Transcript Document

Repeated Measures Design
Group 5: Jian Wang, Xi Zhang, Yan Cao, Qichao Sun,
Jixiang Zhang, Zishan Chen, Xing Peng, Xiao Liu
Introduction to repeated
measures design
Jian Wang
Review of paired sample t-test
• A paired sample t-test is used to determine whether
there is a significant difference between the average
values of the same measurement made under two
different conditions. The usual null hypothesis is that
the difference in the mean values is zero.
• －－－repeated measures design is actually an
extension of paired sample t－test or we can say
paired t test is a special case in the repeated measures
design(which involved only two related measures)
What is repeated measures design
A repeated-measures design is one in
which multiple, or repeated,
measurements are made on each
experimental unit under different
conditions.
Simple Example
• Students were asked to rate their stress on a
50 point scale in the week before, the week of,
or the week after their midterm exam.
Advantages－why use repeated
measures design
• Require fewer participants－－This design is
economical because each member is measured
under all conditions.
• Eliminating individual differences－－ collecting data
from the same participants under repeated
conditions so it reduces individual difference and the
test becomes more powerful.
• Longitudinal analysis—Repeated measure design is
especially good for researchers to monitor how
participants change over time.
Disadvantage
• Carry－over effect：an effect that “carries over”
from one experimental condition to another when
subjects perform in more than one condition.
One-Way Repeated Measures
ANOVA
Compared with One-Way ANOVA
• One-Way ANOVA: we look at differences
between different samples that come from
different groups within a single factor.
– That is, 1 factor, k levels  k separate groups are
compared.
• One-Way RM ANOVA: we can answer the
same question looking at just 1 sample that is
exposed to different manipulations within a
factor.
– That is, 1 factor, k levels  only 1 group is
compared across k conditions.
• For normal One-Way ANOVA, we use a different groups
to test these a levels. So the observations between
treatments are independent.
• But for One-way RM ANOVA, we use only one group to
test these a levels. So the observations between
treatments are dependent.
Compared with One-Way ANOVA
• MODEL:
Yij= μi+εij
μi: the mean of the ith treatment.
εij: the random error which follows N(0,σow2)
Compared with One-Way ANOVA
• MODEL:
Yij= μi+Bj+ε’ij
μi: the mean of the ith treatment.
Bj: a parameter associated with jth subject
which follows N(0,σB2).
ε’ij: the random error which follows
N(0,σRMOW2) and is independent with Bj
Compared with One-Way ANOVA
• One-Way ANOVA TABLE
Source
DF
Treatment
a-1
Error
N-a
Total
N-1
SS
MS
σOW2
Compared with One-Way ANOVA
• One-Way RM ANOVA TABLE
Source
DF
Treatment
a-1
Subject
n-1
Error
(a-1)(n-1)
Total
N-1
SS
MS
σRMOW2
Compared with One-Way ANOVA
• Partition of Sum of Square
Standard ANOVA
SSTOTAL
SSTOTAL
SStreatment
Repeated Measures ANOVA
SSerror
SSbetween subject
SSwithin subject
SStreatment
SSerror
Compared with paired sample t-test
• Example (2 treatments)
Compared with paired sample t-test
• Hypothesis
• T-TEST
Compared with paired sample t-test
• T-TEST
Since
So
Compared with paired sample t-test
• F-TEST
Compared with paired sample t-test
• F-TEST
Compared with paired sample t-test
•
The paired sample t-test is just the One-way
Repeated measures design.
•
Repeated measures design are considered an
extension of the paired sample t-test when
comparisons between more than two
repeated measures are needed.
one-way repeated measure ANOVA
Repeated measures one-way ANOVA compares the means of
two or more matched groups.
When to use?
repeated measures ANOVA is used when all members of a
random sample are measured under a number of different
conditions
one-way repeated measure ANOVA
Model
Yij = μi +Sj+εij
μi = The fixed effect .
Sj= The random effect of subject j .
εij = The random error independent of Sj .
one-way repeated measure ANOVA
• Repeated Measures ANOVA Table
SS
Df
MS
( 𝑎𝑖𝑗 )2 𝑇 2
−
𝑠
𝑁
( 𝑎𝑖𝑗 )2
2
𝑎𝑖𝑗 −
𝑠
a-1
MSA=𝑎−1
s-1
-Error
( 𝑆𝐼 )2 𝑇 2
−
𝑎
𝑁
𝑆𝑆𝑤𝑖𝑛𝑡𝑖𝑛 − 𝑆𝑆𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠
Total
𝑆𝑆𝑏𝑒𝑡𝑒𝑤𝑤𝑛 + 𝑆𝑆𝑤𝑖𝑡ℎ𝑖𝑛𝑔
N-1
Between
Within
-Subjects
F
𝑆𝑆𝐴
N-a
MSE=
𝑆𝑆𝐸
(𝑎−1)(𝑠−1)
𝑀𝑆𝐴
F=𝑀𝑆𝐸
one-way repeated measure ANOVA
Example:
• We have four drugs (1,2,3 and 4) that relieve pain. Each subject
is given each of the four drugs. The subject’s pain tolerance is
then measured. Enough time is allowed to pass between
successive drug administrations so that we can be sure there’s
no residual effect from the previous drug. Are there any
difference between the four drugs using significant level 𝛼
=0.05?data from the pain experiment:
SUBJECT
DRUG1
DRUG2
DRUG3
DRUG41
1
5
9
6
11
2
7
12
8
9
3
11
12
10
14
4
3
8
5
8
𝐻0:𝜇1=𝜇2=𝜇3=𝜇4
𝐻𝑎:𝐻0 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑙𝑙 𝑒𝑞𝑢𝑎𝑙
one-way repeated measure ANOVA
SUBJECT
DRUG1
DRUG2
DRUG3
DRUG41
1
5
9
6
11
2
7
12
8
9
3
11
12
10
14
4
3
8
5
8
Degrees of Freedom
Total N=16
Treatment I=4
𝑑𝑓𝑏𝑒𝑡𝑤𝑒𝑒𝑛 = 𝐼 − 1 = 4 − 1 = 3
𝑑𝑓𝑊𝑖𝑡ℎ𝑖𝑛 = 𝑑𝑓𝐸𝑟𝑟𝑜𝑟 + 𝑑𝑓𝑠𝑢𝑏𝑗𝑒𝑐𝑡 = 3 − 9 = 12
𝑑𝑓𝑠𝑢𝑏𝑗𝑒𝑐𝑡 = 𝑅 − 1 = 4 − 1 = 3
𝑑𝑓𝐸𝑟𝑟𝑜𝑟 = 𝐼 − 1 ∗ (𝑅 − 1)=3 * 3=9
𝑑𝑓𝑡𝑜𝑡𝑎𝑙 = 𝑁 − 1 = 16 − 1 = 15
Subject R=4
one-way repeated measure ANOVA
SUBJECT
DRUG1
DRUG2
DRUG3
DRUG4
Y.j
1
5
9
6
11
7.75
2
7
12
8
9
9
3
11
12
10
14
11.75
4
3
8
5
8
6
Yi.
6.5
10.25
7.25
10.5
Y..=8.625
SS Treatment =R× ∑(Yi. − Y..)2= 4 × 12.5625=50.25
SS Subject = I × ∑(Y.j − Y..)2= 4 × 17.5625 = 70.25
SS Total = ∑ ∑(Yij − Y..) 2=133.75
SS Error = ∑ ∑(Yij − Yi. − Y.j + Y..)2 = SS Total − SS Treatment − SS Subject
= 13.25
one-way repeated measure ANOVA
SUBJECT
DRUG1
DRUG2
DRUG3
DRUG4
Y.j
1
5
9
6
11
7.75
2
7
12
8
9
9
3
11
12
10
14
11.75
4
3
8
5
8
6
Yi.
6.5
10.25
7.25
10.5
Y..=8.625
Test Statistic:
𝑀𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛
𝐹0 =
=
𝑀𝑆𝑒𝑟𝑟𝑜𝑟
(𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 )
𝑆𝑆𝑒𝑟𝑟𝑜𝑟
𝑑𝑓𝑏𝑒𝑡𝑤𝑒𝑒𝑛
𝑑𝑓𝑒𝑟𝑟𝑜𝑒
50.25
=
13.25
3 ≃ 11.3774
9
Critical Region:
𝐼𝑓 𝐹0 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝐹 3,9,0.025 = 5.08, 𝑟𝑒𝑗𝑒𝑐𝑡 𝑛𝑢𝑙𝑙 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠.
Now, the 𝐹0 = 11.3774 > 5.08. 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, 𝑤𝑒 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻0 𝑡ℎ𝑎𝑡 𝑚𝑒𝑎𝑛𝑠
𝑡ℎ𝑒 𝑓𝑜𝑢𝑟 𝑑𝑟𝑢𝑔𝑠 𝑎𝑟𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑢𝑠𝑖𝑛𝑔 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑡 𝑙𝑒𝑣𝑒𝑙 𝛼 = 0.05.
one-way repeated measure ANOVA
SAS Code
data pain;
input subj@;
do drug = 1 to 4 ;
input pain @;
output;
end;
datalines;
1 5 9 6 11
2 7 12 8 9
3 11 12 10 14
43858
;
proc anova data = pain;
title'one-way Repeatede
Measures ANOVA';
class subj drug;
model pain = subj drug;
means drug/duncan;
run;
one-way repeated measure ANOVA
• SAS output:
Two –factor experiments with
a repeated measure on one factor
Qichao Sun
Why do we need the two-factor ANOVA with a
repeated measure on one factor?
1）mean scores in each treatment group change over two or
more time points.
2）under two or more different conditions, there are
differences in mean scores in one treatment group.
Example
• To investigate the effect of the drug, which needs several hours to have
effects on patients. There are four groups with different dose of drug.
Obviously, we cannot be sure if the effects are caused by the dose
(maybe time has an effect). So we add the other factor ‘time’, and
measure the observation before and after the treatment.
Control Group
(placebo)
Treatment Group 2
( 1.00mg)
Treatment Group 3
( 1.25mg)
Treatment Group 4
( 1.50mg)
Subject
Before
After
1
80
83
2
85
86
3
83
88
4
82
94
5
87
93
6
84
98
7
85
102
8
87
98
9
83
97
10
86
107
11
82
103
12
88
106
• In this example, we have two factors, the dose of the drug and the
time. Here we call the dose Factor A and the time Factor B. And we
use the same subjects in one level of Factor A before and after the
treatment. So Factor B ‘time’ is the factor with repeated measure.
• This setting is two factor ANOVA with repeated measure on one
factor. The appropriate error term for the test of the dose is
subject|dose. The appropriate error term for time and time*dose is
time*subject|dose (which is the residual error since we do not
include the term in the model).
Model
𝑌𝑖𝑗𝑘 = 𝜇 + 𝛼𝑖 + 𝛽𝑗 + 𝛼𝛽
𝑖𝑗
+ 𝛾𝑘(𝑖) + 𝜀 𝑖𝑗𝑘
𝑌𝑖𝑗𝑘
The observation of the kth subject in the ith level of the dose before the treatment (j=1) or after
the treatment (j=2).
𝜇
Overall mean
𝛼𝑖
Main fixed effect of the dose in the ith level (
𝛽𝑗
Main fixed effect of the time in the jth time point (
𝛼𝛽
𝑖𝑗
𝐼
𝑖=1 𝛼𝑖
= 0. 𝑖 = 1,2,3,4 )
𝐽
𝑗=1 𝛽𝑗
=0)
fixed Interaction effect of dose in the ith level and the time in jth time point (
𝛼𝛽
𝛾𝑘(𝑖)
random effect of the kth subject nested in dose from ith level, which follows 𝑁(0, 𝜎𝑏2 )
𝜀 𝑖𝑗𝑘
Error term, which follows 𝑁(0, 𝜎 2 )
𝑖𝑗
= 0)
Hypothesis
• Main effect of Factor A ‘dose’
𝐻0A : 𝛼1 = 𝛼2 = 𝛼3 = ⋯ = 𝛼I
•
𝐻0𝑎 :
𝐻0 𝑖𝑠 𝑛𝑜𝑡 𝑡𝑟𝑢𝑒
• Main effect of factor B ‘time’
𝐻0B : 𝛽1 = 𝛽2 = 𝛽3 = ⋯ = 𝛽𝐽
•
𝐻0𝑎 :
𝐻0 𝑖𝑠 𝑛𝑜𝑡 𝑡𝑟𝑢𝑒
• Interaction effect of A and B (dose*time)
𝐻0AB : 𝛼𝛽 11 = 𝛼𝛽 12 = 𝛼𝛽 13 = ⋯ = 𝛼𝛽
•
𝐻0𝑎 :
𝐻0 𝑖𝑠 𝑛𝑜𝑡 𝑡𝑟𝑢𝑒
𝐼𝐽
ANOVA Table
Source
Factor A
Dose
Factor B
Time
AB
Interaction
Time*Dose
Subjects
within A
DF
SS
I-1=3
SSA
MS
SSA/(I-1)
J-1=1
SSB
SSB/(J-1)
(I-1)(J-1)=3
SSAB
SSAB/(I-1)(J-1)
I(K-1)=8
SSWA
SSWA/I(K-1)
I(J-1)(K-1)=8
IJK-1
SSE
SST
SSE/I(J-1)(K-1)
subject|d
ose
Error
Total
, where I=4, J=2, K=3
F
MSA
MSWA
MSB
MSE
MSAB
MSE
Sum of Square
• Dose:
• Time point:
SSA=JK (Yi.. − Y)2
SSB=IK (Y.j. − Y)2
• Dose×subject: SSAB = K
(Yij. − Y.j. − Yi.. + Y)2
• Error due to subjects within Dose (subject|dose):
• SSWA=J
(Yi.k − Yi.. )2
• Error: SSE=
(Yijk − Yij. − Yi.k + Yi.. )2
• SST=
Yijk − Y
2
Test statistic
• 𝐹𝐴 =
𝑀𝑆𝐴
~𝐹 𝐼−1 ,𝐼(𝐾−1)
𝑀𝑆𝑊𝐴
• If FA > F
• 𝐹𝐵 =
I−1 ,I K−1 ,α
, Factor A ‘dose’ is significant.
𝑀𝑆𝐵
~𝐹 𝐽−1 ,𝐼(𝐽−1)(𝐾−1)
𝑀𝑆𝐸
• If FB > 𝐹 𝐽−1 ,𝐼 𝐽−1 𝐾−1 ,𝛼 , Factor B ‘time’ is significant.
MSAB
• FAB =
MSE ~𝐹 𝐼−1
𝐽−1 ,𝐼(𝐽−1)(𝐾−1)
• If FAB > 𝐹 𝐼−1 𝐽−1 ,𝐼 𝐽−1 𝐾−1 ,𝛼 , Factor AB ‘dose*time’ is
significant.
The SAS code is following.
data drug;
input subj dose $ before after;
datalines;
1 C 80 83
2 C 85 86
3 C 83 88
4 T2 82 94
5 T2 87 93
6 T2 84 98
7 T3 85 102
8 T3 87 98
9 T3 83 97
10 T4 86 107
11 T4 82 103
12 T4 88 106
;
proc anova data= drug;
class dose;
model before after= dose / nouni;
repeated time 2 (0 1);
means dose;
run;
SAS Output for repeated statement
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time Effect
H = Anova SSCP Matrix for time
E = Error SSCP Matrix
S=1 M=-0.5 N=3
Statistic
Value
F Value
Num DF
Den DF
Pr > F
Wilks' Lambda
0.03764883
204.49
1
8
<.0001
Pillai's Trace
0.96235117
204.49
1
8
<.0001
Hotelling-Lawley
Trace
25.56125000
204.49
1
8
<.0001
Roy's Greatest
Root
25.56125000
204.49
1
8
<.0001
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time*dose Effect
H = Anova SSCP Matrix for time*dose
E = Error SSCP Matrix
S=1 M=0.5 N=3
Statistic
Value
F Value
Num DF
Den DF
Pr > F
Wilks' Lambda
0.12847278
18.09
3
8
0.0006
Pillai's Trace
0.87152722
18.09
3
8
0.0006
Hotelling-Lawley
Trace
6.78375000
18.09
3
8
0.0006
Roy's Greatest
Root
6.78375000
18.09
3
8
0.0006
How to choose Multivariate statistics?
• The output shows that there are four F statistics:
• Wilks’ Lambda
• Pillai’s Trace
• Hotelling- Lawley Trace
• Roy’s Greatest Root
• If there are only very small differences among the P-values, we can
either of them or choose Wilks’ Lambda which is often appropriate.
• If not, find consultant.
The ANOVA Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects
Source
DF
Anova SS
Mean Square
F Value
Pr > F
dose
3
397.4583333
132.4861111
15.59
0.0011
Error
8
68.0000000
8.5000000
The ANOVA Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects
Source
DF
Anova SS
Mean Square
F Value
Pr > F
time
1
852.0416667
852.0416667
204.49
<.0001
time*dose
3
226.1250000
75.3750000
18.09
0.0006
Error(time)
8
33.3333333
4.1666667
Therefore, we can conclude that the dose, Time effects and their interaction are
significant at 𝛼 =0.05 .
The Other method without using
repeated statement
• Two-factor ANOVA without using the REPEATED statement
• SAS code:
data twoway;
set drug;
/* ‘SET’ causes observations to be read from the original data
set*/
length time $ 7;
/*set the length of variable ‘Time’*/
time = ‘before’;
/*creat a new variable ‘Time’ and set the value of Time to
‘before’*/
score =before;
/*creat a new variable ‘Score’ equl to the before value*/
output;
/*the first observation in data set TWOWAY*/
time = ‘after’;
score = after;
output;
keep subj dose time score;
run;
proc anova data=twoway;
class subj dose time;
model score = dose subj(dose) time dose*time time*subj(dose);
/*In the model statement, all sources of variation are included, so SSE
will be 0.*/
means dose|time;
test H=dose E=subj(dose);
test H=time dose*time E=time*subj(dose);
/*the error term for dose is subj(dose), and the error term for time
and dose*time is time*subj(dose).*/
run;
Output for the method without repeated statement
Class Level Information
Class
Levels
Values
subj
12
1234567
8 9 10 11 12
dose
4
C T2 T3 T4
time
2
after before
Source
DF
Sum of Squares Mean
Square
F Value Pr > F
Model
23
1576.958333
68.563406
.
Error
0
0.000000
.
Corrected 23
Total
1576.958333
Model statement contains all sources of variation
about the grand mean. So SSE is 0.
Source
DF
Anova SS
Mean Square F Value
Pr > F
dose
3
397.4583333
132.4861111
.
.
subj(dose)
8
68.0000000
8.5000000
.
.
time
1
852.0416667
852.0416667
.
.
dose*time
3
226.1250000
75.3750000
.
.
33.3333333
4.1666667
.
.
subj*time(dose) 8
.
Level of
dose
N
C
score
Mean
Std Dev
6
84.1666667
2.7868740
T2
6
89.6666667
6.2822501
T3
6
92.0000000
7.9498428
T4
6
95.3333333
11.2011904
Level of
group
Level of
time
N
C
post
C
score
Mean
Std Dev
3
85.6666667
2.51661148
pre
3
82.6666667
2.51661148
T
post
3
95.0000000
2.64575131
T
pre
3
84.3333333
2.51661148
The effect of the dose, the time and their
interaction are significant at 𝛼 =0.05 .
Tests of Hypotheses Using the Anova MS for subj(dose) as an Error Term
Source
DF
Anova SS
Mean Square
F Value
Pr > F
dose
3
397.4583333
132.4861111
15.59
0.0011
Tests of Hypotheses Using the Anova MS for subj*time(dose) as an Error Term
Source
DF
Anova SS
Mean Square
F Value
Pr > F
time
1
852.0416667
852.0416667
204.49
<.0001
dose*time
3
226.1250000
75.3750000
18.09
0.0006
Two-Factor ANOVA with Repeated
Measures on both Factors
Jixiang Zhang
7/17/2015
The model is
Yijk     j   k （ ）jk   i
  ij  ik   ijk
This design is similar to the previous design
except that each subject is measured under all
levels of both factor.
•𝑌𝑖𝑗𝑘 :The observation taken at the kth time point
from jth group in the ith subject
•𝛼𝑗 :Main effect of factor A subject to   j  0
•𝛽𝑘 :Main effect of factor A subject to   k  0
•(𝛼𝛽)𝑗𝑘 :Interaction effect of factor A and B
2
•𝛿𝑖 :Main effect of subjects~ N (0,  i )
• 𝛾𝑖𝑗 :Interaction effect of subjects and factor A~ N (0,  ij2 )
2
N
(
0
,

• 𝜂𝑖𝑘 :Interaction effect of subjects and factor B~
ik )
• 𝜀𝑖𝑗𝑘 :Error term~ N (0,  2 )
ANOVA TABLE
Source
DF
SS
MS
F
Subjects
n-1
SSS
SSS/I-1
MSS/MSE
Factor A
a-1
SSA
SSA/(a-1)
MSA/MSA*S ～ F(a-1),(n1)(a-1)
Factor B
b-1
SSB
SSB/(b-1)
MSB/MSB*S ～ F(b-1),(n1)(b-1)
SSAB/((a-1)(b- MSAB/MSE～F(a-1)(b-1),(n1))
1)(a-1)(b-1)
AB Interaction
(a-1)(b-1)
SSAB
A*Subjects
(n-1)(a-1)
SSA*S SSWA/((n-1)a) SSA*S/MSE
F(a-1)(n-1),(n-1)(a-1)(b-1)
B*Subjects
(n-1)(b-1)
SSB*S SSWB/((n-1)b) SSA*S/MSE
F(n-1)(b-1),(n-1)(a-1)(b-1)
Error
Total
(n-a)(a-1)(b- SSE
1)
nab-1
SST
SSE/((n-1)(a1)(b-1))
Example:
A group of subjects is treated in the
morning and afternoon of two different days.
On one of the days, the subjects receive a
strong sleeping aid the night before the
experiment is to be conducted; on the other,
a placebo.
A diagram of the experiment
Treatment
Time
A.M.
P.M
Control
Subject
1
2
3
1
2
3
Drug
Reaction Subject
65
1
72
2
90
3
55
1
64
2
80
3
Reaction
70
78
97
60
68
85
•We would like to see if the drug had any
effect on the reaction time and if the effect
was the same for the whole day.
•We can use the AM/PM measurements on
the control day as a comparison for the
AM/PM changes on the drug days.
•Since each subject is measured under all
levels of treatment(PLACEBO or DRUG)
and TIME(AM/PM), we can treat this
experiment as a SUBJ by TREATMENT
by TIME factorial design. However, we
must specify the error terms to test our
hypotheses.
•In any design where all factors are
repeated, such as this one, we can treat the
SUBJ variable as being crossed by all
other factors.
•However, by including the SUBJ term in
our model, the error term will be zero.
•The error terms to test each hypothesis are
simpler to remember: For factor X, the
error term is SUBJ*X.
SAS CODE
data repeat;
proc anova data=repeat;
input react1-react4;
model react1-react4=
datalines;
/nouni;
65 70 55 60
72 78 64 68
repeated time 2, treat 2
90 97 80 85
/nom;
;
run;
Run;
A portion of output from SAS
Interpretation
• According to the observed p-values, The drug
increase reaction time(F=256.00,P=.0039)
• Reaction time is longer in the morning compared to
the afternoon(F=300.00,P=.0033)
• The interaction of treat and time is not
significant(F=4.00,P=0.1835)
Three-factor
Experiments
with a repeated
measure
Zishan Chen
ANOVA Table
Source
d.f
SS
MS
Factor A
a-1
SSA
MSA=SSA/(a-1)
Factor B
b-1
SSB
MSB=SSB/(b-1)
Factor
C(repeated)
c-1
SSC
MSC=SSC/(c-1)
Subjects(within
AB)
ab(i-1)
SSWAB
MSWAB= SSWAB/(ab(i-1))
AB interaction
(a-1)(b-1)
SSAB
MSAB =SSAB/(a-1)(b-1)
AC interaction
(a-1)(c-1)
SSAC
MSAC = SSAC/(a-1)(c-1)
BC interaction
(b-1)(c-1)
SSBC
MSBC = SSBC/(b-1)(c-1)
ABC interaction
(a-1)(b-1) (c-1) SSABC
MSABC = SSABC/(a-1)(b-1)
(c-1)
Error
ab(c-1)(i-1)
SSE
MSE = SSE/ab(c-1)(i-1)
Total
Abci-1
SST
Example
• There is a marketing experiment.
• Male and female subjects are offered one
of three different brands of coffee: A, B, C.
• Each brand is tasted twice. After breakfast
and After dinner.
• The preference of each brand is measured
on a scale from 1 to 10
The results of the experiment
Bran
d
A
SUB
J
GEN
DER
B
C
Brkfs Dinne SU
r
BJ
Brkfst Dinne SU
r
BJ
Brkfst Dinne
r
M 1
7
8
7
4
6
13
8
9
2
6
7
8
3
5
14
6
9
3
6
8
9
3
5
15
5
8
4
5
7
10
3
4
16
6
9
5
4
7
11
4
4
17
6
9
6
4
6
12
2
3
18
7
8
F
BRAND and GENDER are crossed
MEAL: Repeated Measure Factor
The SAS code
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
data coffee;
input subj brand $ gender $ score_B score_D;
datalines;
1AM78
2AM67
3AM68
4AF57
5AF47
6AF46
7BM46
8BM35
9BM35
10 B F 3 4
11 B F 4 4
12 B F 2 3
13 C M 8 9
14 C M 6 9
15 C M 5 8
16 C F 6 9
17 C F 6 9
18 C F 7 8
;
THE SAS CODE
“nouni” means that we
do not want univariate
statistics for score_b
and score_d
proc anova data=coffee;
title "coffee study";
class brand gender;
model score_b score_d = brand | gender / nouni;
repeated meal;
means brand | gender;
run;
“brand | gender” meams
brand + gender+brand
*gender
Selected output 1
Selected output 2
Selected output 3
Selected output 4
Three-Factor
Experiments With
Repeated Measures
on Two Factors
To test these ideas, the
following experiment was
devised:
• A group of high- and low-SES children is selected for the
experiment. Their reading comprehension is tested each
spring and fall for three consecutive years. A Diagram of
the design is shown below:
Notice that each subject is measured each spring and fall of each year so
that the variables SEASON and YEAR are both repeated measures
factors.
To analyze this experiment, we will use the SAS program:
the REPEATED statement of PROC ANOVA:
• DATA READ
•
INPUT SUBJ SES $ READ1 READ2 READ3 READ4 READ5
READ6;
•
LABEL READ1 = ' SPRING YR 1 '
•
READ2 = ' FALL YR 1 '
•
READ3 = ' SPRING YR 2 '
•
READ4 = ' FALL YR 2 '
•
READ5 = ' SPRING YR 3 '
•
READ6 = ' FALL YR 3 ';
DATALINES;
1 HIGH 61 50 60 55 59 62
2 HIGH 64 55 62 57 63 63
3 HIGH 59 49 58 52 60 58
4 HIGH 63 59 65 64 67 70
5 HIGH 62 51 61 56 60 63
6 LOW 57 42 56 46 54 50
7 LOW 61 47 58 48 59 55
8 LOW 55 40 55 46 57 52
9 LOW 59 44 61 50 63 60
10 LOW 58 44 56 49 55 49
;
PROC ANOVA DATA=READ;
TITLE "READING COMPREHENSION ANALYSIS";
CLASS SES;
MODEL READ1-READ6 = SES / NOUNI;
REPEATED YEAR 3, SEASON 2;
MEANS SES;
RUN;
• Since the REPEATED statement is confusing when we have
more than one repeated factor, it is important for you to know
how to determine the order of the factor names. Look at the
REPEATED statement in this example:
REPEATED YEAR 3, SEASON 2;
This statement instructs the ANOVA procedure to choose the first
level of YEAR(1), then loop through two levels of
SEASON(SPRING FALL), then return to the next level of
YEAR(2), followed by two levels of SEASON, etc.
• High-SES student have higher reading comprehension scores
than low-SES students (F=13.54, p=0.0062).
• Reading comprehension increases with each year (F=26.91,
p=0.0001).
• Students had higher reading comprehension scores in the spring
compared to the following fall (F=224.82, p=0.0001)
• The "slippage" was greater for the low-SES students (there was a
significant SES*SEASON interaction [F=37.05, p=0.0003}).
• "Slippage" decreases as the students get older (YEAR*SEASON
is significant [F=112.95, p=0.0001]).
Independent Measures Design vs.
Repeated Measures Design
in ANOVA
• Experimental design is a method of assigning
observations to different groups in an
experiment.
• Independent-measures design means that
there is a separate sample for each of the
treatments being compared. Assume the data
are independent.
• Repeated-measures design means that the
same sample is tested in all of the different
treatments. Data are not independent.
• Independent measures ANOVA: an extension of the
pooled variance t-test
• Repeated Measures ANOVA: an extension of the
paired sample t-test
• A t-test has more odds of committing an error the
more means are used. Thus, the t-test is used when
determining if two averages or means are the same or
different. The ANOVA is preferred if comparing three
or more averages or means.
Independent Measures Design
Advantage:
• Avoid order effects
Disadvantages:
• Much more time-consuming;
• The difference between observations such as
age, sex, etc. may interfere.
Repeated Measures Design
Advantages:
• Limited number of subjects
• Less variability
• Longitudinal analysis
Disadvantage:
• Order effects
Repeated Measures Design
Solution: Counterbalancing
Counterbalancing means alternating the order
of the conditions. It helps researchers to make
sure that practice effects are distributed equally
across the conditions.
Condition 1
Condition 2 Remarks
Group A
Group A1
Group A2
A performs Condition 1 first
Group B
Group B2
Group B1
B performs Condition 1 first
Acknowledgements
• http://en.wikipedia.org/wiki/Repeated_measu
res_design
• http://ira06.wordpress.com/2012/03/10/inde
pendent-measures-design-vs-repeatedmeasures-design-in-anova/
• APPLIED STATISTICS AND THE SAS
PROGRAMMING LANGUAGE