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SPSS meets SPM
All about Analysis of Variance
• Introduction and definition of terms
• One-way between-subject ANOVA: An example
• One-way repeated measurement ANOVA
• Two-way repeated measurement ANOVA:
• Pooled and partitioned errors
• How to specify appropriate contrasts to test main effects
and interactions
SPSS meets SPM
Two-sample t-test Paired-sample-t-test
ANOVA Repeated ANOVA
between-subject ANOVA within-subject ANOVA
F-test F-test
Factors
Levels
K1 x K2 ANOVA
Two Factors
with K1 levels of one factor and
K2 level of the second factor
Repeated MeasuresSingle Measures
Analysis of Variance
Level
1
Level
2
Level 1 Group
1
Group
2
Level
2
Group
3
Group
4
Factor B
Two-way ANOVA
2 x 2 ANOVA
2 x 2 repeated measurement ANOVA
Level
1
Level
2
Level 1 Subj.
1….12
Subj.
1….12
Level
2
Subj.
1….12
Subj.
1….12
Factor A
Factor B
Factor A
Mixed Design
Drug Placebo
Patient Subj.
1…12
Subj.
1…12
Control Subj.
13...24
Subj.
13...24
Factor AWithin-subject Factor
Factor BBetween-subject
Factor
ImagingDesigns
Fearful Neutral
Implicit Group
1
Group
2
Explicit Group
3
Group
4
Fac
tor
B2 x 2 ANOVA 2 x 2 repeated
measurement ANOVA
Fearful Neutral
Implicit Subj.
1….12
Subj.
1….12
Explicit Subj.
1….12
Subj.
1….12
Factor A
Fac
tor
B
Factor A
Main Effect A
Main E
ffect B
Interaction A X B
Implicit Explicit
Fearful
Neutral
Fearful Neutral Happy
Implicit
Explicit
3 x 2 ANOVA
Contrasts
ijjijX
An individual score is specified by
jj
Grand mean
Treatment effect
jijij X Residual error
One-way between-subject ANOVA
ijjijX FULL MODEL
General Principle of ANOVA
ijijX
Is the full model a significantly better model then the reduced model?
Data represent a random variation around the grand mean
REDUCED MODEL
Total Variation (SStotal)
Treatment effect (SStreat)
Error (SSerror)
errorerror
treattreat
DFSS
DFSSF
/
/
errortreattotal SSSSSS
errortreattotal DFDFDF
Partitions of Sums of Squares
____________________________________
4-different drug treatments (Factor A with p levels)____________________________________
1 2 3 4____________________________________
2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2
_____________________________________Sums(Ai) 10 15 35 20_____________________________________Means(Ai) 2 3 7 4
_____________________________________
One factor with p levels; i = 1…4M subjects with n subjects per level Number of total observations = 20
1.
2.
3.
4.
5.
m
mii
tot GxSS 2)(
pn
AG i
i
*
m
imii
error AxSS 2)(
i
itreat GAnSS 2)(*
1* pndftot
1pdftreat
)1(* npdferror
errorerror
treattreat
DFSS
DFSSF
/
/
420
20351510
G
..)41()42( 22
..)43(*5)42(*5 22
..)21()22( 22
One-way ANOVA between subjects1st levels betas from one voxel in amygdala
Multiple Regression
____________________________________Drug treatment (Factor A with p levels)____________________________________ 1 2 3 4____________________________________ 2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2
Dependent variable = 1st level betas extracted from the amygdala
One way ANOVA
Do the drug treatment affect differentlyaffect differently mean activation in the amygdala ?
Do the drug treatments relaterelate to the mean activation in the amygdala?
213313435068741055532
1st level betas
Drug treatments
11111222223333344444
y = a + bX
11111111111111111111
Multiple Regression
____________________________________Drug treatment (Factor A with p levels)____________________________________ 1 2 3 4____________________________________ 2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2
Dependent variable = 1st level betas extracted from the amygdala
One way ANOVA
Do the drug treatment affect differentlyaffect differently mean activation in the amygdala ?
Do the drug treatments relaterelate to the mean activation in the amygdala?
213313435068741055532
1st level betas
Drug treatments
00000111110000000000
11111111111111111111
00000000001111100000
00000000000000011111
11111000000000000000
y y xx11 xx22 xx33 xx44
Multiple Regression
____________________________________Drug treatment (Factor A with p levels)____________________________________ 1 2 3 4____________________________________ 2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2
Dependent variable = 1st level betas extracted from the amygdala
One way ANOVA
Do the drug treatment affect differentlyaffect differently mean activation in the amygdala ?
Do the drug treatments relaterelate to the mean activation in the amygdala?
213313435068741055532
1st level betas
Drug treatments
00000111110000000000
11111111111111111111
00000000001111100000
00000000000000011111
11111000000000000000
Y Y bb11xx11= bb22xx22 bb33xx33 bb44xx44 bb00
+ + + +
Multiple Regression
____________________________________Teaching Methods (Factor A with p levels)____________________________________ 1 2 3 4____________________________________ 2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2
Dependent variable = reading score
One way ANOVA
1 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 1
ijjijX
y = *
11
21
31
41
51
12
.
.
.
.
.
.
.
.
.
.
.
.
44
54
eeeeee
ee
+
ebXbXbXbXby 044332211
b1
b2
b3
b4
b0
jj bb
;0
Do the drug treatment affect differentlyaffect differently mean activation in the amygdala ?
Do the drug treatments relaterelate to the mean activation in the amygdala?
Two-sample t-test Paired-sample-t-test
ANOVA Repeated ANOVA
between-subject ANOVA within-subject ANOVA
F-test F-test
Repeated MeasuresSingle Measures
Repeated ANOVA
Assumptions
• Homogeneity of Variance
• Homogeneity of Correlations
• Normality
Assumptions
• Homogeneity of Variance
• Normality
• Independence of observations
Drug 1 Drug 2 Drug 3 Placebo
Subj.1
Subj. 2
Subj. 3
….
Subj.1
Subj. 2
Subj. 3
….
Subj.1
Subj. 2
Subj. 3
….
Subj.1
Subj. 2
Subj. 3
…..
Drug 1 Drug 2 Drug 3 Placebo
Group 1 Group2 Group3 Group4
One-way within-subject
ANOVA
ijjijX
An individual score is specified by
One-way between-subject
ANOVA
An individual score is specified by
ijjiijX
jj
Grand mean
Treatment effect
jijij X Residual error
i Grand mean
Subject effect
Residual error
j Treatment effect (within-subject effect)
ij
Total Variation (SStotal)
Treatment effect (SStreat)
Error (SSerror)
Total Variation (SStotal)
Between subj (SSbetween)Subject effects
Subj. x Treat & Error
Within subj. (SSwithin)
Treatment effect (SStreat)
Residual (SSres)
errorerror
treattreat
DFSS
DFSSF
/
/
errortreattotal SSSSSS
errortreattotal DFDFDF
errorresidual
treattreat
DFSS
DFSSF
/
/
residualtreatbetweentotal SSSSSSSS
residualtreatmentbetweentotal DFDFDFDF
Partitions of Sums of Squares
Within subjectsWithin subjectsBetween SubjectsBetween Subjects
ijjijX
1 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 1
y = *
11
21
31
41
51
12
.
.
.
.
.
.
.
.
.
.
.
.
44
54
eeeeee
ee
+
b1
b2
b3
b4
b0
ijjiijX jj
ebXbXbXbXby 044332211 ebXbXbXbXby 044332211
1 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 1
y = *
11
21
31
41
51
12
.
.
.
.
.
.
.
.
.
.
.
.
44
54
eeeeee
ee
+
b1
b2
b3
b4
b0
+
1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 11 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1
1 2 3 4
Drug 1 Drug 2 Drug3 Placebo
1
2
3
4
5
Within subjectsWithin subjectsBetween SubjectsBetween Subjects
ijjijX ijjiijX jj
ebXbXbXbXby 044332211 ebXbXbXbXby 044332211
00000111110000000000
11111111111111111111
00000000001111100000
00000000000000011111
11111000000000000000
00000111110000000000
00000000001111100000
00000000000000011111
11111000000000000000
10000000000000000000
01000000000000000000
00100000000000000000
00010000000000000000
00000000000000000000
11111111111111111111
1 2 3 4
Drug 1 Drug 2 Drug3 Placebo
Level
1
Level
2
Level 1
Subj.
1….12
Subj.
1….12
Level
2
Subj.
1….12
Subj.
1….12
Factor A
Factor B
ijkABjk
Bk
AjiijkX
2 x 2 Repeated Measurement ANOVA
ABijk
Bik
Aij
ABjk
Bk
AjiijkX
Pooled Error
Partitioned Error Interaction between effect and subject
Within-Subjects Two-Way ANOVAWithin-Subjects Two-Way ANOVA
1 2 3 4Fear-implicit neutral-implicit fear-explicit neutral-explicit
ijjiijX
ebXbXbXbXby 044332211
00000111110000000000
00000000001111100000
00000000000000011111
11111000000000000000
10000000000000000000
01000000000000000000
00100000000000000000
00010000000000000000
00000000000000000000
11111111111111111111
Repeated Measurement Repeated Measurement ANOVA in SPMANOVA in SPM
One way ANOVA = 1st level betas 2nd level + subjects effects
Two way ANOVA = 1st level differential effects betweenlevels of a factors for main effects
differences of differential effects for interactions
2nd level (T-test for 2x2 ANOVA F-test for 3x3 ANOVA)
Pooled errors
Partitioned errors
What contrast to take from 1st level?Two way ANOVA (2*2) with repeated measured
Fearful Neutral
Implicit
Explicit
Factor A
Fac
tor
B
Fear/
implicit
Fear/ explicit
Neutral/ implicit
Neutral/ explicit
What contrast to take from 1st level?Two way ANOVA (3*3) with repeated measured
Factor A
Fac
tor
B
semantic perception
Imagery
Picture
Words
Sounds
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