Chapter 14: Repeated-Measures Analysis of Variance

1. Chapter 14: Repeated-Measures Analysis of Variance

2. The Logical Background for a Repeated-Measures ANOVA • Chapter 14 extends analysis of variance to research situations using repeated‑measures (or related‑samples) research designs. • Much of the logic and many of the formulas for repeated‑measures ANOVA are identical to the independent‑measures analysis introduced in Chapter 13. • However, the repeated‑measures ANOVA includes a second stage of analysis in which variability due to individual differences is subtracted out of the error term.

3. The Logical Background for a Repeated-Measures ANOVA (cont.) • The repeated‑measures design eliminates individual differences from the between‑treatments variability because the same subjects are used in every treatment condition. • To balance the F‑ratio the calculations require that individual differences also be eliminated from the denominator of the F‑ratio. • The result is a test statistic similar to the independent‑measures F‑ratio but with all individual differences removed.

4. IM ANOVA: Comparison of components of F formulas for Independent and Repeated Measures ANOVA Variance Between Treatments F = Variance Due to Chance Variance (differences) Between Treatments = Variance (differences) Expected Sampling Error treatment effect + individual differences + error = individual differences + error RM ANOVA: Treatment Effect + Experimental Error F = Experimental Error

5. Total variability 2 stages of variability Stage 1 Within treatments variability Individual Differences Experimental Error Between treatments variability Treatment Effect Experimental Error Numerator of F-Ratio Stage 2 Between subjects variability Individual Differences Error variability Experimental Error Denominator of F-ratio

6. Data table (3 Test Sessions)

7. SSbetween treatments SSbetween subjects SS broken down into two stages Stage 1 SSwithin treatments = SS inside each treatment Stage 2 SSerror = SSwithin treatments - SSbetween subjects

8. df broken down into two stages dftotal = N - 1 Stage 1 dfbetween treatments = k - 1 dfwithin treatments = N - k Stage 2 dfbetween subjects = n - 1 dferror = (N - k) - (n - 1)

9. Data table (time after treatment)

10. Distribution of F scores ( F = 3.86) 3.86

11. Data table (time after treatment) with mean scores

12. Tukey’s Honestly Significant Difference Test (or HSD) for Repeated Measures ANOVA Denominator of F-ratio From Table (Number of treatments, dferror) Number of Scores in each Treatment

13. Advantages of Repeated Measures Design • Economical - fewer SS required • More sensitive to treatment effect - individual differences having been removed

14. Independent vs. Repeated Measures ANOVA Independent: treatment effect + individual differences + experimental error F = individual differences + experimental error vs. Repeated Measures: treatment effect + experimental error F = experimental error Imagine: Treatment Effect = 10 units of variance Individual Differences = 1000 units of variance Experimental Error = 1 unit of variance

15. Disadvantages of Repeated Measures Designs: • Carry over effects (e.g. drug 1 vs. drug 2) • Progressive error (e.g. fatigue, general learning strategies, etc.) *Counterbalancing

16. Assumptions of the Repeated Measures ANOVA • Observations within each treatment condition must be independent • Population distribution within each treatment must be normal • Variances of the population distributions for each treatment must be equivalent (homogeneity of variance) • Homogeneity of covariance.