Repeated Measures ANOVA Notes
Repeated Measures ANOVA
Introduction
- Repeated measures ANOVA builds upon the independent samples ANOVA, paralleling the relationship between independent and repeated measures t-tests.
- 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).
- If only two samples are compared, a t-test is more appropriate.
Examples
- Example 1: Measuring vocabulary size in the same group of infants at 12, 14, 16, and 18 months.
- Example 2: Measuring stress levels in the same people on Monday, Wednesday, and Friday.
Hypotheses
- Null Hypothesis: The population means of all samples are equal (e.g., \mu1 = \mu2 = \mu_3 ).
- Alternative Hypothesis: There is at least one difference among the population means. This hypothesis is vague and requires further investigation to determine the specifics.
Advantages of Repeated Measures Design
- Requires fewer participants since the same individuals are measured multiple times.
- Allows measurement of how a variable changes over time.
- Reduces variability due to individual differences (e.g., IQ, personality).
- By using the same participants in all conditions, factors like inherent vocabulary skills or access to educational resources are controlled.
- Reduces error variance, making it easier to detect significant differences.
Variability in ANOVA
Independent Measures ANOVA
- Variability in the dependent variable is divided into:
- Variability between treatment conditions: Caused by the independent variable, random error, and individual differences.
- Variability within treatment conditions: Caused by error and individual differences.
Repeated Measures ANOVA
- Variability in the dependent variable is divided similarly, but individual differences are removed.
- Variability between treatment conditions: Caused by the independent variable and error (no individual differences).
- Variability within treatment conditions: Caused by error (no individual differences).
F Ratio
- The F ratio for repeated measures ANOVA is:
F = \frac{\text{Treatment effect + Error}}{\text{Error}}
- Numerator: Treatment effect plus error (no individual differences).
- Denominator: Error (variability within conditions).
- Individual differences are measured and statistically subtracted out during the analysis process to reduce variability within conditions.
Elaboration on Variability within Time Points
- When examining infants at a specific age (e.g., 12 months), variability in vocabulary size exists due to individual differences.
- Factors contributing to this variability include access to educational toys and the amount of interaction with adults.
- Repeated measures ANOVA acknowledges and addresses this variability within each time point, isolating the effects of time (the independent variable) on vocabulary growth.
