AMS 572 Group 5 Outline Jia Chen Introduction of repeated measures ANOVA Chewei Lu Oneway repeated measures Wei Xi Twofactor repeated measures Tomoaki Sakamoto Threefactor repeated measures ID: 144224
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Slide1
Repeated Measure Design of ANOVA
AMS 572 Group 5Slide2
Outline
Jia
Chen: Introduction of repeated measures ANOVA
Chewei
Lu: One-way repeated measures
Wei Xi: Two-factor repeated measures
Tomoaki
Sakamoto : Three-factor repeated measures
How-Chung Liu: Mixed models
Margaret Brown: Comparison
Xiao Liu: ConclusionSlide3
Introduction of Repeated Measures ANOVA
Jia
ChenSlide4
What is it ?
Definition:
- It is a technique used to test the equality of means.
Slide5
When To Use It?
It is used when all members of a random sample are measured under a number of different conditions.
As the sample is exposed to each condition in turn, the measurement of dependent variable is repeated.Slide6
Introduction of One-Way Repeated Measures ANOVA
Che
-Wei, Lu
Professor:Wei
ZhuSlide7
One-Way Repeated Measures ANOVA
Definition
A one-way repeated measures ANOVA instead of having one score per subject, experiments are frequently conducted in which multiple score are gathered for each case.
Concept of Repeated Measures
ANOVA
One factor with at least two levels, levels are dependent.
Dependent means that they share variability in some way.
The Repeated Measures ANOVA is extended from standard ANOVA.Slide8
One-Way Repeated Measures ANOVA
When to Use
Measuring performance on the same variable over time
for example looking at changes in performance during training or before and after a specific treatment
The same subject is measured multiple times under different conditions
for example performance when taking Drug A and performance when taking Drug B
The same subjects provide measures/ratings on different characteristics
for example the desirability of red cars, green cars and blue cars
Note how we could do some RM as regular between subjects designs
For example, Randomly assign to drug A or BSlide9
One-Way Repeated Measures ANOVA
Source of Variance in Repeated Measures ANOVA
SStotal
Deviation of each individual score from the grand mean
SSb
/t subjects
Deviation of subjects' individual means (across treatments) from the grand mean.
In the RM setting, this is largely uninteresting, as we can pretty much assume that
‘
subjects differ
’
SSw
/in subjects: How
Ss
vary about their own mean, breaks down into:
SStreatment
As in between subjects ANOVA, is the comparison of treatment means to each other (by examining their deviations from the grand mean)
However this is now a partition of the within subjects variation
SSerror
Variability of individuals
’
scores about their treatment meanSlide10
One-Way Repeated Measures ANOVA
Partition of Sum of Square
Repeated Measures ANOVA
Standard ANOVASlide11
One-Way Repeated Measures ANOVA
Variation
SS
Df
MS
F
Between
a-1
MSA=
F=
Within
N-a
MSE=
Total
N-1
Variation
SS
Df
MS
F
Between
a-1
Within
N-a
Total
N-1
Standard ANOVA Table
Repeated Measures ANOVA Table
SS
Df
MS
F
Between
a-1
MSA=
F=
Within
N-a
-Subjects
s-1
-Error
MSE=
Total
N-A
SS
Df
MS
F
Between
a-1
Within
N-a
-Subjects
s-1
-Error
Total
N-ASlide12
One-Way Repeated Measures ANOVA
Example:
Researchers want to test a new anti-anxiety medication. They measure the anxiety of 7 participants three times: once before taking the medication, once one week after taking the medication, and once two weeks after taking medication. Anxiety is rated on a scale of 1-10,with 10 being ”high anxiety” and 1 being “low anxiety”. Are there any difference between the three condition using significant level
Participants
Before
Week1
Week2
1
9
7
4
2
8
6
3
3
7
6
2
4
8
7
3
5
8
8
4
6
9
7
3786
2Slide13
One-Way Repeated Measures ANOVA
Define Null and Alternative Hypotheses
Participants
Before
Week1
Week2
1
9
7
4
2
8
6
3
3
7
6
2
4
8
7
3
5
8
8
4
6
9
7
3
7
8
6
2Slide14
One-Way Repeated Measures ANOVA
Define Degrees of Freedom
N=21 s=7
-
=18-6=12
Participants
Before
Week1
Week2
1
9
7
4
2
8
6
3
3
7
6
2
4
8
7
3
5
8
8
4
6
9
7
3
7
8
6
2Slide15
One-Way Repeated Measures ANOVA
Analysis of Variance
Participants
Before
Week1
Week2
1
9
7
4
2
8
6
3
3
7
6
2
4
8
7
3
5
8
8
4
6
9
7
3
7
8
6
2Slide16
One-Way Repeated Measures ANOVA
Analysis of
Variance(ANOVA Table)
SS
Df
MS
F
Between
98.67
2
49.34
224.27
Within
10.29
18
-Subjects
7.62
6
-Error
2.67
12
0.22
Total
108.96
20
Error=within-Subjects=10.29-7.62=2.67
Total=
Beteen+Within
=98.67+10.29=108.96
Test Statistic
:
244.27
Slide17
One-Way Repeated Measures ANOVA
Critical Region
Now, the
SS
Df
MS
F
Between
98.67
2
49.34
224.27
Within
10.29
18
-Subjects
7.62
6
-Error
2.67
12
0.22
Total
108.96
20Slide18
One-Way Repeated Measures ANOVA
SAS Code
DATA
REPEAT;
INPUT SUBJ BEFORE WEEK1 WEEK2;
DATALINES;
1 9 7 4
2 8 6 3
3 7 6 2
4 8 7 3
5 8 8 4
6 9 7 3
7 8 6 2
;
PROC
ANOVA
DATA=REPEAT;
TITLE "One-Way ANOVA using the repeated
Statment
";
MODEL BEFORE WEEK1 WEEK2= / NOUNI;
REPEATED TIME
3
(
1
2 3);RUN;
The Before, Week1, and Week2 are the time level at each participants.
In here, we don’t have CLASS statement because our data set does not have an independent variable
The NOUNI(no
univariate
) is a request not to conduct a separate analysis for each of the three times variables.
This indicates the labels we want to printed for each level of timesSlide19
One-Way Repeated Measures ANOVA
SAS ResultSlide20
Two-Factor ANOVA with Repeated Measures
Wei XiSlide21
Stating of the HypothesisSlide22
Two-Factor ANOVA with Repeated Measures on One FactorSlide23
HypothesisSlide24
ANOVA TABLE
Source
DF
SS
MS
F
Factor A
a-1
SSA
SSA/(a-1)
MSA/MSWA
～
F
(a-1),n(a-1)
Factor
B
b-1
SSB
SSB/(b-1)
MSB/MSE
～
F
(b-1),n(a-1)(b-1)
AB Interaction
(a-1)(b-1)
SSAB
SSAB/(a-1)(b-1)MSAB/MSE ～ F
(a-1)(b-1,n(a-1)(b-1)Subjects within A
(n-1)aSSWASSWA/(n-1)a
Error (n-1)a(b-1)SSE
SSE/((n-1)a(b-1)Total nab-1SSTSlide25
Example
The
shape
variable is the repeated variable. This produces an ANOVA with one between-subjects factor. If you were to examine the expected mean squares for this setup, you would find that the appropriate error term for the test of
calib
is
subject|calib
. The appropriate error term for
shape
and
shape#calib
is
shape#subject|calib
(which is the residual error since we do not include the term in the model).Slide26
SAS Code
Data
Q1;
set
pre.Q1;
run
;
proc
anova
data=Q1;
title
' Two-way Anova with a Repeated Measure on One Factor';
class
calib;
model
shape_1 shape_2 shape_3 shape_4 = calib/
nouni
;
repeated shape
4
;
means calib;
run; Slide27
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no shape Effect
H = Anova SSCP Matrix for shape
E = Error SSCP Matrix
S=1 M=0.5 N=0
Statistic
Value
F Value
Num DF
Den DF
Pr > F
Wilks' Lambda
0.02529573
25.69
3
2
0.0377
Pillai's Trace
0.97470427
25.69
3
2
0.0377
Hotelling-Lawley Trace
38.53236607
25.69
3
2
0.0377
Roy's Greatest Root
38.53236607
25.69
3
2
0.0377
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no shape*calib Effect
H = Anova SSCP Matrix for shape*calib
E = Error SSCP Matrix
S=1 M=0.5 N=0
Statistic
Value
F Value
Num DF
Den DF
Pr > F
Wilks' Lambda
0.16750795
3.31
3
2
0.2404
Pillai's Trace
0.83249205
3.31
3
2
0.2404
Hotelling-Lawley Trace
4.96986607
3.31
3
2
0.2404
Roy's Greatest Root
4.96986607
3.31
3
2
0.2404
At
α
=0.05
,we
reject
the hypothesis and conclude that there is shape Effect
At
α
=0.05
,we
cannot
reject
the hypothesis and conclude that there is
no
shape*calib Effect
Analysis of SAS OutputSlide28
Source
DF
Anova SS
Mean Square
F Value
Pr > F
calib
1
51.04166667
51.04166667
11.89
0.0261
Error
4
17.16666667
4.29166667
Source
DF
Anova SS
Mean Square
F Value
Pr > F
Adj Pr > F
G - G
H - F
shape
3
47.45833333
15.81944444
12.80
0.0005
0.0099
0.0011
shape*calib
3
7.45833333
2.48611111
2.01
0.1662
0.2152
0.1791
Error(shape)
12
14.83333333
1.23611111
Univariate Tests of Hypotheses for Within Subject Effects
Tests of Hypotheses for Between Subjects EffectsSlide29
Two-Factor ANOVA with Repeated Measures on both
FactorsSlide30
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),(n-1)(a-1)
Factor B
b-1
SSB
SSB/(b-1)
MSB/MSB*S
～
F(b-1),(n-1)(b-1)
AB Interaction
(a-1)(b-1)
SSAB
SSAB/((a-1)(b-1))
MSAB/MSE
～
F(a-1)(b-1),(n-1)(a-1)(b-1)
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
(n-a)(a-1)(b-1)
SSE
SSE/((n-1)(a-1)(b-1))
Total
nab-1
SST
ANOVA TABLESlide31
Example
Three subjects, each with nine accuracy scores on all combinations of the three different dials and three different periods. With
subject
a random factor and both
dial
and
period
fixed factors, the appropriate error term for the test of
dial
is the
dial#subject
interaction. Likewise,
period#subject
is the correct error term for
period
, and
period#dial#subject
(which we will drop so that it becomes residual error) is the appropriate error term for
period#dial
.
Slide32
SAS Code
Data
Q2;
Input
Mins1-Mins9;
Datalines
;
45 53 60 40 52 57 28 37 46
35 41 50 30 37 47 28 32 41
60 65 75 58 54 70 40 47 50
;
ODS RTF STYLE=
BarrettsBlue
;
Proc
anova
data=Q2;
Model Mins1-Mins9=/
nouni
;
Repeated period 3,
dail
3/nom;
Run;
ods
rtf close;
Slide33
SAS Output
Source
DF
Anova SS
Mean Square
F Value
Pr > F
Adj Pr > F
G - G
H - F
period
2
1072.666667
536.333333
14.45
0.0148
0.0563
0.0394
Error(period)
4
148.444444
37.111111
Greenhouse-
Geisser
Epsilon
0.5364
Huynh-
Feldt
Epsilon
0.6569
Source
DF
Anova SS
Mean Square
F Value
Pr > F
Adj Pr > F
G - G
H - F
dail
2
978.6666667
489.3333333
50.91
0.0014
0.0169
0.0115
Error(
dail
)
4
38.4444444
9.6111111
Greenhouse-
Geisser
Epsilon
0.5227
Huynh-
Feldt
Epsilon
0.5952
Source
DF
Anova SS
Mean Square
F Value
Pr > F
Adj Pr > F
G - G
H - F
period*dail
4
8.66666667
2.16666667
0.30
0.8715
0.6603
0.7194
Error(period*
dail
)
8
58.22222222
7.27777778
Greenhouse-
Geisser
Epsilon
0.2827
Huynh-
Feldt
Epsilon
0.4006
Univariate Tests of Hypotheses for Within Subject Effects
At
α
=0.05
,we
cannot
reject the hypothesis and conclude the there is
no
period*
dail
Effect
At
α
=0.05
,we
reject
the hypothesis and conclude that there is
dail
Effect
At
α
=0.05
,we
reject
the hypothesis and conclude that there is period EffectSlide34
Three-factor Experiments
with a repeated measure
T. SakamotoSlide35
35
/8
7
Example of a
marketing experiment
Experiment
The
subjects
who belong to a region X or Y see the Liquid Crystal Display A, B, or C.
Each type of LCD is seen twice; once in the light and the other in the dark.
The preferences of the LCD are measured by the subjects, on a scale from 1 to 5 (1= lowest, 5=highest).
Case of this example
A company which produces some Liquid Crystal Display wants to examine the characteristics of its prototype products.Slide36
36
/8
7
Experimental Design and Data
Three factors
Type of LCD
Regions to which the specimens belong
In the light / In the dark
Repeted measure factor : In the light / In the dark
Type of LCD
A
B
C
subj
light
dark
subj
light
dark
subj
light
dark
REGION
X
1
5
4
11
4
4
21
5
5
2
4
2
12
5
6
22
5
3
3
5
4
13
3
4
23
3
3
4
3
5
14
5
4
24
4
4
5
5
3
15
4
6
25
4
3
Y
6
4
4
16
5
5
26
3
5
7
3
5
17
4
3
27
4
3
8
4
3
18
5
3
28
5
2
9
2
5
19
5
5
29
3
4
10
5
4
20
4
4
30
4
3Slide37
37
/8
7
data
lcd;
input
subj type $ region $ light dark @@;
datalines
;
1 a 5 4 2 a 4 2 3 a 5 4 4 a 3 5 5 a 5 3
6 a 4 4 7 a 3 5 8 a 4 3 9 a 2 5 10 a 5 4
11 b 4 4 12 b 5 6 13 b 3 4 14 b 5 4 15 b 4 6
16 b 5 5 17 b 4 3 18 b 5 3 19 b 5 5 20 b 4 4
21 c 5 5 22 c 5 3 23 c 3 3 24 c 4 4 25 c 4 3
26 c 3 5 27 c 4 3 28 c 5 2 29 c 3 4 30 c 4 3
;
run
;
proc anova
data
=lcd;
title
’Three-way ANOVA with a Repeated Measure'
;
class
type region;
model
light dark = type | region /
nouni
;
repeated light_dark;means type | region;
run;
SAS PROGRAMSlide38
OUTPUT(Part 1/4):Slide39
OUTPUT(Part 2/4):Slide40
40
/81
OUTPUT(Part 3/4):Slide41
OUTPUT(Part 4/4):Slide42
Mixed Effect Models
How-Chang LiuSlide43
Mixed Models
When we have a model that contains random effect as well as fixed effect, then we are dealing with a mixed model.
From the above definition, we see that mixed models must contain at least two factors.
One having fixed effect and one having random effect.Slide44
Why use mixed models?
When repeated measurements are made on the same statistical units, it would not be realistic to assume that these measurements are independent.
We can take this dependence into account by specifying covariance structures using a mixed modelSlide45
Definition
A mixed model can be represented in matrix notation by:
is the vector of observations
is the vector of fixed effects
is the vector of random effects
is the vector of I.I.D. error terms
and
are matrices relating
and
to
Slide46
Assumptions
R and G are constants
We also assume that
and
are independent
We get V
= ZGZ' +
R, where V is the variance of y
Slide47
How to estimate
and
?
If R and G are given: Using Henderson’s Mixed Model equation, we have:
=
So
=
And
=
)
Slide48
What if G and R are unknown?
We know that both
and
are normally distributed, so the best approach is to use likelihood based methods
There are two methods used by SAS:
1)Maximum likelihood (ML)
2)Restricted/residual maximum likelihood (REML)
Slide49
Example
Below is a table of growth measurements for 11 girls and 16 boys at ages 8, 10, 12, 14:
Person gender age8 age10 age12 age14
1 F 21.0 20.0 21.5 23.0
2 F 21.0 21.5 24.0 25.5
3 F 20.5 24.0 24.5 26.0
4 F 23.5 24.5 25.0 26.5
5 F 21.5 23.0 22.5 23.5
6 F 20.0 21.0 21.0 22.5
7 F 21.5 22.5 23.0 25.0
8 F 23.0 23.0 23.5 24.0
9 F 20.0 21.0 22.0 21.5
10 F 16.5 19.0 19.0 19.5
11 F 24.5 25.0 28.0 28.0
12
M
26.0 25.0 29.0 31.0
13 M 21.5 22.5 23.0 26.5
14 M 23.0 22.5 24.0 27.5
Person
gender age8 age10 age12
age14
15 M 25.5 27.5 26.5 27.0
16
M 20.0 23.5 22.5 26.0 17 M 24.5 25.5 27.0 28.5 18 M 22.0 22.0 24.5 26.5 19
M 24.0 21.5 24.5 25.5 20 M 23.0 20.5 31.0 26.0 21 M 27.5 28.0 31.0 31.5
22 M 23.0 23.0 23.5 25.0 23 M 21.5 23.5 24.0 28.0 24 M 17.0 24.5 26.0 29.5 25 M 22.5 25.5 25.5 26.0 26 M 23.0 24.5 26.0
30.0 27 M 22.0 21.5 23.5 25.0 Slide50
Using SAS
data
pr
;
input
Person
Gender
$ y1 y2 y3 y4;
y=y1
;
Age
=8; output;
y=y2
;
Age
=10; output;
y=y3
;
Age
=12; output;
y=y4
;
Age=14; output; drop y1-y4;
datalines; 1 F 21.0 20.0 21.5 23.0 2 F 21.0 21.5 24.0 25.5
…;Run;Slide51
Using SAS
proc
mixed data=
pr
method=ml
covtest
;
class
Person Gender;
model
y = Gender Age Gender*Age / s;
repeated
/ type=un
subject=Person r;
run
; Slide52
Results
Model Information
Data Set
WORK.PR
Dependent Variable
y
Covariance Structure
Unstructured
Subject Effect
Person
Estimation Method
ML
Residual Variance Method
None
Fixed Effects SE Method
Model-Based
Degrees of Freedom Method
Between-Within
Class Level Information
Class
Levels
Values
Person
27
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
Gender
2
F M
As one can see, the covariance matrix is unstructured, as we are going to estimate it using the maximum likelihood methodSlide53
Dimensions
Covariance Parameters
10
Columns in X
6
Columns in Z
0
Subjects
27
Max Obs Per Subject
4
Number of Observations
Number of Observations Read
108
Number of Observations Used
108
Number of Observations Not Used
0
Iteration History
Iteration
Evaluations
-2 Log Like
Criterion
0
1
478.24175986
1
2
419.47721707
0.00000152
2
1
419.47704812
0.00000000
Convergence criteria met.
As one can see, we do not have a Z matrix for this model
the
convergence of the Newton-
Raphson
algorithm
means that we have found the maximum likelihood estimatesSlide54
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
Standard Error
Z Value
Pr Z
UN(1,1)
Person
5.1192
1.4169
3.61
0.0002
UN(2,1)
Person
2.4409
0.9835
2.48
0.0131
UN(2,2)
Person
3.9279
1.0824
3.63
0.0001
UN(3,1)
Person
3.6105
1.2767
2.83
0.0047
UN(3,2)
Person
2.7175
1.0740
2.53
0.0114
UN(3,3)
Person
5.9798
1.6279
3.67
0.0001
UN(4,1)
Person
2.5222
1.0649
2.37
0.0179
UN(4,2)
Person
3.06241.01353.020.0025
UN(4,3)Person3.82351.25083.060.0022UN(4,4)Person4.61801.2573
3.670.0001The table lists the 10 estimated covariance parameters in order. In other words, these are the estimates for R, the variance of Slide55
Solution for Fixed Effects
Effect
Gender
Estimate
Standard Error
DF
t Value
Pr > |t|
Intercept
15.8423
0.9356
25
16.93
<.0001
Gender
F
1.5831
1.4658
25
1.08
0.2904
Gender
M
0
.
.
.
.
Age
0.8268
0.07911
25
10.45
<.0001
Age*Gender
F
-0.3504
0.1239
25
-2.83
0.0091
Age*Gender
M
0
.
.
.
.
From this table, we see that the boys intercept is at 15.8423, whole the girls intercept is at 15.8423+1.5831=17.42.
The estimate of the boys’ slope is at 0.827, while the girls’ slpe
is at 0.827-0.350=0.477So the girls’ starting point is higher than the girls but their growth rate is only about half of that of the boysSlide56
Type 3 Tests of Fixed Effects
Effect
Num DF
Den DF
F Value
Pr > F
Gender
1
25
1.17
0.2904
Age
1
25
110.54
<.0001
Age*Gender
1
25
7.99
0.0091
This is probably the most important table from our results:
The gender row tests the null hypothesis that girls and boys have a common intercept.
As we can see we cannot reject that hypothesis
The Age tests the null hypothesis that age does not affect the growth rate.
As we can see, we reject the null hypothesis as the F-value is large.
The Age*gender tests reveals that there is a difference in slope at the 1% significance level.Slide57
Repeated Measures ANOVA vs. Independent Measures ANOVA
Magarate
BrownSlide58
Can we just use standard ANOVA with repeated measures data?
No, Independent Measures (standard) ANOVA assumes the data are independent.
Data from a repeated measures experiment
not independentSlide59
How are standard ANOVA and repeated measures ANOVA the same?
Independent measures ANOVA: an extension of the pooled variance t-test
Repeated Measures ANOVA: an extension of the paired sample t-testSlide60
How are standard ANOVA and repeated measures ANOVA the same?
Independent measures ANOVA: assumes the population variances are equal (
homogeneity of variance
)
Repeated Measures ANOVA:
sphericity
assumption that the population variances of all the differences are equalSlide61
How are standard ANOVA and repeated measures ANOVA the same?
Both assume Normality of the populationSlide62
Advantages to using Repeated Measures instead of Independent Measures
Limited number of subjects available
Prefer to limit the number of subjects
Less variability (finger tapping with caffeine example)
Can examine effects over timeSlide63
Drawbacks
Practice effect
Example: subjects get better at performing a task each time with “practice”
Differential transfer: “This
occurs when the effects of one condition persist and affect participants’ experiences during subsequent conditions
.” (format: http://www.psychmet.com/id16.html)
Example: medical treatmentsSlide64
Resources (to be formatted)
http://www.psychmet.com/id16.html
http://www.utexas.edu/courses/schwab/sw388r7_spring_2007/SolvingProblemsInSPSS/Solving%20Repeated%20Measures%20ANOVA%20Problems.pdf
http://en.wikipedia.org/wiki/Repeated_measures
http://www.mhhe.com/socscience/psychology/shaugh/ch07_summary.html