Repeated Measures ANOVA Notes

Repeated Measures ANOVA

Introduction to Repeated Measures ANOVA

• Repeated measures ANOVA builds upon the independent samples ANOVA, similar to how repeated measures t-tests relate to independent samples t-tests.
• The single factor repeated measures ANOVA is used when:
  • There is one variable (single factor).
  • The same subjects are used across all samples (repeated measures).
  • Three or more samples are being compared (ANOVA).

Examples

• Comparing vocabulary size in the same group of infants at 12, 14, 16, and 18 months.
• Measuring stress levels in the same people on Monday, Wednesday, and Friday.

Hypotheses

• Null Hypothesis: All population means are equal ($\mu1 = \mu2 = \mu_3 = …$).
• Alternative Hypothesis: There is at least one difference among the population means.

Advantages of Repeated Measures Design

• Requires fewer participants.
• Allows measurement of how a variable changes over time.
• Reduces variability due to individual differences (e.g., IQ, personality).

Variability in Independent Measures ANOVA

• Variability in the dependent variable is divided into:
  • Variability between treatment conditions:
    • Independent variable (treatment effect).
    • Random error.
    • Individual differences.
  • Variability within treatment conditions:
    • Error in measurement.
    • Individual differences.

Variability in Repeated Measures ANOVA

• Variability in the dependent variable is divided into:
  • Variability between treatment conditions:
    • Independent variable (treatment effect).
    • Error.
    • Note: Individual differences are removed.
  • Variability within treatment conditions:
    • Error.
    • Note: Individual differences are removed.

F Ratio for Repeated Measures ANOVA

• The F ratio is calculated as:
  • F = \frac{\text{Treatment effect + Error}}{\text{Error}}
• Numerator: Treatment effect plus error (no individual differences).
• Denominator: Error.

Individual Differences

• Variability between ages is not due to individual differences because it’s the same group of people at each time point.
• Variability within a given time point (e.g., 12 months) is due to individual differences.
• These individual differences are measured and statistically subtracted out during the analysis.