Repeated-measures ANOVA: Change Over Time

In the realm of research, understanding how variables evolve over time or under different conditions holds immense value. Enter repeated-measures ANOVA (RM ANOVA), a powerful statistical tool designed to analyze data collected from the same subjects across multiple time points or conditions. This guide delves into the intricacies of RM ANOVA, equipping you with the knowledge to unlock valuable insights from your longitudinal data.

Unveiling the Scenario: When to Use RM ANOVA

Imagine studying the effect of a new exercise program on participants’ heart rate. Here, heart rate is your continuous dependent variable, and you measure it before, during, and after the exercise program for each participant. If you aim to analyze how heart rate changes across these time points within each participant, RM ANOVA is your champion.

Its reach extends beyond fitness studies:

• Evaluating the effectiveness of different teaching methods on student learning over multiple semesters.
• Analyzing anxiety levels of patients undergoing therapy at different treatment stages.
• Assessing the impact of environmental factors on plant growth at various time points.

Remember, RM ANOVA thrives on three key pre-requisites:

1. Same subjects: Measurements are collected from the same participants across all conditions or time points.
2. Continuous dependent variable: The outcome variable (e.g., heart rate, student score) must be numerical and ideally follow a normal distribution.
3. At least three conditions or time points: You need to measure the dependent variable at least three distinct points (e.g., before, during, after).

Building the Blocks: The Anatomy of RM ANOVA

At its core, RM ANOVA compares within-subject changes across conditions or time points. Unlike one-way ANOVA, it capitalizes on the relatedness of data from the same subjects, leading to increased statistical power and sensitivity to change.

Here’s how it works:

1. Null Hypothesis (H0): There are no significant differences between means across conditions or time points within each subject. 2. Alternative Hypothesis (Ha): At least two conditions or time points have different means for at least one subject.

3. Partitioning Variance:

• Total Variance (SST): The overall variability in the data.
• Between-Subjects Variance (SSB): Variation due to individual differences between subjects.
• Within-Subjects Variance (SSW): Variation due to changes across conditions or time points within each subject.
• Interaction Variance (SSW_i): Variation due to the interaction between subjects and conditions/time points (optional).

4. F-statistic: We calculate F-statistics for the main effect of conditions/time points (within-subjects effect) and the interaction (if used).

5. Hypothesis Testing:

• Similar to one-way ANOVA, we use F-statistics and p-values to assess the significance of main and interaction effects.

Formulas:

• Total Variance (SST): Similar to one-way ANOVA (Σ(Yi – Y̅)^2)
• Between-Subjects Variance: SSB = Σn_i * (Y̅_i.. – Y̅)^2 (sum of squared deviations of group means from the grand mean, weighted by group size)
• Within-Subjects Variance: SSW = ΣΣ(Yij – Y̅_ij)^2 (sum of squared deviations of individual observations from individual cell means)
• Interaction Variance: SSW_i = ΣΣ(Yij – Y̅i. – Y̅.j. + Y̅)^2 (sum of squared deviations of individual observations from adjusted means, accounting for main effects, weighted by cell size)
• F-statistics: Similar to one-way ANOVA (ratio of between-group or interaction variance to within-group variance, accounting for degrees of freedom)

Degrees of Freedom (df):

• df_between = k – 1 (number of conditions or time points minus 1)
• df_within = (k – 1) * (N – 1) (product of number of conditions and number of subjects minus 1)
• df_interaction = (k – 1) * (N – 1) (same as within-subjects if no interaction term)

Important Notes

• Like one-way ANOVA, RM ANOVA assumes normality and sphericity (equal variances across conditions/time points). Violations require transformations or alternative tests.
• Post-hoc tests: Similar to one-way ANOVA, use appropriate post-hoc tests (e.g., Bonferroni, Tukey’s HSD) to identify specific differences between conditions/time points.
• Greenhouse-Geisser correction: Used when sphericity is violated, adjusting degrees of freedom to maintain accurate p-values