This document discusses repeated measures ANOVA. It explains that repeated measures ANOVA is used when the same participants are measured under different treatment conditions. This allows researchers to remove variability caused by individual differences. The document outlines the components of the repeated measures ANOVA F-ratio, including the numerator which is the variance between treatments and the denominator which is the variance due to chance/error after removing individual differences. It also discusses how to conduct hypothesis testing and calculate effect size for repeated measures ANOVA.

1. CHAPTER 13
Repeated Measures ANOVA

2. Green, Fry, &
Myerson (1994)
The Delayed Discounting Phenomenon
• We are comparing 5 means
• However, each participant was measured in each
treatment condition (5 scores for each individual)
How much would
you accept today
instead of waiting for
a future reward of
$1000 if you had to
wait:
• 1 month?
• 6 months?
• 12 months?
• 24 months?
• 60 months?

3. Research Designs for ANOVA
• Independent-measures design
• Uses a separate group of participants for each of the treatment
conditions being compared
• Repeated-measures design
• Uses one group of participants for all of the treatment
conditions
• Two-factor (or Factorial) design
• To be covered in Chapter 14

4. Types of Single-Factor ANOVA Designs
Independent-Measures
• One IV
• Two or more treatment
conditions
• Independent samples in
each group (each
treatment condition)
Repeated-Measures
• One IV
• Two or more treatment
conditions
• Dependent samples in
each group
• Same participants in each
treatment condition

5. Repeated-Measures ANOVA
• Utilized to evaluate mean differences in two general
conditions:
1. Experimental Study
IV manipulated to create two or more treatment
conditions, with same group of participants tested in all
conditions

6. Repeated-Measures ANOVA
• Utilized to evaluate mean differences in two general
conditions:
1. Experimental Study
2. Non-experimental Study
Same group of
participants is
simply observed
at two or more
times
(no manipulation of IV)

7. CHAPTER 13.2
The Repeated Measures ANOVA

8. RM ANOVA Hypotheses
Hypotheses remain the same as the independent-measures
one-factor ANOVA design: we are still comparing means among
the different treatment conditions
3210 :  H
The Null Hypothesis
The Alternative Hypothesis
:1H At least one treatment mean (µ) is different from another

9. The Alternative Hypothesis
3210 :  H
3210 :  H
3210 :  H

10. How is this different from independent-
measures ANOVA?
We are able to remove the variability caused by individual
differences
Individual differences:
Participant characteristics that vary from person
to person (may influence the measurements
obtained for each person)
Since the same individuals are measured in
each treatment, we can calculate individual
differences

11. The F-ratio for ANOVA
Variance between treatments
Variance expected by chance/error
Treatment effect + chance/error
Chance/error
excluding individual differences
excluding individual differences

12. Components of the RM ANOVA F-ratio
Numerator: Variance between treatments
• Treatment effect: Systematic differences caused by the
treatments
• The treatment conditions have different effects, which cause an
individual’s score to be higher/lower in different conditions
• Error/chance: Random, unsystematic differences
• Even with no treatment effect, it is still possible for scores in one
treatment to be different than scores in another (even though
individual differences are eliminated from the numerator by the
nature of the research design)

13. Components of the RM ANOVA F-ratio
Denominator: Variance due to chance/error
• Start with variance within treatments
• How much variance is reasonable to expect by chance alone
• Subtract the variance attributed to individual
differences
• This provides a measure of pure error

14. CHAPTER 13.3
Hypothesis Testing and Effect Size
with the Repeated-Measures ANOVA

15. An Illustration
of the Overall
Structure of the
RMANOVA
• Residual variance
(or the error variance)
• How much variance is expected if there are no
systematic treatment effects and no individual
differences contributing to the variability of the scores
Stage 1 is identical to
the ANOVA we just
covered in chapter 12
Stage 2 is performed to
remove the individual
differences from the
denominator of the ratio

16. RM ANOVA Notation
Notation for RM ANOVA is identical to notation for ANOVA:
• k = the number of levels of the factor
• n = the number of scores in each treatment
• N = the total number of scores in the study
• T = the sum of scores (∑X) for a specific treatment
• G = the sum of all scores (∑T) in the study
P = the
total for
all scores
for each
person in
the study

17. RM ANOVA Hypothesis Testing
Stage 1
Partition total variance into:
• Between-treatments variance
• Within treatments variance
Stage 2
Remove individual differences (between-subjects
variance) from within treatments variance to leave:
• Residual/error variance (how much variability is
reasonable to expect by chance after individual
differences have been removed)

18. Stage 1 is the same as the ANOVA
SS computation
remains the same
df computation
remains the same
N
G
XSStotal
2
2
 
 
N
G
n
T
SSbetween
22
treatmenteachinwithin SSSS ..
withintotalbetween SSSSSS 
1 Ndftotal
1 kdfbetween
menteach treatindfdfwithin 

19. Stage 2: Remove individual differences
Remove individual differences from the denominator of the
F-ratio to get a measure of pure error
1.
..
22
.
..





ndf
dfdfdf
N
G
k
P
SS
SSSSSS
subjectsbetween
subjectsbetweentreatmentswithinerror
subjectsbetween
subjectsbetweentreatmentswithinerror

20. Calculate Variances (Mean Squares)
),(. .
.
.
.
.
errortreatmentsbetweenlevel
error
treatmentsbetween
error
error
error
treatmentsbetween
treatmentsbetween
treatmentsbetween
dfdfFcrit
MS
MS
F
df
SS
MS
df
SS
MS






21. The F-ratio
error
treatmentsbetween
MS
MS
F .

),(. . errortreatmentsbetweenlevel dfdfFcrit 

22. Formulas for the RM ANOVA

23. ANOVA Summary Table
Source SS df MS F
Between treatments 50 3 16.67 F(3,12) = 24.88
Within treatments 32 16
Between subjects 24 4
Error 8 12 0.67
Total 82 19

24. Effect size for RM ANOVA
subjectsbetweentotal
treatmentsbetween
SSSS
SS
.
.2


The proportion of variability in the data (except for
the individual differences) accounted for by the
differences between treatments
Independent-measures ANOVA:
denominator is SS total only

25. Post-hoc tests for RM ANOVA
• Substitute MS error for
MS within treatments
• Use dferror in place of
dfwithin treatments when
locating critical value
• Steps and interpretation
remain the same
n
MS
qsHSDTukey error
'
error
between
between
between
between
between
MS
MS
F
df
SS
MS
N
G
n
T
SS
eScheff





22

26. Assumptions of the RM ANOVA
1. The observations within each treatment must be
independent
2. The population distribution within each treatment must
be normal
• Only important with small samples
3. The variances of the population distributions for each
treatment should be equivalent