1. Chapter 13 Repeated-Measures and Two-Factor Analysis of Variance
PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau

2. Chapter 13 Learning Outcomes
Understand logic of repeated-measures ANOVA study 2
Compute repeated-measures ANOVA to evaluate mean differences for single-factor repeated-measures study 3
Measure effect size, perform post hoc tests and evaluate assumptions required for single-factor repeated-measures ANOVA

3. Ch 13 Learning Outcomes (continued)
Understand logic of two-factor study and matrix of group means 4
Describe main effects and interactions from pattern of group means in two-factor ANOVA 5
Compute two-factor ANOVA to evaluate means for two-factor independent-measures study 6 7
Measure effect size, interpret results and articulate assumptions for two-factor ANOVA

4. Tools You Will Need
Independent-Measures Analysis of Variance (Chapter 12)
Repeated-Measures Designs (Chapter 11)
Individual Differences

5. 13.1 Overview Analysis of Variance Complex Analysis of Variance
Evaluated mean differences for two or more groups
Limited to one independent variable (IV)
Complex Analysis of Variance
Samples are related; not independent (Repeated-measures ANOVA)
Two independent variables are manipulated (Factorial ANOVA; only Two-Factor in this text)

6. 13.2 Repeated-Measures ANOVA
Independent-measures ANOVA uses multiple participant samples to test the treatments
Participant samples may not be identical
If groups are different, what was responsible?
Treatment differences?
Participant group differences?
Repeated-measures solves this problem by testing all treatments using one sample of participants

7. Repeated-Measures ANOVA
Repeated-Measures ANOVA used to evaluate mean differences in two general situations
In an experiment, compare two or more manipulated treatment conditions using the same participants in all conditions
In a nonexperimental study, compare a group of participants at two or more different times
Before therapy; After therapy; 6-month follow-up
Compare vocabulary at age 3, 4 and 5

8. Repeated-Measures ANOVA Hypotheses
Null hypothesis: in the population there are no mean differences among the treatment groups
Alternate hypothesis: there is one (or more) mean differences among the treatment groups
H1: At least one treatment mean μ differs from another

9. General structure of the ANOVA F-Ratio
F ratio based on variances
Numerator measures treatment mean differences
Denominator measures treatment mean differences when there is no treatment effect
Large F-ratio  greater treatment differences than would be expected with no treatment effects

10. Individual differences
Participant characteristics may vary considerably from one person to another
Participant characteristics can influence measurements (Dependent Variable)
Repeated measures design allows control of the effects of participant characteristics
Eliminated from the numerator by the research design
Must be removed from the denominator statistically

11. Structure of the F-Ratio for Repeated-Measures ANOVA
The biggest change between independent-measures ANOVA and repeated-measures ANOVA is the addition of a process to mathematically remove the individual differences variance component from the denominator of the F-ratio

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

13. Figure 13.1 Structure of the Repeated-Measures ANOVA
FIGURE The partitioning of variance for a repeated measures experiment.

14. Repeated-Measures ANOVA Stage One Equations

15. Two Stages of the Repeated-Measures ANOVA
First stage
Identical to independent samples ANOVA
Compute SStotal, SSbetween treatments and SSwithin treatments
Second stage
Done to remove the individual differences from the denominator
Compute SSbetween subjects and subtract it from SSwithin treatments to find SSerror (also called residual)

16. Repeated-Measures ANOVA Stage Two Equations

17. 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

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

19. F-Ratio General Structure for Repeated-Measures ANOVA

20. 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 or

21. In the Literature
Report a summary of descriptive statistics (at least means and standard deviations)
Report a concise statement of the ANOVA results
E.g., F (3, 18) = 16.72, p<.01, η2 = .859

22. Repeated Measures ANOVA post hoc tests (posttests)
Significant F indicates that H0 (“all populations means are equal”) is wrong in some way
Use post hoc test to 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

23. Repeated-Measures ANOVA Assumptions
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

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

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

26. 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 could be written as [(N–k) – (n–1)]
The first stage of the repeated-measures ANOVA is the same as the independent-measures ANOVA

27. Learning Check - Answer
True
dferror = dfw/i treatments – dfbetwn subjects
Within treatments df = N-k; between subjects df = n-1
After the first stage analysis, the second stage analysis adjusts for individual differences

28. Repeated-Measures ANOVA Advantages and Disadvantages
Advantages of repeated-measures designs
Individual differences among participants do not influence outcomes
Smaller number of participants needed to test all the treatments
Disadvantages of repeated-measures designs
Some (unknown) factor other than the treatment may cause participant’s scores to change
Practice or experience may affect scores independently of the actual treatment effect

29. 13.3 Two-Factor ANOVA
Both independent variables and quasi-independent variables may be employed as factors in Two-Factor ANOVA
An independent variable (factor) is manipulated in an experiment
A quasi-independent variable (factor) is not manipulated but defines the groups of scores in a nonexperimental study

30. 13.3 Two-Factor ANOVA Factorial designs
Consider more than one factor
We will study two-factor designs only
Also limited to situations with equal n’s in each group
Joint impact of factors is considered
Three hypotheses tested by three F-ratios
Each tested with same basic F-ratio structure

31. 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

32. 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
Interactions are one of the most difficult topics for students to grasp. A vivid and memorable example of an interaction may be used as an example to demonstrate an interaction by instructors with some basic chemistry knowledge. Hydrochloric acid (HCl) is a highly caustic and dangerous substance which will dissolve many materials including metals, woods, plastics and flesh. Sodium Hydroxide (NaOH) is a highly caustic substance which will dissolve hair, fats, and flesh (which makes it a common ingredient in drain cleaners).
The concentration of both substances determines how caustic it is: the higher concentration, the more caustic the substance. This represents a main effect: higher concentrations  more caustic effects. But what would happen these two highly caustic substances were combined? How caustic would they be together? Most students (particularly those without a chemistry background) will assume additivity. In other words, if substance A is caustic and substance B is caustic, then adding the two together will be doubly caustic. They expect to predict the joint effect from knowledge of the separate, individual (main) effects. Instructors with a minimum gag reflex might pose these questions: “For 1 million dollars, would you drink a shot glass of HCL?”; “For 1 million dollars, would you drink a shot glass of NaOH?”; and “For 1 million dollars, would you drink a double shot glass of HCL + NaOH?” Generally students respond with a resounding “No!” to all three questions—particularly after the instructor has provided some examples of how caustic these substance are or pictures of burns suffered by people who have accidently spilled either substance one themselves. However, provided a trained chemist provides the exact concentrations and amounts of each substance, the last offer could be a relatively safe path to 1 million dollars. Why? Because of the way acids and bases interact. The interaction of HCl + NaOH produces NaCl + H2O—salty water. A double shot glass of salty water might make you vomit, but it would not harm you like HCl or NaOH. Interactions are different from what would be predicted from simply “adding” together the individual (main) effects of each factor.

33. Interpreting Interactions
Dependence of factors
The effect of one factor depends on the level or value of the other
Sometimes called “non-additive” effects because the main effects do not “add” together predictably
Non-parallel lines (cross, converge or diverge) in a graph indicate interaction is occurring
Typically called the A x B interaction

34. Figure 13.2 Group Means Graphed without (a) and with (b) Interaction
FIGURE (a) Graph showing the treatment means from Table 13.5, for which there is no interaction. (b) Graph for Table 13.6, for which there is an interaction.

35. Structure of the Two-Factor Analysis of Variance
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
Results of one are independent of the others

36. 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

37. Figure 13.3 Structure of the Two-Factor Analysis of Variance
FIGURE Structure for the analysis for a two-factor ANOVA.

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

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

40. 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

41. Mean squares and F-ratios for the Two-Factor ANOVA

42. Two-Factor ANOVA Summary Table Example
Source SS df MS F
Between treatments
200 3
Factor A 40 1 4
Factor B 60 *6
A x B
100
**10
Within Treatments
300 20 10
Total
500 23
(N = 24; n = 6)

43. Two-Factor ANOVA Effect Size
η2, is computed to show the percentage of variability not explained by other factors

44. In the Literature
Report mean and standard deviations (usually in a table or graph due to the complexity of the design)
Report results of hypothesis test for all three terms (A & B main effects; A x B interaction)
For each term include F, df, p-value & η2
E.g., F (1, 20) = 6.33, p<.05, η2 = .478

45. Interpreting the Results
Focus on the overall pattern of results
Significant interactions require particular attention because even if you understand the main effects, interactions go beyond what main effects alone can explain.
Extensive practice is typically required to be able to clearly articulate results which include a significant interaction

46. Figure 13.4 Sample means for Example 13.4
FIGURE Sample means for the data in Example The data are quiz scores from a two-factor study examining the effect of studying text on paper versus on a computer screen for either a fixed time or a self-regulated time.

47. Two-Factor ANOVA Assumptions
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)

48. 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

49. 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

50. 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 participants’ scores if it were examined in an experiment by itself

51. Learning Check - Answers
False
Main effects in Two-Factor ANOVA are identical to results of two One-Way ANOVAs; but Two-Factor ANOVA provides Interaction results too!
The two-factor ANOVA allows you to determine the effect of one variable controlling for the effect of the other

52. Figure 13.5 Independent-Measures Two-Factor Formulas
FIGURE The ANOVA for an independent-measures two-factor design.

53. Figure 13.6 Example 13.1 SPSS Output for Repeated-Measures
FIGURE Portions of the SPSS output for the repeated-measures ANOVA for the study evaluating different strategies for studying in Example 13.1.

54. Figure 13.7 Example 13.4 SPSS Output for Two-Factor ANOVA
FIGURE Portions of the SPSS output for the two-factor ANOVA for the study in Example 13.4.

55. Any Questions?
Concepts?
Equations?