The One-Way Repeated-Measures ANOVA
(For Within-Subjects Designs)
Repeated-Measures ANOVA
Logic of the Repeated-Measures ANOVA
• The repeated-measures ANOVA extends the analysis of
variance to research situations using repeated-measures (or
related-samples) research designs
• Much of the logic and many of the formulas for
repeated-measures ANOVA are identical to the
independent-measures analysis introduced in the previous
lecture
• However, the repeated-measures ANOVA includes a second
stage of analysis in which variability due to individual
differences is subtracted out of the error term.
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Logic of the Repeated-Measures ANOVA
• The repeated-measures design eliminates individual
differences from the between-treatments variability because
the same subjects are used in every treatment condition.
• To balance the F-ratio, the calculations require that individual
differences also be eliminated from the denominator of the
F-ratio.
• The result is a test statistic similar to the
independent-measures F-ratio but with all individual
differences removed.
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Comparing Independent-Measures & Repeated
Measures ANOVAs
• The independent-measures analysis is used in research
situations for which there is a separate sample for each
treatment condition.
• The analysis compares the mean square (MS) between
treatments to the mean square within treatments in the form
of a ratio
F
MSbetween treatment effect + error (including individual diffs.)

MSwithin
error (including individual diffs.)
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Comparing Independent-Measures & Repeated
Measures ANOVAs
• In the repeated-measures study, there are no individual
differences between treatments because the same individuals
are tested in every treatment.
• This means that variability due to individual differences is not
a component of the numerator of the F ratio.
• Therefore, the individual differences must also be removed
from the denominator of the F ratio to maintain a balanced
ratio with an expected value of 1.00 when there is no
treatment effect.
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Comparing Independent-Measures & Repeated
Measures ANOVAs
• That is, we want the repeated-measures F-ratio to have the
following structure:
F
treatment effect + random, unsystematic differences
random, unsystematic differences
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Logic of the Repeated-Measures ANOVA
• This is accomplished by a two-stage analysis. In the first
stage, total variability (SStotal) is partitioned into variability
between-treatments (SSbetween) and within-treatments
(SSwithin).
– Individual differences do not appear in SSbetween because the same
sample of subjects serves in every treatment.
– On the other hand, individual differences do play a role in SStotal
because the sample contains different subjects.
• In the second stage of the analysis, we measure the individual
differences by computing the variability between subjects, or
SSsubjects
• This value is subtracted from SSwithin leaving a remainder,
variability due to sampling error, SSerror
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Logic of the Repeated-Measures ANOVA
Total Variance
SSbetween  SStotal  SS within
dfbetween  dftotal  df within
Between
Treatments
Variance
SSwithin  SStotal  SSbetween
df within  dftotal  dfbetween
SStotal  SSbetween  SSwithin
dftotal  dfbetween  df within
Within
Treatments
Variance
Between
Subjects
Variance
SSsubjects  SSwithin  SSerror
01:830:200:10-13 Spring 2013
df subjects  df within  df error
Error Variance
SSerror  SSwithin  SSsubjects
df error  df within  df subjects
Repeated-Measures ANOVA
Computations for the Repeated-Measures ANOVA
• Start by computing
compute:
 x and  x for each group, then
2
• Grand total: The overall total, computed over all scores in all
groups (samples)
k
n
i
j
 x   x
T
ij
• Total sum of squared scores: The sum of squared scores
computed over all scores in all groups
k
n
i
j
 x   x
2
T
01:830:200:10-13 Spring 2013
2
ij
Repeated-Measures ANOVA
Computations for the Repeated-Measures ANOVA
• In addition, you will have to compute the sum of scores
(across all k conditions) for each subject and/or the mean for
each subject
x
k
subject
M subject
01:830:200:10-13 Spring 2013
  xsubjecti
i
x


subject
k
Repeated-Measures ANOVA
Computations for the ANOVA: SS terms
• SStotal : The sum of squared deviations of all observations
from the grand mean
SStotal    x  M T    xT2 
2
  xT 
2
N
or
SStotal  SSbetween  SSwithin
– Not strictly needed for computing the F ratio, but it makes computing the
needed SS terms much easier
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Computations for the ANOVA: SS terms
• SSbetween: The sum of squared deviations of the sample
means from the grand mean multiplied by the number of
observations
k
k
SSbetween  n  M i  M T   
i
2
i

 xi
n

2

  xT 
N
2
or
SSbetween  SStotal  SSwithin
• SSwithin: The sum of squared deviations within each sample
k
SSwithin   SS j  SS1  SS2  ...  SSk
j
01:830:200:10-13 Spring 2013
or
SSwithin  SStotal  SSbetween
Repeated-Measures ANOVA
Computations for the Repeated-Measures ANOVA
• SSsubjects : The sum of squared deviations of the subject
means from the grand mean multiplied by the number of
conditions (k)
SSsubjects  k   M subject  M T   
2

 x subject
k
   x 
2
2
T
N
• SSerror: The sum of squared deviations due to sampling
error
SSerror  SSwithin  SSsubjects
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Computations for the Repeated-Measures ANOVA
• dftotal = N-1 :
– degrees of freedom associated with SStotal
– N is the total number of scores
• dfbetween (dfgroup) = k-1 :
– degrees of freedom associated with SSbetween
– k is the number of groups
• dfsubjects = n-1
– Degrees of freedom associated with SSsubjects
– n is the number of subjects
• dferror= dftotal - dfbetween -dfsubjects= N-k-n+1 :
– degrees of freedom associated with SSerror
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Computing the F-statistic
MSbetween
MSerror
SSbetween

dfbetween
SSerror

df error
01:830:200:10-13 Spring 2013
MSbetween
F  dfbetween , df error  
MSerror
Repeated-Measures ANOVA
The Repeated-Measures ANOVA: Steps
1. State Hypotheses
2. Compute F-ratio statistic:
–
F  dfbetween , df error  
MSbetween
MSerror
For data in which I give you raw scores, you will have to compute:
•
•
•
Sample means & subject means
SStotal, SSbetween, SSwithin, SSsubjects, & SSerror
dftotal, dfbetween, dfwithin, dfsubjects, & dferror
3. Use F-ratio distribution table to find critical F-value representing
rejection region
4. Make a decision: does the F-statistic for your sample fall into the
rejection region?
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Repeated-Measures ANOVA: Example
Does giving students a pedometer cause them to walk more?
• Measure their initial average daily number of steps over a week
• Follow up for 12 weeks
• x1 : number of steps (in thousands) during week 1
• x2 : number of steps during week 6
• x3 : number of steps during week 12
• Null Hypothesis H0: µ1 = µ2 = µ3
• Research Hypothesis H1: one of the population means is different
• Do we accept or reject the null hypothesis?
– Assume α = 0.05
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Set up a summary ANOVA table:
Thousands of Steps
Student
A
B
C
D
E
F
X1
6
4
5
1
0
2
X2
8
5
5
2
1
3
X3
10
6
5
3
2
4
𝑀𝑔𝑟𝑜𝑢𝑝
3
4
5
SStotal  112
𝑀𝑠𝑢𝑏𝑗
8
5
5
2
1
3
Source
df
SS
Between groups
Within groups
subjects
error
Total
112
MT  4
1. Compute degrees of freedom
dftotal  N  1  17
dfbetween  k  1  2
df subjects  n  1  5
df within  dftotal  dfbetween  15
df error  df within  df subjects  10
01:830:200:10-13 Spring 2013
MS
F
Repeated-Measures ANOVA
Set up a summary ANOVA table:
Thousands of Steps
Student
A
B
C
D
E
F
X1
6
4
5
1
0
2
X2
8
5
5
2
1
3
X3
10
6
5
3
2
4
𝑀𝑔𝑟𝑜𝑢𝑝
3
4
5
SStotal  112
𝑀𝑠𝑢𝑏𝑗
8
5
5
2
1
3
Source
df
Between groups
2
Within groups
15
subjects
5
error
10
Total
SS
17
MS
112
MT  4
2. Compute SSbetween (or SSwithin) directly
SSbetween  n  M  M T 
2
2
2
2
 6  3  4    4  4    5  4  


 6 1  0  1  6(2)  12
01:830:200:10-13 Spring 2013
F
Repeated-Measures ANOVA
Set up a summary ANOVA table:
Thousands of Steps
Student
A
B
C
D
E
F
X1
6
4
5
1
0
2
X2
8
5
5
2
1
3
X3
10
6
5
3
2
4
𝑀𝑔𝑟𝑜𝑢𝑝
3
4
5
SStotal  112
𝑀𝑠𝑢𝑏𝑗
8
5
5
2
1
3
Source
df
SS
Between groups
2
12
Within groups
15
subjects
5
error
10
Total
17
MS
F
112
MT  4
3. Compute the missing top-level SS value
(SSbetween or SSwithin) via subtraction
SSwithin  SStotal  SSbetween
 112  12  100
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Set up a summary ANOVA table:
Thousands of Steps
Student
A
B
C
D
E
F
X1
6
4
5
1
0
2
X2
8
5
5
2
1
3
X3
10
6
5
3
2
4
𝑀𝑔𝑟𝑜𝑢𝑝
3
4
5
SStotal  112
𝑀𝑠𝑢𝑏𝑗
8
5
5
2
1
3
Source
df
SS
Between groups
2
12
Within groups
15
100
subjects
5
error
10
Total
17
MS
F
112
MT  4
4. Compute SSsubjects:
SS subjects  k   M subj  M T 
2
2
2
2
2
2
2
 3  8  4    5  4    5  4    2  4   1  4    3  4  


 3 16  1  1  4  9  1  3(32)  96
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Set up a summary ANOVA table:
Thousands of Steps
Student
A
B
C
D
E
F
X1
6
4
5
1
0
2
X2
8
5
5
2
1
3
X3
10
6
5
3
2
4
𝑀𝑔𝑟𝑜𝑢𝑝
3
4
5
SStotal  112
𝑀𝑠𝑢𝑏𝑗
8
5
5
2
1
3
Source
df
SS
Between groups
2
12
Within groups
15
100
subjects
5
96
error
10
Total
17
MS
112
MT  4
5. Compute SSerror by subtraction
SSerror  SSwithin  SSsubjects
 100  96  4
01:830:200:10-13 Spring 2013
F
Repeated-Measures ANOVA
Set up a summary ANOVA table:
Thousands of Steps
Student
A
B
C
D
E
F
X1
6
4
5
1
0
2
X2
8
5
5
2
1
3
X3
10
6
5
3
2
4
𝑀𝑔𝑟𝑜𝑢𝑝
3
4
5
SStotal  112
𝑀𝑠𝑢𝑏𝑗
8
5
5
2
1
3
Source
df
SS
Between groups
2
12
Within groups
15
100
subjects
5
96
error
10
4
17
112
Total
MS
MT  4
6. Compute the MS values needed to
compute the F ratio:
MSbetween 
MSerror 
01:830:200:10-13 Spring 2013
SSbetween 12
  6.0
dfbetween
2
SSerror
4
  0.4
df error 10
F
Repeated-Measures ANOVA
Set up a summary ANOVA table:
Thousands of Steps
Student
A
B
C
D
E
F
X1
6
4
5
1
0
2
X2
8
5
5
2
1
3
X3
10
6
5
3
2
4
𝑀𝑔𝑟𝑜𝑢𝑝
3
4
5
SStotal  112
𝑀𝑠𝑢𝑏𝑗
8
5
5
2
1
3
Source
df
SS
MS
Between groups
2
12
6.0
Within groups
15
100
subjects
5
96
error
10
4
17
112
Total
MT  4
7. Compute the F ratio:
F  dfbetween , df error  
F  2,10  
01:830:200:10-13 Spring 2013
MSbetween
MSerror
6.0
 15.0
0.4
0.4
F
Repeated-Measures ANOVA
dfnumerator
F table for α=0.05
dferror
reject H0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
22
24
26
28
30
40
50
60
120
200
500
1000
01:830:200:10-13 Spring 2013
1
2
3
4
5
6
7
8
9
10
161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88
18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40
10.13
9.55
9.28
9.12
9.01
8.94
8.89
8.85
8.81
8.79
7.71
6.94
6.59
6.39
6.26
6.16
6.09
6.04
6.00
5.96
6.61
5.79
5.41
5.19
5.05
4.95
4.88
4.82
4.77
4.74
5.99
5.14
4.76
4.53
4.39
4.28
4.21
4.15
4.10
4.06
5.59
4.74
4.35
4.12
3.97
3.87
3.79
3.73
3.68
3.64
5.32
4.46
4.07
3.84
3.69
3.58
3.50
3.44
3.39
3.35
5.12
4.26
3.86
3.63
3.48
3.37
3.29
3.23
3.18
3.14
4.96
4.10
3.71
3.48
3.33
3.22
3.14
3.07
3.02
2.98
4.84
3.98
3.59
3.36
3.20
3.09
3.01
2.95
2.90
2.85
4.75
3.89
3.49
3.26
3.11
3.00
2.91
2.85
2.80
2.75
4.67
3.81
3.41
3.18
3.03
2.92
2.83
2.77
2.71
2.67
4.60
3.74
3.34
3.11
2.96
2.85
2.76
2.70
2.65
2.60
4.54
3.68
3.29
3.06
2.90
2.79
2.71
2.64
2.59
2.54
4.49
3.63
3.24
3.01
2.85
2.74
2.66
2.59
2.54
2.49
4.45
3.59
3.20
2.96
2.81
2.70
2.61
2.55
2.49
2.45
4.41
3.55
3.16
2.93
2.77
2.66
2.58
2.51
2.46
2.41
4.38
3.52
3.13
2.90
2.74
2.63
2.54
2.48
2.42
2.38
4.35
3.49
3.10
2.87
2.71
2.60
2.51
2.45
2.39
2.35
4.30
3.44
3.05
2.82
2.66
2.55
2.46
2.40
2.34
2.30
4.26
3.40
3.01
2.78
2.62
2.51
2.42
2.36
2.30
2.25
4.23
3.37
2.98
2.74
2.59
2.47
2.39
2.32
2.27
2.22
4.20
3.34
2.95
2.71
2.56
2.45
2.36
2.29
2.24
2.19
4.17
3.32
2.92
2.69
2.53
2.42
2.33
2.27
2.21
2.16
4.08
3.23
2.84
2.61
2.45
2.34
2.25
2.18
2.12
2.08
4.03
3.18
2.79
2.56
2.40
2.29
2.20
2.13
2.07
2.03
4.00
3.15
2.76
2.53
2.37
2.25
2.17
2.10
2.04
1.99
3.92
3.07
2.68
2.45
2.29
2.18
2.09
2.02
1.96
1.91
3.89
3.04
2.65
2.42
2.26
2.14
2.06
1.98
1.93
1.88
3.86
3.01
2.62
2.39
2.23
2.12
2.03
1.96
1.90
1.85
3.85
3.00
2.61
2.38
2.22
2.11
2.02
1.95
1.89
1.84
Repeated-Measures ANOVA
dfnumerator
F table for α=0.05
dferror
reject H0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
22
24
26
28
30
40
50
60
120
200
500
1000
01:830:200:10-13 Spring 2013
1
2
3
4
5
6
7
8
9
10
161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88
18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40
10.13
9.55
9.28
9.12
9.01
8.94
8.89
8.85
8.81
8.79
7.71
6.94
6.59
6.39
6.26
6.16
6.09
6.04
6.00
5.96
6.61
5.79
5.41
5.19
5.05
4.95
4.88
4.82
4.77
4.74
5.99
5.14
4.76
4.53
4.39
4.28
4.21
4.15
4.10
4.06
5.59
4.74
4.35
4.12
3.97
3.87
3.79
3.73
3.68
3.64
5.32
4.46
4.07
3.84
3.69
3.58
3.50
3.44
3.39
3.35
5.12
4.26
3.86
3.63
3.48
3.37
3.29
3.23
3.18
3.14
4.96
4.10
3.71
3.48
3.33
3.22
3.14
3.07
3.02
2.98
4.84
3.98
3.59
3.36
3.20
3.09
3.01
2.95
2.90
2.85
4.75
3.89
3.49
3.26
3.11
3.00
2.91
2.85
2.80
2.75
4.67
3.81
3.41
3.18
3.03
2.92
2.83
2.77
2.71
2.67
4.60
3.74
3.34
3.11
2.96
2.85
2.76
2.70
2.65
2.60
4.54
3.68
3.29
3.06
2.90
2.79
2.71
2.64
2.59
2.54
4.49
3.63
3.24
3.01
2.85
2.74
2.66
2.59
2.54
2.49
4.45
3.59
3.20
2.96
2.81
2.70
2.61
2.55
2.49
2.45
4.41
3.55
3.16
2.93
2.77
2.66
2.58
2.51
2.46
2.41
4.38
3.52
3.13
2.90
2.74
2.63
2.54
2.48
2.42
2.38
4.35
3.49
3.10
2.87
2.71
2.60
2.51
2.45
2.39
2.35
4.30
3.44
3.05
2.82
2.66
2.55
2.46
2.40
2.34
2.30
4.26
3.40
3.01
2.78
2.62
2.51
2.42
2.36
2.30
2.25
4.23
3.37
2.98
2.74
2.59
2.47
2.39
2.32
2.27
2.22
4.20
3.34
2.95
2.71
2.56
2.45
2.36
2.29
2.24
2.19
4.17
3.32
2.92
2.69
2.53
2.42
2.33
2.27
2.21
2.16
4.08
3.23
2.84
2.61
2.45
2.34
2.25
2.18
2.12
2.08
4.03
3.18
2.79
2.56
2.40
2.29
2.20
2.13
2.07
2.03
4.00
3.15
2.76
2.53
2.37
2.25
2.17
2.10
2.04
1.99
3.92
3.07
2.68
2.45
2.29
2.18
2.09
2.02
1.96
1.91
3.89
3.04
2.65
2.42
2.26
2.14
2.06
1.98
1.93
1.88
3.86
3.01
2.62
2.39
2.23
2.12
2.03
1.96
1.90
1.85
3.85
3.00
2.61
2.38
2.22
2.11
2.02
1.95
1.89
1.84
Repeated-Measures ANOVA
Set up a summary ANOVA table:
Thousands of Steps
Student
A
B
C
D
E
F
X1
6
4
5
1
0
2
X2
8
5
5
2
1
3
X3
10
6
5
3
2
4
𝑀𝑔𝑟𝑜𝑢𝑝
3
4
5
SStotal  112
𝑀𝑠𝑢𝑏𝑗
8
5
5
2
1
3
Source
df
SS
MS
F
Between groups
2
12
6.0
15.0
Within groups
15
100
subjects
5
96
error
10
4
17
112
Total
0.4
MT  4
8. Compare computed F statistic with Fcrit
and make a decision
Fcrit  4.1
15  4.1, reject H 0
Conclusion: Giving students pedometers influences the
amount of walking that they do.
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Compute MSbetween , MSerror & F
SSbetween 12
 6
dfbetween
2
MSbetween 
SSbetween 12
 6
dfbetween
2
SSerror 4
  0.4
df error 10
MSwithin 
SSwithin 100

 6.67
df within
15
MSbetween 
MSerror 
Note that if we had computed F without
accounting for the repeated-measures
design of the study:
F  dfbetween , df error  
MSbetween
MSerror
F  dfbetween , df within  
F  2,10  
6
 15
0.4
F  2,17  
01:830:200:10-13 Spring 2013
MSbetween
MS within
6
 0.90
6.67
Repeated-Measures ANOVA
Effect Size for the Repeated-Measures ANOVA
• For repeated-measures ANOVAs, effect sizes are usually
indicated using partial eta-squared  p2 
– Partial eta-squared is like the eta-squared that we use to measure effect
sizes in the independent-samples ANOVA, except that it removes the
effect of between-subjects variability
 p2 
variability explained by treatment effect
total variability (- subject variability)
For our example:
 p2 
SSbetween
SSbetween
12.0


 0.75
SStotal  SS subject SSbetween  SSerror 16.0
01:830:200:10-13 Spring 2013
Repeated-Measures ANOVA
Post hoc test: Example (Fisher’s LSD)
ANOVA Summary Table
Source
df
SS
MS
F
M1  3
Between groups
2
12
6.0
15.0
M2  4
Within groups
15
100
M3  5
subjects
5
96
n1  n2  n3  6
error
10
4
17
112
Total
0.4
Let’s do all possible comparisons: {1,2},{1,3},{2,3}
t-statistic for Fisher’s LSD test
when comparing {A,B}:
t  df error  
MA  MB
M  MB
 A
MSerror MSerror
2MSerror

n
n
n
01:830:200:10-13 Spring 2013
Again, note that the denominator is the same for
all comparisons:
t 10  
MA  MB
2  0.4 
6

MA  MB
0.365
Repeated-Measures ANOVA
t-Distribution Table
α
t
One-tailed test
α/2
α/2
-t
t
Two-tailed test
01:830:200:10-13 Spring 2013
df
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
100
0.25
0.2
0.5
1.000
0.816
0.765
0.741
0.727
0.718
0.711
0.706
0.703
0.700
0.697
0.695
0.694
0.692
0.691
0.690
0.689
0.688
0.688
0.687
0.686
0.686
0.685
0.685
0.684
0.684
0.684
0.683
0.683
0.683
0.681
0.679
0.677
0.4
1.376
1.061
0.978
0.941
0.920
0.906
0.896
0.889
0.883
0.879
0.876
0.873
0.870
0.868
0.866
0.865
0.863
0.862
0.861
0.860
0.859
0.858
0.858
0.857
0.856
0.856
0.855
0.855
0.854
0.854
0.851
0.849
0.845
Level of significance for one-tailed test
0.15
0.1
0.05
0.025
0.01
Level of significance for two-tailed test
0.3
0.2
0.1
0.05
0.02
1.963
3.078
6.314
12.706 31.821
1.386
1.886
2.920
4.303
6.965
1.250
1.638
2.353
3.182
4.541
1.190
1.533
2.132
2.776
3.747
1.156
1.476
2.015
2.571
3.365
1.134
1.440
1.943
2.447
3.143
1.119
1.415
1.895
2.365
2.998
1.108
1.397
1.860
2.306
2.896
1.100
1.383
1.833
2.262
2.821
1.093
1.372
1.812
2.228
2.764
1.088
1.363
1.796
2.201
2.718
1.083
1.356
1.782
2.179
2.681
1.079
1.350
1.771
2.160
2.650
1.076
1.345
1.761
2.145
2.624
1.074
1.341
1.753
2.131
2.602
1.071
1.337
1.746
2.120
2.583
1.069
1.333
1.740
2.110
2.567
1.067
1.330
1.734
2.101
2.552
1.066
1.328
1.729
2.093
2.539
1.064
1.325
1.725
2.086
2.528
1.063
1.323
1.721
2.080
2.518
1.061
1.321
1.717
2.074
2.508
1.060
1.319
1.714
2.069
2.500
1.059
1.318
1.711
2.064
2.492
1.058
1.316
1.708
2.060
2.485
1.058
1.315
1.706
2.056
2.479
1.057
1.314
1.703
2.052
2.473
1.056
1.313
1.701
2.048
2.467
1.055
1.311
1.699
2.045
2.462
1.055
1.310
1.697
2.042
2.457
1.050
1.303
1.684
2.021
2.423
1.047
1.299
1.676
2.009
2.403
1.042
1.290
1.660
1.984
2.364
0.005
0.0005
0.01
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.704
2.678
2.626
0.001
636.619
31.599
12.924
8.610
6.869
5.959
5.408
5.041
4.781
4.587
4.437
4.318
4.221
4.140
4.073
4.015
3.965
3.922
3.883
3.850
3.819
3.792
3.768
3.745
3.725
3.707
3.690
3.674
3.659
3.646
3.551
3.496
3.390
Repeated-Measures ANOVA
Post hoc tests: Example (Fisher’s LSD)
M1  3
M2  4
M3  5
tcrit  2.228
Apply the t-test formula to all comparisons:
{1,2}
M1  M 2
0.365
3 4

0.365
1

 2.74
0.365
t 10  
01:830:200:10-13 Spring 2013
{1,3}
M1  M 3
0.365
35

0.365
2

 5.48
0.365
t 10  
{2,3}
M2  M3
0.365
45

0.365
1

 2.74
0.365
t 10  