other factors

Multivariate Analysis of Variance (MANOVA)

• Extension of ANOVA used when there are multiple dependent variables
• Tests for significant differences between groups across multiple dependent variables simultaneously
• Accounts for correlations among the dependent variables
• Provides a more holistic understanding of the effects of the independent variables on the combined set of dependent variables
• Helps control the familywise error rate by reducing the number of individual tests conducted

Assumptions and Corrections

Mauchly's Test of Sphericity

• Assesses the assumption of sphericity in repeated measures ANOVA
  • Sphericity assumes equal variances of the differences between all possible pairs of within-subject conditions
• Significant result indicates a violation of the sphericity assumption
• Corrections (Greenhouse-Geisser or Huynh-Feldt) can be applied to adjust the degrees of freedom and p-values when sphericity is violated

Corrections for Sphericity Violations

• Greenhouse-Geisser correction
  • Conservative adjustment to the degrees of freedom when sphericity is violated
  • Multiplies the numerator and denominator degrees of freedom by the Greenhouse-Geisser epsilon ($\epsilon$)
• Huynh-Feldt correction
  • Less conservative adjustment to the degrees of freedom when sphericity is violated
  • Multiplies the numerator and denominator degrees of freedom by the Huynh-Feldt epsilon ($\tilde{\epsilon}$)
• Both corrections help maintain the validity of the F-test when the sphericity assumption is not met

Post-hoc Analysis

Conducting Post-hoc Tests

• Used to determine which specific groups or conditions differ significantly from each other after a significant omnibus test (ANOVA or MANOVA)
• Control the familywise error rate to maintain the overall Type I error rate at the desired level (usually $\alpha = .05$)
• Common post-hoc tests for repeated measures designs
  • Bonferroni correction adjusts the significance level by dividing $\alpha$ by the number of comparisons
  • Holm-Bonferroni correction sequentially adjusts the significance level based on the number of remaining comparisons
  • Sidak correction adjusts the significance level using the formula $1 - (1 - \alpha)^{1/k}$, where $k$ is the number of comparisons

Interpreting Pairwise Comparisons

• Pairwise comparisons identify specific differences between pairs of groups or conditions
• Significance values (p-values) indicate whether the difference between the pair is statistically significant
• Mean differences and confidence intervals provide information about the magnitude and direction of the differences
• Plotting the means and confidence intervals can help visualize the pairwise differences (line graphs, bar charts with error bars)