1. Repeated Measures ANOVA
Quantitative Methods in HPELS
440:210

2. Agenda Introduction The Repeated Measures ANOVA
Hypothesis Tests with Repeated Measures ANOVA
Post Hoc Analysis
Instat
Assumptions

3. Introduction
Recall  There are two possible scenarios when obtaining two sets of data for comparison:
Independent samples: The data in the first sample is completely INDEPENDENT from the data in the second sample.
Dependent/Related samples: The two sets of data are DEPENDENT on one another. There is a relationship between the two sets of data.

4. Introduction Three or more data sets?
If the three or more sets of data are independent of one another  Analysis of Variance (ANOVA)
If the three or more sets of data are dependent on one another  Repeated Measures ANOVA

5. Agenda Introduction The Repeated Measures ANOVA
Hypothesis Tests with Repeated Measures ANOVA
Post Hoc Analysis
Instat
Assumptions

6. Repeated Measures ANOVA
Statistical Notation  Recall for ANOVA:
k = number of treatment conditions (levels)
nx = number of samples per treatment level
N = total number of samples
N = kn if sample sizes are equal
Tx = SX for any given treatment level
G = ST
MS = mean square = variance

7. Repeated Measures ANOVA
Additional Statistical Notation:
P = total score for each subject (personal total)
Example: If a subject was assessed three times and had scores of 3, 4, 5  P = 12

8. Repeated Measures ANOVA
Formula Considerations  Recall for ANOVA:
SSbetween = ST2/n – G2/N
SSwithin = SSSinside each treatment
SStotal = SSwithin + SSbetween
SStotal = SX2 – G2/N

9. ANOVA Formula Considerations: dftotal = N – 1 dfbetween = k – 1
dfwithin = S(n – 1)
dfwithin = Sdfin each treatment

10. ANOVA Formula Considerations:
MSbetween = s2between = SSbetween / dfbetween
MSwithin = s2within = SSwithin / dfwithin
F = MSbetween / MSwithin

11. Repeated Measures ANOVA
New Formula Considerations:
SSbetween  SSbetween treatments = ST2/n – G2/N
SSbetween subjects = SP2/k – G2/N
SSwithin  SSwithin treatments = SSSinside each treatment
SSerror = SSwithin treatments – SSbetween subjects

12. Repeated Measures ANOVA
New Formula Considerations:
dfbetween  dfbetween treatments = k – 1
dfwithin  dfwithin treatments = N – k
dfbetween subjects = n – 1
dferror = (N – k) – (n – 1)

13. Repeated Measures ANOVA
MSbetween treatments=SSbetween treatments/dfbetween treatments
MSerror = SSerror / dferror
F = MSbetween treatments / MSerror

14. Repeated Samples Designs
One-group pretest posttest (repeated measures) design:
Perform pretest on all subjects
Administer treatments followed by posttests
Compare pretest to posttest scores and posttest to posttest scores
O X O X O

15. Agenda Introduction The Repeated Measures ANOVA
Hypothesis Tests with Repeated Measures ANOVA
Post Hoc Analysis
Instat
Assumptions

16. Hypothesis Test: Repeated Measuers ANOVA
Example 14.1 (p 457)
Overview:
Researchers are interested in a behavior modification technique on outbursts in unruly children
Four students (n=4) are pretested on the # of outbursts during the course of one day
Teachers begin using “cost-response” technique
Students are posttested one week later, one month later and 6 months later

17. Hypothesis Test: ANOVA
Questions:
What is the experimental design?
What is the independent variable/factor?
How many levels are there?
What is the dependent variable?

18. Step 1: State Hypotheses
Non-Directional
H0: µpre = µ1week = µ1month = µ6months
H1: At least one mean is different than the others
Table B.4 (p 693)
Critical value = 3.86
Step 2: Set Criteria
Alpha (a) = 0.05
Critical Value:
Use F Distribution Table
Appendix B.4 (p 693)
Information Needed:
dfbetween treatments = k – 1 = 4 – 1 = 3
dferror = (N-k)-(n-1) = (16-4)-(4-1) = 9

19. Step 3: Collect Data and Calculate Statistic
Total Sum of Squares
SStotal = SX2 – G2/N
SStotal = 222 – 442/20
SStotal =
SStotal = 101
Sum of Squares Between each Treatment
SSbetween treatment = ST2/n – SG2/N
SSbetween treatment = 262/4+82/4+62/4+42/4 – 442/20
SSbetween treatment = ( ) - 121
SSbetween treatment = 77
Sum of Squares Within each Treatment
SSwithin = SSSinside each treatment
SSwithin =
SSwithin = 24
Sum of Squares Error
SSerror = SSwithin treatments – SSbetween subjects
SSerror =
SSwithin = 11
Sum of Squares Between each Subject
SSbetween subjects = SP2/k – SG2/N
SSbetween subjects = (122/4+62/4+102/4+162/4) - 442/16
SSbetween subjects = ( ) – 121
SSbetween subjects = 13
Raw data can be found in Table14.3 (p 457)

20. Step 3: Collect Data and Calculate Statistic F-Ratio
F = MSbetween treatment / MSerror
F = / 1.22
F = 21.04
Mean Square Between each Treatment
MSbetween treatment = SSbetween treatment / dfbetween treatment
MSbetween treatment = 77 / 3
MSbetween = 25.67
Step 4: Make Decision
Mean Square Error
MSerrorn = SSerror / dferror
MSerror = 11 / 9
MSwithin = 1.22

21. Agenda Introduction Repeated Measures ANOVA
Hypothesis Tests with Repeated Measures ANOVA
Post Hoc Analysis
Instat
Assumptions

22. Post Hoc Analysis What ANOVA tells us: What ANOVA doesn’t tell us:
Rejection of the H0 tells you that there is a high PROBABILITY that AT LEAST ONE difference exists SOMEWHERE
What ANOVA doesn’t tell us:
Where the differences lie
Post hoc analysis is needed to determine which mean(s) is(are) different

23. Post Hoc Analysis
Post Hoc Tests: Additional hypothesis tests performed after a significant ANOVA test to determine where the differences lie.
Post hoc analysis IS NOT PERFORMED unless the initial ANOVA H0 was rejected!

24. Post Hoc Analysis  Type I Error
Type I error: Rejection of a true H0
Pairwise comparisons: Multiple post hoc tests comparing the means of all “pairwise combinations”
Problem: Each post hoc hypothesis test has chance of type I error
As multiple tests are performed, the chance of type I error accumulates
Experimentwise alpha level: Overall probability of type I error that accumulates over a series of pairwise post hoc hypothesis tests
How is this accumulation of type I error controlled?

25. Two Methods Bonferonni or Dunn’s Method: Specific post hoc tests:
Perform multiple t-tests of desired comparisons or contrasts
Make decision relative to a / # of tests
This reduction of alpha will control for the inflation of type I error
Specific post hoc tests:
Note: There are many different post hoc tests that can be used
Our book only covers two (Tukey and Scheffe)

26. Repeated Measures ANOVA
Bonferronni/Dunn’s method is appropriate with following consideration:
Use related-samples T-tests
Tukey’s and Scheffe is appropriate with following considerations:
Replace MSwithin with MSerror in all formulas
Replace dfwithin with dferror in all formulas
Note: Statisticians are not in agreement with post hoc analysis for Repeated Measures ANOVA

27. Agenda Introduction The Repeated Measures ANOVA
Hypothesis Tests with Repeated Measures ANOVA
Post Hoc Analysis
Instat
Assumptions

28. Instat Label three columns as follows:
Block: This groups your data by each subject.
Example: If you conducted a pretest and 2 posttests (3 total) on 5 subjects, the block column will look like:
Treatment: This tells you which treatment level/condition occurred for each data point.
Example: If each subject (n=5) received three treatments, the treatment column will look like:
Response: The data for each subject and treatment condition

29. Instat Convert the “Block” and “Treatment” columns into “factors”:
Choose “Manage”
Choose “Column Properties”
Choose “Factor”
Select the appropriate column to be converted
Indicate the number of levels in the factor
Example: Block (5 levels, n = 5), Treatment (3 levels, k = 3)
Click OK

30. Instat Choose “Statistics” Choose “General” Response variable:
Choose “Analysis of Variance”
Choose “General”
Response variable:
Choose the Response variable
Treatment factor:
Choose the Treatment variable
Blocking factor:
Choose the Block variable
Click OK.
Interpret the p-value!!!

31. Instat Post hoc analysis:
Perform multiple related samples t-Tests with the Bonferonni/Dunn correction method

32. Reporting ANOVA Results
Information to include:
Value of the F statistic
Degrees of freedom:
Between treatments: k – 1
Error: (N – k) – (n – 1)
p-value
Examples:
A significant treatment effect was observed (F(3, 9) = 21.03, p = 0.002)

33. Reporting ANOVA Results
An ANOVA summary table is often included
Source SS df MS
Between 77 3
25.67
F = 21.03
Within Treatments 24 12
Between subjects 13
Error 11 9
1.22
Total
101 15

34. Agenda Introduction The Analysis of Variance (ANOVA)
Hypothesis Tests with ANOVA
Post Hoc Analysis
Instat
Assumptions

35. Assumptions of ANOVA Independent Observations Normal Distribution
Scale of Measurement
Interval or ratio
Equal variances (homogeneity)
Equal covariances (sphericity)
If violated a penalty is incurred

36. Violation of Assumptions
Nonparametric Version  Friedman Test (Not covered)
When to use the Friedman Test:
Related-samples design with three or more groups
Scale of measurement assumption violation:
Ordinal data
Normality assumption violation:
Regardless of scale of measurement

37. Textbook Assignment
Problems: 5, 7, 10, 23 (with post hoc)