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

2. Chapter 14 Learning Outcomes
Explain logic of repeated-measures ANOVA study 2
Use repeated-measures ANOVA to evaluate mean differences 3
Explain logic of two-factor study and matrix of group means 4
Describe main effects and interactions from pattern of group means 5
Compute two-factor ANVOA to evaluate mean differences

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

4. 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)

5. 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.

6. 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

7. 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

8. 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

9. 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

10. Figure 14.1 Structure of the Repeated-Measures ANOVA

11. Stage One Repeated-Measures ANOVA equations

12. 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

13. Stage Two Repeated-Measures ANOVA equations

14. 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

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

16. 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

17. 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

18. 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.

19. 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? A 15 B 14 C 13 D 7

20. 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? A 15 B 14 C 13 D 7

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

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

23. 14.2 Two-Factor ANOVA Factorial designs
Consider more than one factor
Joint impact of factors is considered.
Three hypotheses tested by three F-ratios
Each tested with same basic F-ratio structure

24. Main effects Mean differences among levels of one factor
Differences are tested for statistical significance
Each factor is evaluated independently of the other factor(s) in the study

25. Interactions between factors
The mean differences between individuals treatment conditions, or cells, are different from what would be predicted from the overall main effects of the factors
H0: There is no interaction between Factors A and B
H1: There is an interaction between Factors A and B

26. Interpreting Interactions
Dependence of factors
The effect of one factor depends on the level or value of the other
Non-parallel lines (cross or converge) in a graph
Indicate interaction is occurring
Typically called the A x B interaction

27. Figure 14.2 Graph of group means with and without interaction

28. Structure of the Two-Factor Analysis
Three distinct tests
Main effect of Factor A
Main effect of Factor B
Interaction of A and B
A separate F test is conducted for each

29. Two Stages of the Two-Factor Analysis of Variance
First stage
Identical to independent samples ANOVA
Compute SSTotal, SSBetween treatments and SSWithin treatments
Second stage
Partition the SSBetween treatments into three separate components, differences attributable to Factor A, to Factor B, and to the AxB interaction

30. Figure 14.3 Structure of the Two- Factor Analysis of Variance

31. Stage One of the Two-Factor Analysis of Variance

32. Stage 2 of the Two Factor Analysis of Variance
This stage determines the numerators for the three F-ratios by partitioning SSbetween treatments

33. Degrees of freedom for Two-Factor ANOVA
dftotal = N – 1
dfwithin treatments = Σdfinside each treatment
dfbetween treatments = k – 1
dfA = number of rows – 1
dfB = number of columns– 1
dferror = dfwithin treatments – dfbetween subjects

34. Mean squares and F-ratio for the Two-Factor ANOVA

35. Effect Size for Two-Factor ANOVA
η2, is computed as the percentage of variability not explained by other factors.

36. Figure 14.4 Sample means for Example 14.3

37. Assumptions for the Two-Factor ANOVA
The validity of the ANOVA presented in this chapter depends on three assumptions common to other hypothesis tests
The observations within each sample must be independent of each other
The populations from which the samples are selected must be normally distributed
The populations from which the samples are selected must have equal variances (homogeneity of variance)

38. Learning Check
If a two-factor analysis of variance produces a statistically significant interaction, then you can conclude that _____. A
either the main effect for factor A or the main effect for factor B is also significant B
neither the main effect for factor A nor the main effect for factor B is significant C
both the man effect for factor A and the main effect for factor B are significant D
the significance of the main effects is not related to the significance of the interaction

39. Learning Check - Answer
If a two-factor analysis of variance produces a statistically significant interaction, then you can conclude that _____. A
either the main effect for factor A or the main effect for factor B is also significant B
neither the main effect for factor A nor the main effect for factor B is significant C
both the man effect for factor A and the main effect for factor B are significant D
the significance of the main effects is not related to the significance of the interaction

40. Learning Check
Decide if each of the following statements is True or False.
T/F
Two separate single-factor ANOVAs provide exactly the same information that is obtained from a two-factor analysis of variance.
A disadvantage of combining 2 factors in an experiment is that you cannot determine how either factor would affect the subjects' scores if it were examined in an experiment by itself.

41. Answer
False
The two-factor ANOVA allows you to determine the effect of one variable controlling for the effect of the other.
Either main effect can be examined in a Oneway ANOVA, but the Two-Factor ANOVA provides more information, not less.

42. Any Questions?
Concepts?
Equations?