Why N

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Why N How (I forgot the title)

• Donald G. McLaren, Ph.D.Department of
  Neurology, MGH/HMS
• GRECC, ERNM Veterans Hospital
• http//www.martinos.org/mclaren
• 11/15/2012

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Types of Data
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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??

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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

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Overview

• From a line to the GLM and matrices
• Statistical Tests
• Contrasts
• Designs
• Power
• Caveats

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General Linear Model(GLM)
YaXb
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GLM Theory
Is Activity correlated with Age?
Activity
Dependent Variable, Measurement
Subject 1
Subject 2
HRF Amplitude IQ, Height, Weight
Age
Of course, youd need more then two subjects
Independent Variable
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Linear Model
System of Linear Equations y1 1b x1m y2
1b x2m
Intercept Offset
X Design Matrix b Regression Coefficients
Parameter estimates betas
Intercepts and Slopes
Y Xb
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Hypotheses and Contrasts
Is Activity correlated with Age? Does m 0? Null
Hypothesis H0 m0
C0 1 Contrast Matrix
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Hypotheses and Contrasts
Is Activity different from 0? Does b 0? Null
Hypothesis H0 b0
C1 0 Contrast Matrix
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Hypotheses and Contrasts
Is Activity different from 0? Does b 0? Null
Hypothesis H0 b0
C1 0 Contrast Matrix
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Hypotheses and Contrasts
Is Activity different from 0? Does b 0? Null
Hypothesis H0 b0
C1 0 Contrast Matrix
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Hypotheses and Contrasts
Is Activity different from 0? Does b 0? Null
Hypothesis H0 b0
C1 0 Contrast Matrix
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More than Two Data Points
Activity
Intercept b
Slope m
Age
Y Xbn
y1 1b x1m y2 1b x2m y3 1b
x3m y4 1b x4m

• Model Error
• Noise
• Uncertainty

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The General Linear Model
observed predicted
random error
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Summary of the GLM
Y X .
ß e
Observed data 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
Parameters Define the contribution of each
component of the design matrix to the value of
Y Estimated so as to minimise the error, e, i.e.
least sums of squares
Error Difference between the observed data, Y,
and that predicted by the model, Xß?. Not assumed
to be spherical in fMRI
Design matrix 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)

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Brain Imaging

• From the beginning (almost).

5 6 7 5
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Spatial Normalization, Atlas Space
Native Space
MNI305 Space
Subject 1
Subject 1
MNI305
Subject 2
Subject 2
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Group Analysis
Does not have to be all positive!
Contrast Amplitudes Variances (Error Bars)
Contrast Amplitudes
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Mass Univariate Analyses

• (1) Run the GLM for each voxel.
• (2) Compute the statistic from the GLM for each
  voxel
• (3) Inferences

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Statistical Parametric Map (SPM)
Significance t-Map (p,z,F) (Thresholded
plt.01) sig-log10(p)
Contrast Amplitude CON, COPE, CES
Contrast Amplitude Variance (Error
Bars) VARCOPE, CESVAR
Massive Univariate Analysis -- Analyze
each voxel separately
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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.

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Types Of Analysis
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Random Effects (RFx) Analysis
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Random Effects (RFx) Analysis

• Model Subjects as a Random Effect
• Variance comes from a single source variance
  across subjects
• 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

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Mixed Effects (MFx) Analysis
MFx
RFx

• Down-weight each subject based on variance.
• Weighted Least Squares vs (Ordinary LS)

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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
  variance (easier to compute)

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Fixed Effects (FFx) Analysis
FFx
RFx
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Fixed Effects (FFx) Analysis

• As if all subjects treated as a single subject
  (fixed effect)
• Small error bars (with respect to RFx)
• Large DOF
• Same mean as RFx
• Huge areas of activation
• Not generalizable beyond sample.

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Population vs Sample
Group Population (All members) Hundreds? Thousands
? Billions?

• Do you want to draw inferences beyond your
  sample?
• Does sample represent entire population?
• Random Draw?

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fMRI Analysis Overview
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Second-Level Modeling

• These are all random effects (because of variance
  corrections and using betas 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??)

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Statistical Tests
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Implementing the T-test
c 1 0 0 0 0 0 0 0
t-test H0 cT? 0

• Variance Estimate
• Sqrt(VarcT(XTX)-1c)

contrast ofestimatedparameters
T
varianceestimate
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Implementing the F-test
0 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00
0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1
H0 cT? 0
c
additionalvarianceaccounted forby effects
ofinterest
F
errorvarianceestimate
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Contrasts and the Full Model
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T/r/F Notes

• If F is a single row contrast, then FT2
• 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. cohens 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.

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Contrasts

• Identify the Null Hypothesis
• Ho AB
• Make the Null Hypothesis equal 0
• Ho A-B0
• Identify the columns for A and B, apply their
  weights
• Ho 1A(-1)B
• Contrast ? 1 -1

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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 (A1A2)/20
• ½ ½ 0 0
• B is the average B1 B2, or Ho (B1B2)/20
• 0 0 ½ ½
• ½ ½ -½ -½

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Higher Level GLM Analysis
y X b
1 1 1 1 1
Vector of Regression Coefficients (Betas)

Observations (Low-Level Contrasts)
Contrast Matrix C 1 Contrast Cb bG
Design Matrix (Regressors)
Data from one voxel
One-Sample Group Mean (OSGM)
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Two Groups GLM Analysis
y X b
1 1 1 0 0
0 0 0 1 1

Observations (Low-Level Contrasts)
Data from one voxel
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Contrasts Two Groups GLM Analysis
1. Does Group 1 by itself differ from 0? Ho
bG10 Contrast Cb bG1 C 1 0
2. Does Group 2 by itself differ from 0? Ho
bG20 Contrast Cb bG2 C 0 1
3. Does Group 1 differ from Group 2? Ho bG1
bG2 Contrast Cb 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) Concatenation of
contrasts 1 and 2
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One Group, One Covariate (Age)
y X b
1 1 1 1 1
21 33 64 17 47

Observations (Low-Level Contrasts)
Data from one voxel
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Contrasts One Group, One Covariate

• Does Group offset/intercept differ from 0?
• Does Group mean differ from 0 regressing out age?
• Ho bG0 Contrast Cb bG C 1 0, (Treat
  age as nuisance)

2. Does Slope differ from 0? Ho bAge0 Contrast
Cb bAge C 0 1
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Contrasts One Group, One Mean-Centered Covariate

• Does Group offset/intercept differ from 0?
• Does Group mean differ from 0 regressing out age?
• Ho bG0 Contrast Cb bG C 1 0, (Treat
  age as nuisance)

2. Does Slope differ from 0? Ho bAge0 Contrast
Cb bAge C 0 1, Same effect as
non-mean centered covariate
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Group Effects

• Does Activity vary with Disease Status?
• Does Activity vary with Gender?
• Is there an Interaction between DS and G?

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2x2 Group ANOVA
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5
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While this design matrix was generated in SPM,
you could generate it in any of the MRI Analysis
packagees or statistical programs.
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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 MaleFemale
• 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)

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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 havent 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.

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Contrasts

• Is there an interaction?
• Ho DS-Females-DS-Males DSFemales-DSMales
• Ho (DS-Females-DS-Males) (DSFemales-DSMales)
  0
• Ho DS-Females-DS-Males DSFemalesDSMales0
• 1 -1 -1 1 or
• Are the groups different?
• Ho DS-FemalesDS-MalesDSFemalesDSMales
• F-test
• DS-FemalesDS-Males ? 1 -1 0 0
• DS-MalesDSFemales ? 0 1 -1 0
• DSFemalesDSMales ? 0 0 1 -1
• 1 -1 0 0 0 1 -1 0 0 0 1 -1

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Contrasts

• If there is an interaction, you can not interpret
  the effects of the individual factors (e.g.
  disease and gender)

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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 youre
  actually interested in

covariates
conditions effects of interest
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Contrasts Two Groups GLM Analysis
1. Does Group 1 by itself differ from 0? Ho
bG10 Contrast Cb bG1 C 1 0
2. Does Group 2 by itself differ from 0? Ho
bG20 Contrast Cb bG2 C 0 1
3. Does Group 1 differ from Group 2? Ho bG1
bG2 Contrast Cb 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) Concatenation of
contrasts 1 and 2
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One Group, One Covariate (Age)
y X b
1 1 1 1 1
21 33 64 17 47

Observations (Low-Level Contrasts)
Data from one voxel
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Contrasts One Group, One Covariate

• Does Group offset/intercept differ from 0?
• Does Group mean differ from 0 regressing out age
  (mean-centered)?
• Ho bG0 Contrast Cb bG C 1 0, (Treat
  age as nuisance)

2. Does Slope differ from 0? Ho bAge0 Contrast
Cb bAge C 0 1
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One Group, One Covariate
(http//mumford.fmripower.org/mean_centering/)
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Two Groups
Do groups differ in Intercept? Do groups differ
in Slope?
Is average slope different than 0?
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Two Groups
Y Xb
y11 1b1 0b2 x11m1 0m2 y12 1b1
0b2 x12m1 0m2 y21 0b1 1b2
0m1 x21m2 y22 0b1 1b2 0m1
x22m2
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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?

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Two Groups, One (Nuisance) Covariate
Is there a difference between the group means?
Synthetic Data
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Two Groups, One (Nuisance) Covariate
Effect After Age Regressed Out (e.g. Age0)
Raw Data
Effect of Age

• 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!!!
• Depends on mean-centering

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Two Groups, One (Nuisance) Covariate
y X b
bG1 bG2 bAge
1 1 1 0 0
0 0 0 1 1
21 33 64 17 47

Observations (Low-Level Contrasts)
One regressor for Age.
Data from one voxel
Different Offset Same Slope (DOSS)
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Two Groups, One (Nuisance) Covariate
One regressor for Age indicates that groups have
same slope makes difference between group
means/intercepts independent of age.
bG1 bG2 bAge
1 1 1 0 0
0 0 0 1 1
21 33 64 17 47

Different Offset Same Slope (DOSS)
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Contrasts Two Groups Covariate
1. Does Group 1 intercept/mean differ from 0
(after regressing out effect of age)? HobG10,
Contrast Cb bG1, C 1 0 0
2. Does Group 2 intercept/mean differ from
0 (after regressing out effect of age)? HobG20,
Contrast Cb bG2, C 0 1 0
3. Does Group 1 intercept/mean differ from Group
2 intercept/mean (after regressing out effect of
age)? Ho bG1bG2, , Contrast Cb 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 bAge0, Contrast Cb bAge, C
0 0 1
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Two-Groups, One Covariate, Same Slope
Model from previous slide
3
4
1,2
(http//mumford.fmripower.org/mean_centering/)
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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

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Group/Covariate Interaction
y X b
bG1 bG2 bAge1 bAge2
1 1 1 0 0
0 0 0 1 1
21 33 64 0 0
0 0 0 17 47

Observations (Low-Level Contrasts)
Group-by-Age Interaction
Data from one voxel
Different Offset Different Slope (DODS)
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Group/Covariate Interaction

• Does Slope differ between groups?
• Is there an interaction between group and age?
• Ho bAge1bAge2, Contrast Cb bAge1- bAge2, C
  0 0 1 -1,

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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 Cb bG1- bG2, C 1 -1 0 0
Very tricky! This tests for difference at
Age0 What about Age 12? What about Age 20?
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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
• Analyze with interaction model (DODS)
• Test for a difference in slopes
• If there is no difference, re-analyze with single
  regressor model (DOSS)
• If there is a difference, proceed with caution

Freesurfer terms
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Group/Covariate Interaction
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1
(http//mumford.fmripower.org/mean_centering/)
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Mean Centering

• Across ALL subjects
• Covariate-adjusted group means
• Within each group
• Each group would have the same mean as a
  one-sample t-test
• Why does it matter?
• The interpretation changes
• Correlation between group and covariate (e.g.
  MMSE and Alzheimers diagnosis)

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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.

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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

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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?

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Longitudinal
Subject 1, Visit 1
Subject 1, Visit 2
Paired Differences Between Subjects
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Longitudinal Paired Analysis
y X b
1 1 1 1 1
Observations (V1-V2 Differences in Low-Level
Contrasts)

Ho bDV0 Contrast Cb bDV Contrast
Matrix C 1
Design Matrix (Regressors)
Paired Diffs from one voxel
One-Sample Group Mean (OSGM) Paired t-Test
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GLM Paired T-Test
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GLM Repeated Measures
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Constructing Contrasts
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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

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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.)

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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

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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
• C1ones(1,6)/12 ones(1,5)/10 1/2 1/2 1 0 0 1/2
  0 0 1/2 0 0
• C1ones(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

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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 80-90
  power in your study.
• Estimate your effect size, specify your power,
  determine the sample size needed.
• CANNOT BE DONE POST-HOC!!!

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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

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Power Calculations - 7600 study
(Mumford et al. 2008)
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Programs

• GPower
• http//fmripower.org/
• http//fmri.wfubmc.edu/cms/talkPowerSampleSizeCalc
  ulation ? voxel-wise

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Caveat 1 What is analyzed

• Missing Data
• NaN
• Zeros

Also AFNI/FSL
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Caveat 2 Designs

• Between-subject Designs
• Within-subject Designs
• Mixed Designs

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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).
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Pick your design Carefully
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Variance Corrections

• The issue of non-sphericity

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Repeated Measures in FSL

• Limited to designs that have no violations of
  sphericity.

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Misc. Considerations
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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)

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Clustering

• Choose a voxel/vertex-wise threshold
• Eg, 2 (plt.01), or 3 (plt.001)
• Sign (pos, neg, abs)
• A cluster is a group of connected (neighboring)
  voxels/vertices above a threshold
• Cluster has a size (volume in mm3 and area in mm2)

plt.01 (-log10(p)2) Negative
plt.0001 (-log10(p)4) Negative
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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)

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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.

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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/fs
  wiki/FreeSurferSupport
• CARET -- http//brainvis.wustl.edu/wiki/index.php/
  CaretMailing_List
• I highly recommend reading the posts on these
  lists as they will save you time in the future.