1. Chapter 14 Repeated Measures
PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Seventh Edition by Frederick J. Gravetter and Larry B. Wallnau

2. Concepts to review
Independent-measures analysis of variance (Chapter 13)
Repeated measures designs (Chapter 11)
Individual differences

3. 14.1 Overview Analysis of Variance Complex Analysis of Variance
Evaluated mean differences of two or groups
Complex Analysis of Variance
Samples are related not independent (Repeated measures ANOVA)
More than one factor is tested (Factorial ANOVA, here Two-Factor)

4. 14.2 Repeated Measures ANOVA
Advantages of repeated measures designs
Individual differences among participants do not influence outcomes
Smaller number of subjects needed to test all the treatments
Repeated Measures ANOVA
Compares two or more treatment conditions with the same subjects tested in all conditions
Studies same group of subjects at two or more different times.

5. Hypotheses for repeated measures ANOVA
Null hypothesis: in the population, there are no mean differences among the treatment groups
Alternate hypothesis states that there are mean differences among the treatment groups.
H1: At least one treatment mean μ is different from another

6. Individual Differences in the Repeated Measures ANOVA
F ratio based on variances
Same structure as independent measures
Variance due to individual differences is not present

7. Individual differences
Participant characteristics that vary from one person to another.
Not systematically present in any treatment group or by research design
Characteristics may influence measurements on the outcome variable
Eliminated from the numerator by the research design
Must be removed from the denominator statistically

8. Logic of repeated measures ANOVA
Numerator of the F ratio includes
Systematic differences caused by treatments
Unsystematic differences caused by random factors (reduced because same individuals in all treatments)
Denominator estimates variance reasonable to expect from unsystematic factors
Effect of individual differences is removed
Residual (error) variance remains

9. Figure 14.1 Structure of the Repeated-Measures ANOVA

10. Stage One Repeated-Measures ANOVA equations

11. Two Stages of the Repeated-Measures ANOVA
First stage
Identical to independent samples ANOVA
Compute SSTotal, SSBetween treatments and SSWithin treatments
Second stage
Removing the individual differences from the denominator
Compute SSBetween subjects and subtract it from SSWithin treatments to find SSError

12. Stage Two Repeated-Measures ANOVA equations

13. Degrees of freedom for Repeated Measures ANOVA
dftotal = N – 1
dfwithin treatments = Σdfinside each treatment
dfbetween treatments = k – 1
dfbetween subjects = n – 1
dferror = dfwithin treatments – dfbetween subjects

14. Mean squares and F-ratio for Repeated-Measures ANOVA

15. Effect size for the Repeated-Measures ANOVA
Percentage of variance explained by the treatment differences
Partial η2 is percentage of variability that has not already been explained by other factors

16. Post hoc tests with Repeated Measures ANOVA
Determine exactly where significant differences exist among more than two treatment means
Tukey’s HSD and Scheffé can be used
Substitute SSerror and dferror in the formulas

17. Assumptions of the Repeated Measures ANOVA
The observations within each treatment condition must be independent.
The population distribution within each treatment must be normal.
The variances of the population distribution for each treatment should be equivalent.

18. Learning Check
A researcher obtains an F-ratio with df = 2, 12 from an ANOVA for a repeated-measures research study. How many subjects participated in the study? 15 A 14 B 13 C 7 D

19. Learning Check - Answer
A researcher obtains an F-ratio with df = 2, 12 from an ANOVA for a repeated-measures research study. How many subjects participated in the study? 15 A 14 B 13 C 7 D

20. Learning Check
Decide if each of the following statements is True or False.
For the repeated-measures ANOVA, degrees of freedom for SSerror is (N–k) – (n–1).
T/F
The first stage of the repeated-measures ANOVA is the same as the independent-measures ANOVA.

21. Answer
N is the number of scores and n is the number of participants
False
After the first stage, the second stage adjusts for individual differences
True

22. Any Questions?
Concepts?
Equations?