Repeated measures ANOVA is a powerful statistical tool used in biostatistics to analyze data from experiments where subjects are measured multiple times. It extends one-way ANOVA to account for within-subject variability, making it ideal for longitudinal studies and treatment effect assessments over time.

This method offers increased statistical power by reducing individual differences, as each participant serves as their own control. It allows for smaller sample sizes while maintaining robust analysis, making it particularly valuable in clinical trials with limited patient populations.

Overview of repeated measures ANOVA

• Analyzes data from experimental designs where subjects are measured multiple times
• Extends one-way ANOVA to account for within-subject variability in longitudinal studies
• Crucial in biostatistics for assessing treatment effects over time or under different conditions

Within-subjects vs between-subjects designs

• Within-subjects designs measure each participant under all conditions, reducing individual differences
• Between-subjects designs assign different groups to each condition, controlling for order effects
• Repeated measures ANOVA primarily used for within-subjects designs, offering increased statistical power
• Allows for smaller sample sizes while maintaining robust statistical analysis

Assumptions of repeated measures ANOVA

Sphericity assumption

• Requires equality of variances of the differences between all possible pairs of groups
• Assessed using Mauchly's test of sphericity
• Violation leads to increased Type I error rate
• Corrections (Greenhouse-Geisser, Huynh-Feldt) applied when sphericity is violated

Normality assumption

• Assumes normally distributed data within each level of the within-subjects factor
• Checked using visual inspection (Q-Q plots) or statistical tests (Shapiro-Wilk)
• Robust to minor violations with sufficiently large sample sizes

Homogeneity of variance

• Assumes equal variances across all levels of the within-subjects factor
• Tested using Levene's test or Bartlett's test
• Less critical in repeated measures designs due to within-subjects nature

Conducting repeated measures ANOVA

Data organization

• Requires long format data structure for most statistical software
• Each row represents a single observation for a participant at a specific time point
• Includes columns for participant ID, time point or condition, and dependent variable

Calculation of F-statistic

• Compares the variance between conditions to the variance within conditions
• F-statistic calculated as: $F = \frac{MS_{between}}{MS_{within}}$
• MS represents Mean Square, obtained by dividing Sum of Squares by degrees of freedom

Degrees of freedom

• Between-subjects df = k - 1 (k = number of conditions)
• Within-subjects df = N - k (N = total number of observations)
• Error df = (n - 1)(k - 1) (n = number of participants)

Post-hoc tests for repeated measures

Pairwise comparisons

• Conducted to determine which specific conditions differ significantly
• Common methods include t-tests or Tukey's Honestly Significant Difference (HSD)
• Helps identify patterns or trends in the data across different time points or conditions

Bonferroni correction

• Adjusts p-values to control for Type I error rate in multiple comparisons
• Calculated by dividing the desired alpha level by the number of comparisons
• Conservative approach, may increase Type II error rate
• Alternative methods include Holm's sequential Bonferroni or False Discovery Rate (FDR)

Effect size in repeated measures ANOVA

Partial eta squared

• Measures the proportion of variance explained by the factor, excluding other factors
• Calculated as: $\eta_p^2 = \frac{SS_{effect}}{SS_{effect} + SS_{error}}$
• Values typically interpreted as small (0.01), medium (0.06), or large (0.14)

Cohen's f

• Alternative effect size measure, particularly useful for power analysis
• Calculated as: $f = \sqrt{\frac{\eta^2}{1 - \eta^2}}$
• Interpreted as small (0.1), medium (0.25), or large (0.4)

Advantages of repeated measures design

Increased statistical power

• Requires fewer participants to detect significant effects compared to between-subjects designs
• Controls for individual differences by using each participant as their own control
• Particularly beneficial in clinical trials with limited patient populations

Reduced subject variability

• Eliminates between-subject variability from the error term
• Increases sensitivity to detect treatment effects
• Allows for more precise estimation of within-subject changes over time

Limitations and considerations

Carryover effects

• Occur when the effect of one condition influences subsequent conditions
• Mitigated through counterbalancing or randomization of condition order
• May require washout periods in certain experimental designs (pharmacological studies)

Practice effects

• Improvement in performance due to repeated exposure to tasks or measures
• Can confound treatment effects, especially in cognitive or skill-based assessments
• Addressed through careful experimental design and statistical control methods

Repeated measures ANOVA vs other methods

One-way ANOVA vs repeated measures

• One-way ANOVA used for between-subjects designs, repeated measures for within-subjects
• Repeated measures offers higher statistical power and requires fewer participants
• One-way ANOVA assumes independence of observations, not suitable for longitudinal data

Paired t-test vs repeated measures

• Paired t-test limited to two time points or conditions
• Repeated measures ANOVA extends to three or more time points or conditions
• Repeated measures more efficient for multiple comparisons, reducing Type I error rate

Reporting results

Tables and figures

• Present descriptive statistics (means, standard deviations) for each condition in a table
• Use line graphs or box plots to visualize changes across time points or conditions
• Include error bars (standard error or confidence intervals) to represent variability

Interpretation of findings

• Report F-statistic, degrees of freedom, p-value, and effect size
• Describe the nature and direction of significant effects
• Discuss practical implications of findings in the context of the research question
• Address any violations of assumptions and their potential impact on results

Software implementation

SPSS for repeated measures ANOVA

• Accessed through the General Linear Model > Repeated Measures menu
• Requires specification of within-subjects factor(s) and dependent variable
• Offers options for post-hoc tests, effect sizes, and assumption checks
• Provides syntax for reproducibility and customization of analyses

R for repeated measures ANOVA

• Conducted using packages like ez, afex, or lme4
• ezANOVA() function in ez package specifically designed for repeated measures
• Allows for flexible modeling of complex designs and custom contrasts
• Provides tools for visualization (ggplot2) and advanced post-hoc analyses

Real-world applications

Clinical trials

• Assessing drug efficacy over multiple time points (baseline, 1 month, 3 months, 6 months)
• Evaluating changes in physiological measures (blood pressure, cholesterol) pre- and post-intervention
• Comparing different treatment regimens in crossover designs

Longitudinal studies

• Tracking developmental changes in cognitive abilities across childhood and adolescence
• Monitoring disease progression or recovery in patients over extended periods
• Investigating the long-term effects of public health interventions on population health metrics