1. PSY 307 – Statistics for the Behavioral Sciences
Chapter 17 – Repeated Measures ANOVA (F-Test)

2. Kinds of ANOVAs Repeated Measures ANOVA Factorial ANOVA (Two-way)
Used with 2 or more paired samples
Factorial ANOVA (Two-way)
Use with 2 or more independent variables
Both IVs are independent samples
Mixed ANOVA
A two-way ANOVA with both between-subjects and repeated measures IVs

3. Repeated Measures ANOVA
Repeated Measures ANOVA – one-way ANOVA but the same subjects are measured in each group.
Estimates of variability are no longer inflated by random error due to individual differences.
A more powerful F-test
A way to make the denominator smaller.

4. The Repeated Measures F-Ratio
The numerator contains the MS for between-groups, as for one-way ANOVA.
The denominator contains only the error variance (noise), not the entire within-group variance.
Variance for individual subjects is subtracted from the entire within-group variance, leaving the error variance.

5. Splitting Up the Variance: Repeated Measures ANOVA
Total Variance
SStotal
Between Groups
SSbetween
Within Groups
SSwithin
Between Subjects
SSsubject
Error
SSerror

6. Finding SSsubject
Subject 24 48
XSubject A 3 6 B 4 8 C 2 10
Group Mean 5
5 (Grand Mean)
SSsubject is calculated using row totals for each subject.
SSerror is the amount remaining when SSsubject is subtracted from SSwithin.

7. Calculating SSerror and MSerror
This is the same as for one-way ANOVA

8. Effect Size for Repeated Measures ANOVA
The effect size for repeated measures is called the partial squared curvilinear correlation.
It is called partial because the effects of individual differences have been removed.

9. Comparison to One-Way ANOVA
Other assumptions hold (e.g., normality, equal variance), but sphericity is an added assumption.
Sphericity means data are uncorrelated.
Counterbalancing may be needed.
h2 is interpreted the same way as for one-way ANOVA.
Post-hoc t-tests need to be for paired samples, not independent groups.

10. Two-Way ANOVA
Two-way (two factor) ANOVA – tests hypotheses about two independent variables (factors).
Three null hypotheses are tested:
Main effect for first independent variable
Main effect for second independent variable
Test for an interaction between the two variables.

11. Two Factors: Age and Sex
Male
Female
Young 5 10
Old 20 15
Main effect for Age
H0: myoung = mold
H1: H0 is false
Main effect for Sex
H0: mmale = mfemale
H1: H0 is false

12. What About Within the Cells?
Male
Female
Young 5 10
Old 20 15
Main effect for Age
H0: myoung = mold
H1: H0 is false
Main effect for Sex
H0: mmale = mfemale
H1: H0 is false
An interaction occurs when the pattern within the cells is different depending on the level of the IVs.
H0: There is no interaction
H1: H0 is false

13. Interaction
Two factors interact if the effects of one factor on the dependent variable are not consistent for all levels of a second factor.
Interactions provide important information about the question at hand.
Interactions must be discussed in your interpretation of your results because they modify main effects.

14. Applet Demonstrating Two-Way ANOVA
Try this at home to understand what an interaction looks like when graphed and in a data table.

15. Splitting Up the Variance: Two-Way ANOVA
Total Variance
SStotal
Between Cells
Within Cells
SSbetween
SSwithin
Between Columns
Between Rows
Interaction
SScolumns
SSrows
SSInteraction

16. Calculating MScolumns and MSwithin
dfcolumns = c-1, where c is the number of columns
This is the same as for one-way ANOVA but with different df:
dfwithin = N-(c)(r), where c is columns and r is rows

17. Calculating MSrows and MSwithin
dfrows = r-1, where r is the number of rows
This is the same as for one-way ANOVA but with different df:
dfwithin = N-(c)(r), where c is columns and r is rows

18. Calculating MSint and MSwithin
dfint = (c-1)(r-1), where c is the number of columns and r is the number of rows
This is the same as for one-way ANOVA but with different df:
dfwithin = N-(c)(r), where c is columns and r is rows

19. A Sample Two-Way ANOVA Table
Source SS Df MS F
Column 72 2
72/2 = 36
36/5.33 = 6.75*
Row
192 1
192/1 = 192
192/5.33 = 36.02*
Interaction 56
56/2 = 28
28/5.33 = 5.25*
Within 32 6
32/6 = 5.33
Total
352 11
* Significant at the .05 level

20. Proportion of Explained Variance
h2 estimates the proportion of the total variance attributable to one of the two factors or the interaction.
There is an h2 value for each main effect and one for the interaction.
Cohen’s rule for interpreting h2 :
.01 small effect
.09 medium effect
.25 large effect

21. Calculating h2

22. What is h2 used for?
When a main effect is non-significant due to small sample size, the size of the effect can be examined.
Some journals require effect sizes to always be reported along with inferential tests.

23. Simple Effects
A simple effect is the comparison of groups for each level of one IV, at a single level of the other IV.
Example: Young and old males.
Example: Young males and females.
Calculation of the simple effect is the same as doing a one-way ANOVA on just one column or row.
Use t-tests to follow up.

24. Assumptions Similar to those for one-way ANOVA:
Normal distribution, equal variances
All cells should have equal sample sizes.
Use Cohen’s guidelines for effect size (h2).
Use t-tests for multiple comparisons within cells.

25. Mixed ANOVAs One variable is between subjects.
One or more variables are within subjects (paired or repeated measures).
A mixed ANOVA is performed using the Repeated Measures General Linear Model menu choice on SPSS.
Formulas are complex and beyond the scope of this course.