1. Statistics for the Social Sciences
Psychology 340
Fall 2013
Thursday, October 17, 2013
Repeated Measures Analysis of Variance (ANOVA)

2. Homework Assignment Due 10/22
Chapter 12: 14, 15, 21, 22 (Use SPSS for #21 & 22. Print out your output, identify the relevant statistics and probabilities on the output, and write out responses to all parts of each question – if you submit electronically, include a word document or similar that identifies all of the relevant information from the output to fully answer each question)
Chapter 13: 1, 4, 7, 8

3. Last Time Brief review of one-way ANOVA Assumptions in ANOVA
Post-hoc and planned comparisons
Effect sizes in ANOVA
ANOVA in SPSS

4. Today More practice with SPSS ANOVA in research articles
ANOVA structural model
Repeated Measures ANOVA

5. One-Way ANOVA in SPSS
Enter the data: similar to independent samples t-test, observations in one column, a second column for group assignment
Analyze: compare means, 1-way ANOVA
Your grouping variable is the “factor” and your continuous (outcome) variable goes in the “dependent list” box
Specify any comparisons or post hocs at this time too
Planned Comparisons (contrasts): are entered with 1, 0, & -1
Post-hoc tests: make sure that you enter your α-level
Under “options,” you can request descriptive statistics (e.g., to see group means)
SPSS does not give effect size under this procedure (can get it by using analyzeGeneral Linera ModelUnivariate), but it’s easy to calculate by hand

6. Demonstration with “Majors” data
Watch the demonstration learn how to use SPSS to find out whether there are significant differences among engineering, computer sciences, and “other sciences” in terms of average SAT scores.
Now you try it using the same data set, but test whether there is a significant difference between majors in college GPA. Use a post-hoc test to see which groups are different.
Save your output and submit it to me using the hand-in link on the class website.

7. ANOVA in Research Articles
F(3, 67) = 5.81, p < .01
Means given in a table or in the text
Follow-up analyses
Planned comparisons
Using t tests

8. 1 factor ANOVA Reporting your results The observed difference
Kind of test
Computed F-ratio
Degrees of freedom for the test
The “p-value” of the test
Any post-hoc or planned comparison results
“The mean score of Group A was 12, Group B was 25, and Group C was 27. A 1-way ANOVA was conducted and the results yielded a significant difference, F(2,25) = 5.67, p < Post hoc tests revealed that the differences between groups A and B and A and C were statistically reliable (respectively t(13) = 5.67, p < 0.05 & t(13) = 6.02, p <0.05). Groups B and C did not differ significantly from one another”

9. The structural model and ANOVA
The structural model is all about deviations
Score
(X)
Group mean
(M)
Grand mean
(GM)
Group’s mean’s
deviation from
grand mean
(M-GM)
Score’s deviation
from group mean
(X-M)
Score’s deviation from grand mean
(X-GM)

10. Statistical analysis follows design
The 1 factor between groups ANOVA:
More than two
Independent & One score per subject
1 independent variable

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

12. 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

13. 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
Drug A
Drug B
Drug C
Placebo

14. Within-subjects ANOVAMB MA MC MD
Drug A
Drug B
Drug C
Placebo 4 6 7 3 1 5 2
n = 5 participants
Each participates in every condition (4 of these)

15. Within-subjects ANOVA
Hypothesis testing: a four step program
Step 1: State your hypotheses
Step 2: Set decision criteria
Step 3: Compute your test statistics
Compute your estimated variances (2 steps of partitioning used)
Compute your F-ratio
Step 4: Make a decision about your null hypothesis

16. 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 conditions
Within conditions MB MA MC MD
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)

17. Partitioning the variance
Total variance
Stage 1
Between cond. variance
Within cond. variance

18. Partitioning the variance
Total variance
Stage 1
Between cond. variance
Within cond. variance
Between subjects variance
Error variance
Stage 2

19. 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 cond. variance
Within cond. 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)

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

21. Partitioning the variance
Total variance
Stage 1
Between cond. variance
Within cond. 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)

22. Partitioning the variance
Total variance
Stage 1
Between cond. variance
Within cond. variance

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

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

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

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

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

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

29. Partitioning the variance
Total variance
Stage 1
Between cond. variance
Within cond. variance
Stage 2
Between subjects variance
Error variance

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

31. Partitioning the variance
Total variance
Stage 1
Between cond. variance
Within cond. variance
Stage 2
Between subjects variance
Error variance

32. 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 row in each cell 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 = 3.49.
So we reject H0 and conclude that not all groups are the same

33. Computational Formulas
T = Group Total
G = Grand Total
P = Person Total
n = number of participants
k = number of conditions
N = number of scores
Drug A
Drug B
Drug C
Placebo
Person Totals 4 6 7 3
P1 = 20
P2 = 12 1 5 2
P3 = 12
P4 = 8
P5 = 4
TA = 10
TB = 20
TC = 25
TD = 5
n = 5
SSA = 8
SSB = 6
SSC = 10
SSD = 8
k = 4
N = 20
G = 60

34. Computational Formulas
Drug A
Drug B
Drug C
Placebo
Person Totals 4 6 7 3
P1 = 20
P2 = 12 1 5 2
P3 = 12
P4 = 8
P5 = 4
TA = 10
TB = 20
TC = 25
TD = 5
n = 5
SSA = 8
SSB = 6
SSC = 10
SSD = 8
k = 4
N = 20
G = 60

35. From Before
Drug A
Drug B
Drug C
Placebo 4 6 7 3 1 5 2

36. Computational Formulas
Drug A
Drug B
Drug C
Placebo
Person Totals 4 6 7 3
P1 = 20
P2 = 12 1 5 2
P3 = 12
P4 = 8
P5 = 4
TA = 10
TB = 20
TC = 25
TD = 5
n = 5
SSA = 8
SSB = 6
SSC = 10
SSD = 8
k = 4
N = 20
G = 60

37. From Before
Drug A
Drug B
Drug C
Placebo 4 6 7 3 1 5 2

38. Computational Formulas
Drug A
Drug B
Drug C
Placebo
Person Totals 4 6 7 3
P1 = 20
P2 = 12 1 5 2
P3 = 12
P4 = 8
P5 = 4
TA = 10
TB = 20
TC = 25
TD = 5
n = 5
SSA = 8
SSB = 6
SSC = 10
SSD = 8
k = 4
N = 20
G = 60

39. Between subjects variance
From Before
Drug A
Drug B
Drug C
Placebo 4 6 7 3 1 5 2
Between subjects variance

40. Computational Formulas
Drug A
Drug B
Drug C
Placebo
Person Totals 4 6 7 3
P1 = 20
P2 = 12 1 5 2
P3 = 12
P4 = 8
P5 = 4
TA = 10
TB = 20
TC = 25
TD = 5
n = 5
SSA = 8
SSB = 6
SSC = 10
SSD = 8
k = 4
N = 20
G = 60

41. Between subjects variance
From Before
Total variance
Stage 1
Between cond. variance
Within cond. variance
Stage 2
Between subjects variance
Error variance

42. Within-subjects ANOVA in SPSS
Setting up the file
Running the analysis
Looking at the output