Why N’ How
(I forgot the title)
Donald G. McLaren, Ph.D.
Department of Neurology, MGH/HMS
GRECC, ERNM Veteran’s Hospital
http://www.martinos.org/~mclaren
11/15/2012
Types of Data
Types of Data – Dependent
Variable
• Task Data
– Single Condition
– Multiple Conditions
– Multiple Predictors Per Condition
•
•
•
•
•
•
Functional Connectivity – Correlation
Functional Connectivity -- ICA
Context-Dependent Connectivity
VBM
DTI
Other??
Factors, Levels, Groups, Classes
Continuous Variables/Factors: Age, IQ, Volume,
Behavioral measures (emotional scale, memory
ability), Images, etc.
Discrete Variables/Factors: Gender, Handedness,
Diagnosis
Levels of Discrete :
Handedness: Left and Right
Gender: Male and Female
Diagnosis: Normal, MCI, AD
Group or Class: Specification of All Discrete Factors:
• Left-handed Male MCI
• Right-handed Female Normal
Overview
• From a line to the GLM and matrices
• Statistical Tests
• Contrasts
• Designs
• Power
• Caveats
General Linear Model
(GLM)
Y=aX+b
GLM Theory
Is Activity correlated with Age?
Activity
Dependent
Variable,
Measurement
HRF Amplitude
IQ, Height, Weight
Subject 1
y1
Subject 2
y2
Age
x1
x2
Independent Variable
Of course, you’d
need more then two
subjects …
Activity
Linear Model
Intercept: b System of Linear Equations
Slope: m
y1
Matrix Formulation
y2
Age
x1
Intercept = Offset
y1 = 1*b + x1*m
y2 = 1*b + x2*m
x2
X = Design Matrix
b = Regression Coefficients
= Parameter estimates
= “betas”
= Intercepts and Slopes
b
y1
1 x1
=
*
m
y2
1 x2
Y = X*b
b=
b
m
Hypotheses and Contrasts
Is Activity correlated with Age?
Does m = 0?
Null Hypothesis: H0: m=0
b
y1
1 x1
=
*
m
y2
1 x2
m= [0 1]*b
Activity
m
Intercept: b
Slope: m
y1
b=
b
m
g = C*b = 0
?
C=[0 1]: Contrast Matrix
y2
Age
x1
x2
Hypotheses and Contrasts
Is Activity different from 0?
Does b = 0?
Null Hypothesis: H0: b=0
b
y1
1 x1
=
*
m
y2
1 x2
b= [1 0]* b
Activity
m
Intercept: b
Slope: m
y1
b=
b
m
g = C*b = 0
?
C=[1 0]: Contrast Matrix
y2
Age
x1
x2
Hypotheses and Contrasts
Is Activity different from 0?
Does b = 0?
Null Hypothesis: H0: b=0
b
y1
1 x1
=
*
m
y2
1 x2
b= [1 0]* b
Slope: m
Activity
m
b=
y1
Intercept: b
b
m
g = C*b = 0
?
C=[1 0]: Contrast Matrix
y2
Age
x1
x2
Hypotheses and Contrasts
Is Activity different from 0?
Does b = 0?
Null Hypothesis: H0: b=0
b
y1
1 x1
=
*
m
y2
1 x1
b= [1 0]* b
Activity
m
b=
y1
Intercept: b
b
m
g = C*b = 0
?
C=[1 0]: Contrast Matrix
y2
Age
x1
x2
Hypotheses and Contrasts
Is Activity different from 0?
Does b = 0?
Null Hypothesis: H0: b=0
y1
1
=
y2
1
*
b
b= [1 ]* b
Activity
b=
y1
Intercept: b
b
g = C*b = 0
?
C=[1 0]: Contrast Matrix
y2
Age
x1
x2
More than Two Data Points
Activity
Intercept: b
Slope: m
Age
y1 = 1*b + x1*m
y2 = 1*b + x2*m
y3 = 1*b + x3*m
y4 = 1*b + x4*m
b
y1
1 x1
=
*
m
y2
1 x2
y3
1 x3
y4
1 x4
Y = X*b+n
• Model Error
• Noise
• Uncertainty
The General Linear Model
Y = b0  b1 X1  b2 X 2  b p X p  
Y1 = b 0  b1 x1,1  b 2 x1, 2    b p x1, p   1
Y2 = b 0  b1 x2,1  b 2 x2, 2    b p x2, p   2
Y3 = b 0  b1 x3,1  b 2 x3, 2    b p x3, p   3

Yn = b 0  b1 xn ,1  b 2 xn , 2    b p xn , p   n
observed

Yˆ
Y=
=
predicted
In Matrix Form
+
Y = Xb + e
random error
Summary of the GLM
Y
=
X
.
Observed data:
Design matrix:
Imaging uses a
mass univariate
approach – that is
each voxel is
treated as a
separate column
vector of data.
Y is Dependent
Brain Value at
various
subjects/time
points at a single
voxel
Several components which
explain the observed data, i.e.
the BOLD time series for the
voxel
Timing info: onset vectors,
Omj, and duration vectors, Dmj
HRF, hm, describes shape of
the expected BOLD response
over time
Other regressors, e.g.
realignment parameters
At the group level: these are
covariates or grouping
columns (see later slide)
β
+
ε
Parameters:
Error:
Define the
contribution of each
component of the
design matrix to the
value of Y
Estimated so as to
minimise the error, ε,
i.e. least sums of
squares
Difference
between the
observed
data, Y, and
that predicted
by the model,
Xβ .
Not assumed
to be spherical
in fMRI
Brain Imaging
• From the beginning (almost)….
[5 6 7 5]
Spatial Normalization, Atlas Space
Native Space
MNI305 Space
Subject 1
Subject 1
MNI305
Subject 2
25
Subject 2
Group Analysis
Does not have to
be all positive!
26
Contrast Amplitudes
Contrast Amplitudes Variances
(Error Bars)
Mass Univariate Analyses
(1) Run the GLM for each voxel.
(2) Compute the statistic from the GLM for each voxel
(3) Inferences
Statistical Parametric Map (SPM)
+3%
0%
-3%
Contrast Amplitude Contrast Amplitude
CON, COPE, CES
Variance
(Error Bars)
“Massive Univariate Analysis”
28
-- Analyze
each voxel separately
VARCOPE, CESVAR
Significance
t-Map (p,z,F)
(Thresholded
p<.01)
sig=-log10(p)
SPM/FSL/AFNI/CUSTOM
• It is important to recognize that all programs
that utilize the GLM will produce the same
result. However, if your design matrices or
variance correction methods are different,
then you will see differences.
• Some slides show illustrations from FSL,
others show illustrations from SPM, MATLAB,
or other software. These can be done in all
programs.
Types Of Analysis
“Random Effects (RFx)” Analysis
t=
bG
 b2
G
G
bG
 b2 =
G
NG 1
DOF = N G  1
RFx
32
2


b

b
 G i
“Random Effects (RFx)” Analysis
• Model Subjects as a Random Effect
• Variance comes from a single source:
variance across subjects
G
bG
– Mean at the population mean
– Variance of the population variance
• Does not take first-level noise into
account (assumes 0)
• “Ordinary” Least Squares (OLS)
• Usually less activation than individuals
33
RFx
“Mixed Effects (MFx)” Analysis
MFx
RFx
• Down-weight each subject based on variance.
• Weighted Least Squares vs (“Ordinary” LS)
34
“Mixed Effects (MFx)” Analysis
• Down-weight each subject based on variance.
• Weighted Least Squares vs (“Ordinary” LS)
• Protects against unequal variances across
group or groups (“heteroskedasticity”)
• May increase or decrease significance with
respect to simple Random Effects
• More complicated to compute
• “Pseudo-MFx” – simply weight by first-level
MFx
variance (easier to compute)
35
“Fixed Effects (FFx)” Analysis
t=
bG
FFx
 b2
G
 b2 =
G

2
i
2

 i
N G 2
DOF =  DOFi
RFx
36
“Fixed Effects (FFx)” Analysis
• As if all subjects treated as a single subject (fixed
effect)
• Small error bars (with respect to RFx)
b
t= G
• Large DOF
 b2
• Same mean as RFx
2

 b2 =  i2
• Huge areas of activation
N G 
• Not generalizable beyond sample.
G
G
DOF =  DOFi
FFx
37
Population vs Sample
Group Population
(All members)
Hundreds?
Thousands?
Billions?
Sample
18 Subjects
38
• Do you want to draw
inferences beyond your
sample?
• Does sample represent
entire population?
• Random Draw?
fMRI Analysis Overview
Subject 1
Raw Data
Subject 2
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
Preprocessing
MC, STC, B0
Smoothing
Normalization
C
First Level
GLM Analysis
X
C
Higher Level GLM
Subject 3
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
C
Subject 4
Preprocessing
MC, STC, B0
Smoothing
Normalization
Raw Data
39
First Level
GLM Analysis
X
C
X
C
Second-Level Modeling
• These are all random effects (because of
variance corrections and using beta’s from the
first level)
• Mean across subjects divided by variance
across subjects.
– Low subjects with very low variance between
them can lead to a significant finding, even if no
subject was significant at the single subject level
– Implications for analysis (e.g. SLBT??)
Statistical Tests
Implementing the T-test
c = +1 0 0 0 0 0 0 0
t-test H0: cTb = 0
contrast of
estimated
parameters
T=
variance
estimate
Variance Estimate
Sqrt(Var*cT(XTX)-1c)
Implementing the F-test
c=
00100000
00010000
00001000
00000100
00000010
00000001
H 0 : cT b = 0
additional
variance
accounted for
by effects of
interest
F=
error
variance
estimate
Contrasts and the Full Model
y = Xb  n, y = s  n, n ~ N (0,  n2 )
bˆ = ( X T X ) 1 X T y Parameter Estimates
T
ˆ
n
nˆ
2
ˆ n =
DOF
gˆ = Cbˆ
Residual Variance
Contrast


1
ˆ
ˆ g = g = C ( X T X ) 1 C T ˆ n2 Contrast Variance Estimate
J
J = Rows in C
gˆ
Cbˆ
t DOF =
=
t - Test (univariat e)
T

1
T
2
ˆ g
C ( X X ) C ˆ n
2

FDOF,J = gˆ T ˆ g 1gˆ

F - Test (multivari ate)
T/r/F Notes
• If F is a single row contrast, then F=T^2
• An F-test has no direction
• In many programs, T-tests are one-tailed, thus have a
p-value half of the same F-test
• There are formulas to convert between T/r and other
statistics (e.g. cohen’s d)
• To avoid double-dipping, when you extract an ROI to
plot the correlation and get the correlation value, DO
NOT make inferences from the plots, but from the
voxel-wise analysis.
Contrasts
• Identify the Null Hypothesis
– Ho: A=B
• Make the Null Hypothesis equal 0
– Ho: A-B=0
• Identify the columns for A and B, apply their
weights
– Ho: 1*A+(-1)*B
– Contrast  [1 -1]
Contrasts
• What if A and B are not individual columns as
in the case of A1,A2,B1,B2…
– [1 1 -1 -1] would work, but will over estimate the
magnitude of the effect
– A is the average A1 A2, or Ho: (A1+A2)/2=0
• [½ ½ 0 0]
– B is the average B1 B2, or Ho: (B1+B2)/2=0
• [0 0 ½ ½]
– [½ ½ -½ -½]
Higher Level GLM Analysis
(Low-Level Contrasts)
Observations
y
=
X * b
1
1
1
1
1
Design Matrix
(Regressors)
Data from
one voxel
51
=
One-Sample Group Mean (OSGM)
G
bG
bG
Vector of
Regression
Coefficients
(“Betas”)
Contrast Matrix:
C = [1]
Contrast = C*b = bG
Two Groups GLM Analysis
(Low-Level Contrasts)
Observations
y
Data from
one voxel
52
=
X * b
=
1
1
1
0
0
0
0
0
1
1
bG1
bG2
Contrasts: Two Groups GLM
Analysis
1. Does Group 1 by itself differ from 0?
Ho: bG1=0; Contrast = C*b = bG1; C = [1 0]
2. Does Group 2 by itself differ from 0?
Ho: bG2=0; Contrast = C*b = bG2; C = [0 1]
53
=
3. Does Group 1 differ from Group 2?
Ho: bG1= bG2; Contrast = C*b = bG1- bG2;
C = [1 -1]
4. Does either Group 1 or Group 2 differ from 0?
C has two rows: F-test (vs t-test)
1
0
Concatenation of contrasts #1 and
C=
#2 0 1
1
1
1
0
0
0
0
0
1
1
bG1
bG2
One Group, One Covariate (Age)
y
=
X * b
(Low-Level Contrasts)
Observations
Contrast
=
Data from
one voxel
54
1
1
1
1
1
21 bG
33 bAge
64
17
47
Intercept: bG
Slope: bAge
Age
Contrasts: One Group, One
Covariate
1. Does Group offset/intercept differ
from 0?
Does Group mean differ from 0
regressing out age?
Ho: bG=0; Contrast = C*b = bG; C = [1
0], (Treat age as nuisance)
Contrast
2. Does Slope differ from 0?
Ho: bAge=0; Contrast = C*b =
bAge; C = [0 1]
=
1
1
1
1
1
21
33
64
17
47
Intercept: bG
Slope: bAge
Age
55
bG
bAge
Contrasts: One Group, One MeanCentered Covariate
1. Does Group offset/intercept differ
from 0?
Does Group mean differ from 0
regressing out age?
Ho: bG=0; Contrast = C*b = bG; C = [1
0], (Treat age as nuisance)
Contrast
2. Does Slope differ from 0?
Ho: bAge=0; Contrast = C*b =
bAge; C = [0 1], ** Same effect as
non-mean centered covariate
=
1
1
1
1
1
-15
-3
28
-20
11
Mean: bG
Slope: bAge
Age
56
bG
bAge
Group Effects
1. Does Activity vary with Disease Status?
2. Does Activity vary with Gender?
1. Is there an Interaction between DS and G?
2x2 Group ANOVA
10
5
While this design
matrix was generated in
SPM, you could
generate it in any of the
MRI Analysis packagees
or statistical programs.
13
9
Contrasts
• Does Activity vary by Disease Status?
–
–
–
–
Ho: DS-=DS+
Ho: DS- - DS+ =0
[½ ½ -½ -½]; (group difference based on subgroups) or
[10/15 5/15 -13/22 -9/22] (pure average of subjects)
• Does Activity vary by Gender?
–
–
–
–
Ho: Male=Female
Ho: Male - Female =0
[½ -½ ½ -½]; or (group difference based on subgroups) or
[10/23 -5/14 13/23 -9/14] (pure average of subjects)
Contrasts
• Average of Subgroups versus Average of Individuals
– If you have drawn a random sample and want to talk generally about
all subjects in a group, use the contrast weighted by group size.
– If you haven’t drawn a random sample or want to look at the average
effect of the group, then you want to use the contrast that is not
weighted by group size.
Contrasts
• Is there an interaction?
–
–
–
–
Ho: DS-Females-DS-Males= DS+Females-DS+Males
Ho: (DS-Females-DS-Males) – (DS+Females-DS+Males)=0
Ho: DS-Females-DS-Males – DS+Females+DS+Males=0
[1 -1 -1 1]; or
• Are the groups different?
–
–
–
–
–
–
Ho: DS-Females=DS-Males=DS+Females=DS+Males
F-test
DS-Females=DS-Males  [1 -1 0 0]
DS-Males=DS+Females  [0 1 -1 0]
DS+Females=DS+Males  [0 0 1 -1]
[1 -1 0 0; 0 1 -1 0; 0 0 1 -1]
Contrasts
• If there is an interaction, you can not interpret the
effects of the individual factors (e.g. disease and
gender)
GLM
• Important to model all known variables,
even if not experimentally interesting:
– e.g. head movement,
block and subject effects
– minimise residual error
variance for better stats
– effects-of-interest are the
regressors you’re actually
interested in
covariates
conditions:
effects of
interest
Contrasts: Two Groups GLM
Analysis
1. Does Group 1 by itself differ from 0?
Ho: bG1=0; Contrast = C*b = bG1; C = [1 0]
2. Does Group 2 by itself differ from 0?
Ho: bG2=0; Contrast = C*b = bG2; C = [0 1]
64
=
3. Does Group 1 differ from Group 2?
Ho: bG1= bG2; Contrast = C*b = bG1- bG2;
C = [1 -1]
4. Does either Group 1 or Group 2 differ from 0?
C has two rows: F-test (vs t-test)
1
0
Concatenation of contrasts #1 and
C=
#2 0 1
1
1
1
0
0
0
0
0
1
1
bG1
bG2
One Group, One Covariate (Age)
y
=
X * b
(Low-Level Contrasts)
Observations
Contrast
=
Data from
one voxel
65
1
1
1
1
1
21 bG
33 bAge
64
17
47
Intercept: bG
Slope: bAge
Age
Contrasts: One Group, One
Covariate
1. Does Group offset/intercept differ
from 0?
Does Group mean differ from 0
regressing out age (mean-centered)?
Ho: bG=0; Contrast = C*b = bG; C = [1
0], (Treat age as nuisance)
Contrast
2. Does Slope differ from 0?
Ho: bAge=0; Contrast = C*b =
bAge; C = [0 1]
=
1
1
1
1
1
21
33
64
17
47
Intercept: bG
Slope: bAge
Age
66
bG
bAge
One Group, One Covariate
(http://mumford.fmripower.org/mean_centering/)
Two Groups
Activity
Intercept: b1
Slope: m1
Do groups differ in
Intercept?
Do groups differ in Slope?
Is average slope different
than 0?
…
Slope: m2
Age
Intercept: b2
Two Groups
Activity
Intercept: b1
Slope: m1
Slope: m2
Age
Intercept: b2
y11
1 0
y12
1 0
=
y21
0 1
y22
0 1
x11 0 *
x12 0
0 x21
0 x22
b1
b2
m1
m2
Y = X*b
y11 = 1*b1 + 0*b2 + x11*m1 + 0*m2
y12 = 1*b1 + 0*b2 + x12*m1 + 0*m2
y21 = 0*b1 + 1*b2 + 0*m1 + x21*m2
y22 = 0*b1 + 1*b2 + 0*m1 + x22*m2
Two Groups, One Covariate
• Somewhat more complicated design
• Slopes may differ between the groups
• What are you interested in?
• Differences between intercepts? Ie, treat
covariate as a nuisance?
• Differences between slopes? Ie, an
interaction between group and covariate?
70
Two Groups, One (Nuisance)
Covariate
Is there a difference between
the group means?
Synthetic
Data
71
Two Groups, One (Nuisance)
Covariate
Raw Data
Effect of Age
Effect After Age
“Regressed Out”
(e.g. Age=0)
• No difference between groups
• Groups are not well matched for age
• No group effect after accounting for age
• Age is a “nuisance” variable (but important!)
• Slope with respect to Age is same across groups
•If age was mean-centered, there might be a group effect!!!
72
•Depends on mean-centering…
Two Groups, One (Nuisance)
Covariate
(Low-Level Contrasts)
Observations
y
Data from
one voxel
73
Different Offset Same Slope (DOSS)
=
X * b
=
1
1
1
0
0
0
0
0
1
1
21
33
64
17
47
bG1
bG2
bAge
One regressor for Age.
Two Groups, One (Nuisance)
Covariate
=
74
1
1
1
0
0
0
0
0
1
1
Different Offset Same Slope (DOSS)
21
33
64
17
47
bG1
bG2
bAge
One regressor for Age indicates that
groups have same slope – makes
difference between group
means/intercepts independent of
age.
Contrasts: Two Groups + Covariate
1. Does Group 1 intercept/mean differ from
0 (after regressing out effect of age)?
Ho:bG1=0, Contrast = C*b = bG1, C = [1 0 0]
2. Does Group 2 intercept/mean differ from 0
(after regressing out effect of age)?
Ho:bG2=0, Contrast = C*b = bG2, C = [0 1 0]
=
1
1
1
0
0
0
0
0
1
1
21
33
64
17
47
3. Does Group 1 intercept/mean differ from Group 2
intercept/mean (after regressing out effect of age)?
Ho: bG1=bG2, , Contrast = C*b = bG1- bG2, C = [1 -1 0]
4. Does Slope differ from 0 (after regressing out the effect of
group)? Does not have to be a “nuisance”!
Ho: bAge=0, Contrast = C*b = bAge, C = [0 0 1]
75
bG1
bG2
bAge
Two-Groups, One Covariate, Same
Slope
Model from previous
slide
3
4
1,2
(http://mumford.fmripower.org/mean_centering/)
Group/Covariate Interaction
Two Groups, One Covariate, Different Slopes
• Slope with respect to Age differs between
groups
• Interaction between Group and Age
• Intercept different as well
77
Group/Covariate Interaction
(Low-Level Contrasts)
Observations
y
Data from
one voxel
78
Different Offset Different Slope (DODS)
=
X * b
=
1
1
1
0
0
0
0
0
1
1
21 0
33 0
64 0
0 17
0 47
bG1
bG2
bAge1
bAge2
Group-by-Age Interaction
Group/Covariate Interaction
1. Does Slope differ between groups?
Is there an interaction between group and
age?
Ho: bAge1=bAge2, Contrast = C*b = bAge1bAge2, C = [0 0 1 -1],
79
1
1
1
= 0
0
0 21 0
0 33 0
0 64 0
1 0 17
1 0 47
bG1
bG2
bAge1
bAge2
Group/Covariate Interaction
Does this contrast make sense?
2. Does Group 1 intercept/mean differ
from Group 2 mean (after regressing
out effect of age)?
Ho: bG1- bG2, Contrast = C*b = bG1- bG2,
C = [1 -1 0 0]
Very tricky!
This tests for difference at Age=0
What about Age = 12?
What about Age = 20?
80
1
1
1
= 0
0
0 21 0
0 33 0
0 64 0
1 0 17
1 0 47
bG1
bG2
bAge1
bAge2
Group/Covariate Interaction
If you are interested in the difference between
the means but you are concerned there
could be a difference (interaction) in the
slopes:
1. Analyze with interaction model (DODS*)
2. Test for a difference in slopes
3. If there is no difference, re-analyze with
single regressor model (DOSS*)
4. If there is a difference, proceed with
caution
81
* Freesurfer terms
Group/Covariate Interaction
Model from
previous slide
2
1
(http://mumford.fmripower.org/mean_centering/)
Mean Centering
• Across ALL subjects
– Covariate-adjusted group means
• Within each group
– Each group would have the same mean as a one-sample ttest
• Why does it matter?
– The interpretation changes
– Correlation between group and covariate (e.g. MMSE and
Alzheimer’s diagnosis)
Covariates
• If you have a single group:
– Demeaning covariate will not change the slope
– Demeaning makes the group term the mean of
the group; whereas not demeaning makes the
group term the intercept.
Covariates
• If you have a multiple groups:
– Demeaning covariate will not change the slope, no
matter how you demean it
– Demeaning within each group  controlling for
the covariate, but group means are uneffected
– Demeaning across everyone  controlling for the
covariate, but group means are effected. If you do
this, you should refer to group tests as a
comparison of covariate-adjusted means
Longitudinal/Repeated-Measures
Did something change between visits?
• Drug or Behavioral Intervention?
• Training?
• Disease Progression?
• Aging?
• Injury?
• Scanner Upgrade?
• Multiple tasks in the same session?
87
Longitudinal
Subject 1, Visit 1
Subject 1, Visit 2
Paired Differences
Between Subjects
88
Longitudinal Paired Analysis
(V1-V2 Differences in
Low-Level Contrasts)
Observations
y
=
X * b
=
1
1
1
1
1
Design Matrix
(Regressors)
Paired Diffs
from one voxel
89
One-Sample Group Mean (OSGM): Paired t-Test
bDV
Ho: bDV=0
Contrast = C*b = bDV
Contrast Matrix:
C = [1]
GLM – Paired T-Test
GLM – Repeated Measures
Constructing Contrasts
Constructing Contrasts
• What is the null hypothesis?
• Make the null hypothesis equal to 0
• Label the columns based on the weighting of
the components of the null hypothesis
– For repeated measures, form the sub-elements of
the contrast, then apply the weights
Constructing Contrasts
•
•
•
•
•
S1G1C1: [1 zeros(1,10) 1 0 1 0 0 1 0 0 0 0 0]
S1G1C2: [1 zeros(1,10) 1 0 0 1 0 0 1 0 0 0 0]
S2G1C1: [0 1 zeros(1,9) 1 0 1 0 0 1 0 0 0 0 0]
G1: [ones(1,6)/6 zeros(1,5) 1 0 1/3 1/3 1/3 1/3 1/3 1/3 0 0 0]
G1vsG2: [ones(1,6)/6 ones(1,5)/5 1 -1 0 0 0 1/3 1/3 1/3 -1/3 -1/3 1/3]
– (NOTE: This is not a valid contrast, even though it can be constructed.)
Contrast Validity
• Do you only have between-subject factors?
– All contrasts valid
• Do you only have within-subject factors?
– Any contrast comparing levels of a factor/interaction is
valid
– Effect of a single level is not valid
• Do you have between- and within-subject factors?
– Any contrast comparing levels of a factor/interaction is
valid
– Interaction contrasts are valid
– Group/between-subject effects are not valid (e.g. G1vG2)
– Effect of a single level is not valid
Constructing Contrasts
•
•
•
•
•
•
•
•
•
S1G1C1: [1 zeros(1,10) 1 0 1 0 0 1 0 0 0 0 0]
S1G1C2: [1 zeros(1,10) 1 0 0 1 0 0 1 0 0 0 0]
S2G1C1: [0 1 zeros(1,9) 1 0 1 0 0 1 0 0 0 0 0]
G1C1: [ones(1,6)/6 zeros(1,5) 1 0 1 0 0 1 0 0 0 0 0]
G2C1: [zeros(1,6) ones(1,5)/5 0 1 1 0 0 0 0 0 1 0 0]
*C1:[ones(1,6)/12 ones(1,5)/10 1/2 1/2 1 0 0 1/2 0 0 1/2 0 0]
*C1:[ones(1,11)/11 5/11 6/11 0 0 5/11 0 0 6/11 0 0]
C1vsC2: [zeros(1,11) 0 0 1 -1 0 1/2 -1/2 0 1/2 -1/2 0 ]
C1vsC2: [zeros(1,11) 0 0 1 -1 0 5/11 -5/11 0 6/11 -6/11 0 ]
Power Calculations
• The probability that the test will reject the null
hypothesis, when the null hypothesis is false.
• In general, you want to say that you have 8090% power in your study.
• Estimate your effect size, specify your power,
determine the sample size needed.
• CANNOT BE DONE POST-HOC!!!
Power Calculations
• Estimate your effect size
– Which brain region?
• Minimum N to achieve % power in a set of regions
(McLaren et al. 2010)
– Where to find effect sizes?
• Previous studies, pilot studies
• Specify your power (option A)
– The higher the better, but more power means a
larger N
• Specify your N (option B)
– Increasing N will increase the power
Power Calculations - $7600 study
(Mumford et al. 2008)
Programs
• G*Power
• http://fmripower.org/
• http://fmri.wfubmc.edu/cms/talkPowerSampl
eSizeCalculation  voxel-wise
Caveat 1: What is analyzed…
• Missing Data
– NaN
– Zeros
Also AFNI/FSL
Caveat 2: Designs
• Between-subject Designs
• Within-subject Designs
• Mixed Designs
Pick your design Carefully
All of these designs test the same
effect; however only the top 2 give
you the correct RFX results and are
generalizable to the population. The
top right model is a variant of the
GLM that creates a second error
term (more on this next week).
Pick your design Carefully
Variance Corrections
• The issue of non-sphericity
Repeated Measures in FSL
• Limited to designs that have no violations of
sphericity.
Misc. Considerations
Correction for Multiple
Comparisons
• Cluster-based
– Monte Carlo simulation
– Permutation Tests
– Surface Gaussian Random Fields (GRF)
• There but not fully tested
• False Discovery Rate (FDR) – built into tksurfer
and QDEC. (Genovese, et al, NI 2002)
Clustering
1. Choose a voxel/vertex-wise threshold
•
•
Eg, 2 (p<.01), or 3 (p<.001)
Sign (pos, neg, abs)
2. A cluster is a group of connected
(neighboring) voxels/vertices above a
threshold
3. Cluster has a size (volume in mm3 and area in
mm2)
p<.01 (-log10(p)=2)
Negative
p<.0001 (-log10(p)=4)
Negative
What to report in papers
• Be explicit about the model
– What are the factors
– What are the covariates
– What did you set as the variance and dependence for each
factor
• Be explicit about the contrast you are using
• Be explicit about how to interpret the contrast
– Group means, group intercepts, covariate adjusted group means
• Be explicit about the thresholds used
– Corrections for multiple comparisons
– Small Volume Correction (corrected in SPM8 in late Feb. 2012)
SPM/FSL/AFNI/CUSTOM
• It is important to recognize that all programs
that utilize the GLM will produce the same
result. However, if your design matrices or
variance correction methods are different,
then you will see differences.
• Some slides show illustrations from FSL,
others show illustrations from SPM, MATLAB,
or other software. These can be done in all
programs.
Useful Mailing Lists
• SPM – http://www.jiscmail.ac.uk/list/spm.html
• FSL -- http://www.jiscmail.ac.uk/list/fsl.html
• Freesurfer -http://surfer.nmr.mgh.harvard.edu/fswiki/FreeSurferSupport
• CARET -http://brainvis.wustl.edu/wiki/index.php/Caret:Mailing_List
• I highly recommend reading the posts on these lists as they
will save you time in the future.