1. Statistics for the Social Sciences
Psychology 340
Spring 2010
Within Groups ANOVA

2. Outline
Questions regarding One factor between groups ANOVA (covered last week)?
Basics of within groups ANOVA
Repeated measures
Matched samples
Computations
Within groups ANOVA in SPSS

3. Example
Suppose that you want to compare three brand name pain relievers.
Give each person a drug, wait 15 minutes, then ask them to keep their hand in a bucket of cold water as long as they can. The next day, repeat (with a different drug)
Dependent variable: time in ice water
Independent variable: 4 levels, within groups
Placebo
Drug A
Drug B
Drug C

4. Statistical analysis follows design
More than 2 scores per subject
One group
The 1 factor within groups ANOVA:
Repeated measures

5. Statistical analysis follows design
The 1 factor within groups ANOVA:
Repeated measures
More than 2 groups
One group
Matched groups
More than 2 scores per subject
- OR -
Matched samples

6. Within-subjects ANOVAXB XA XC XP
Placebo
Drug A
Drug B
Drug C 3 4 6 7 2 1 5
n = 5 participants
Each participates in every condition (4 of these)

7. Within-subjects ANOVA
Hypothesis testing: a five step program
Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data
Step 4: Compute your test statistics
Compute your estimated variances (2 steps of partitioning used)
Compute your F-ratio
Compute your degrees of freedom (there are even more now)
Step 5: Make a decision about your null hypothesis

8. Step 4: Computing the F-ratio
Analyzing the sources of variance
Describe the total variance in the dependent measure
Why are these scores different?
Sources of variability
Between groups
Within groups XB XA XC XP
Because we use the same people in each condition, we can figure out how much of the variability comes from the individuals and remove it from the analysis
Individual differences
Left over variance (error)

9. Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance

10. Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
Between subjects variance
Error variance
Stage 2

11. Partitioning the variance
Total variance
Because we use the same people in each condition, none of this variability comes from having different people in different conditions
Stage 1
Between groups variance
Within groups variance
Treatment effect
Error or chance (without individual differences)
Individual differences
Other error
Between subjects variance
Error variance
Stage 2
Individual differences
Other error (without individual differences)

12. Step 4: Computing the F-ratio
Ratio of the between-groups variance estimate to the population error variance estimate
Observed variance
Variance from chance
F-ratio =

13. Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
Treatment effect
Error or chance (without individual differences)
Individual differences
Other error
Between subjects variance
Error variance
Stage 2
Individual differences
Other error (without individual differences)

14. Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance

15. Partitioning the variance
Placebo
Drug A
Drug B
Drug C 3 4 6 7 2 1 5

16. Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
Stage 2
Between subjects variance
Error variance

17. Partitioning the variance
Placebo
Drug A
Drug B
Drug C 3 4 6 7 2 1 5
What is ?
The average score for each person
Between subjects variance

18. Partitioning the variance
Placebo
Drug A
Drug B
Drug C 3 4 6 7 2 1 5
What is ?
The average score for each person
Between subjects variance

19. Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
Stage 2
Between subjects variance
Error variance

20. Partitioning the variance
Placebo
Drug A
Drug B
Drug C 3 4 6 7 2 1 5
Error variance

21. Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
Stage 2
Between subjects variance
Error variance

22. Partitioning the variance
Now we return to
variance. But, we
call it
Means Square (MS)
Mean Squares (Variance)
Between groups variance
Error variance
Recall:

23. Partitioning the variance
Total variance
Stage 1
Between groups variance
Within groups variance
Stage 2
Between subjects variance
Error variance

24. Within-subjects ANOVA
The F table
Need two df’s
dfbetween (numerator)
dferror (denominator)
Values in the table correspond to critical F’s
Reject the H0 if your computed value is greater than or equal to the critical F
Separate tables for 0.05 & 0.01
Do we reject or fail
to reject the H0?
From the table (assuming 0.05) with 3 and 12 degrees of freedom the critical F = 3.89.
So we reject H0 and conclude that not all groups are the same

25. Effect sizes in ANOVA The effect size for ANOVA is r2
Sometimes called η2 (“eta squared”)
The percent of the variance in the dependent variable that is accounted for by the independent variable
So 86% of the variance in the DV is accounted for by the IV (except for individual diffs)

26. Within-subjects ANOVA in SPSS
Setting up the file
Each within groups condition goes into a separate column
Running the analysis
Analyze -> General Linear Model -> Repeated Measures
Factor (your IV), how many levels, click ‘add’, then ‘Define’
Need to tell SPSS which columns have your observations
Looking at the output
Tests of within subjects effects