Download - GLM for fMRI Emily Falk, Ph.D. University of Pennsylvania Thanks to Thad Polk and Elliot Berkman 1
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GLM for fMRI
Emily Falk, Ph.D.
University of Pennsylvania
Thanks to Thad Polk and Elliot Berkman
Review of preprocessing
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Y X
2nd level (Group)
GLM
Acquire Structurals
(T1)
Acquire Functional
s
Determine Scanning Parameters
Template
Slice Timing CorrectPredictors(De-noise)
RealignSmooth
Co-Register
All subjects
Normalize
β
y = Xβ + ε1st level (Subject) GLM
Contrast
Threshold
βhow - βwhy
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Determine Scanning Parameters
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Determine Scanning parameters
Signal-Noise Ratio (SNR)
Temporal Resolution
Coverage/ Field of
View
Spatial Resolution
-
- - -
-
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Acquire Structurals
(T1)
Acquire Functionals
Determine Scanning Parameters
Slice Timing Correct
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(De-noise)
Time
Slice
TR #1 TR #2 TR #3 TR #4
Slice 1
Slice 2
Slice 3
Slice 4
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Acquire Structurals
(T1)
Acquire Functional
s
Determine Scanning Parameters
Slice Timing Correct
(De-noise)Realign
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Realignment
Minimize sum of squared diff
reslice
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Acquire Structurals
(T1)
Acquire Functional
s
Determine Scanning Parameters
Template
Slice Timing Correct
(De-noise)Realign
Co-Register
Normalize
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Co-registration
Can’t use minimized squared difference on different image types (different tissue -> signal intensity mapping)
Instead use mutual information (maximize)
reference moved image
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normalization
12 parameter affine transformation
Trans x Pitch x Roll x Yaw x Zoom x Sheer
SheerZoom
Even better with segmentation!12
Smoothing
Y X
Acquire Structurals
(T1)
Acquire Functional
s
Determine Scanning Parameters
Template
Slice Timing CorrectPredictors(De-noise)
RealignSmooth
Co-Register
Normalize
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How
Why
X =
Forming predictors
Events Basis Function Predictors (X)
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Y X
2nd level (Group)
GLM
Acquire Structurals
(T1)
Acquire Functional
s
Determine Scanning Parameters
Template
Slice Timing CorrectPredictors(De-noise)
RealignSmooth
Co-Register
All subjects
Normalize
β
y = Xβ + ε1st level (Subject) GLM
Contrast
Threshold
βhow - βwhy
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Y X
2nd level (Group)
GLM
Acquire Structurals
(T1)
Acquire Functional
s
Determine Scanning Parameters
Template
Slice Timing CorrectPredictors(De-noise)
RealignSmooth
Co-Register
All subjects
Normalize
β
y = Xβ + ε1st level (Subject) GLM
Contrast
Threshold
βhow - βwhy
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Lecture Outline
1. Hypothesis Testing (covered Sunday)Null hypothesis vs. alternative hypothesis; Testing hypotheses about population based on a sample; Sampling distributions & Central Limit Theorem; t-statistic, t-distribution, t-tests, p-values; Interpreting results, Type I error, Type II error; One-tailed vs. two-tailed tests; Multiple comparisons
2. General Linear ModelRegression, multiple regression, model fitting, matrix notation, design matrix, example, issues
3. Overview of fMRI data analysisBuild design matrix, fit model to get betas, contrasts, statistical parametric maps, threshold for significance (correcting for multiple comparisons)
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In fMRI, we experimentally manipulate various independent variables (e.g., task, stimulus) while scanning
We are interested in constructing a model of the predicted brain activity that can be used to explain the observed fMRI data in terms of the independent variables.
In fitting the model to the data, we obtain parameter estimates and make inferences about their consistency with the null hypothesis.
Statistical Model
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The General linear model (GLM) approach treats the data as a linear combination of predictor variables plus noise (error).
The predictors are assumed to have known shapes, but their amplitudes are unknown and need to be estimated.
The GLM framework encompasses many of the commonly used techniques in fMRI data analysis (and data analysis more generally).
General Linear Model
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The GLM Family
One continuous
Predictors
Repeated measures
ContinuousOne predictor
Categorical1 pred., 2 levels
More than two measures
Multiple Regression
2-sample t-test
One-way ANOVA
Analysis
Repeated measures ANOVA
DV
General Linear Model
Paired t-test
ContinuousTwo+ preds
Regression
Categorical1 p., 3+ levels
Categorical2+ predictors
Factorial ANOVA
Two measures, one factor
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A simple linear model with one predictor (a made up, non-fMRI example)
• Fit a straight line to the data, the “best fit”• This line is a simplification, a model with two parameters:
intercept and slope• Can use the model to make predictions• Can test the slope parameter against a null hypothesis of
zero, and make inferences about whether there is a statistically significant relationship
Attractiveness
Blu
shin
g
(b
lood
flow
to
cheeks)
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• Vector notation: y, x, and e are vectors of N values corresponding to N observations
The Regression Model
Outcome
(DV)
Intercept
(constant)slope
Predictor value
Error (residual)
For point i:
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• The line is being pulled by vertical rubber bands attached to each point. Vector of red lines is e
• Minimize squares of vertical distance to lines. Minimize:
Fitting and Residual Variance
Attractiveness
Blu
shin
g
(b
lood
flow
to
cheeks)
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• Structural Model for Regression
The Multiple Regression ModelBasic Model for the GLM
DV
intercept
Slope 1
Slope 2
Slope k
Error
Pred1 Pred2 Predk
Matrix notation • solve for beta vector
• minimize sum of squared residuals
Variables
Parameters
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• Alternatively, we can write
Matrix Notation
as
Observed Data Design matrix
Model parameters
Residuals
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Matrix Notation
Is this same as
Observed Data Design matrix
Model parameters
Residuals
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In fMRI the design matrix specifies how the factors of the model change over time.
The design matrix is an np matrix where n is the number of observations over time and p is the number of model parameters
Design Matrix
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• Does exercise predict life-span?• Control for other variables that might be
important. i.e., gender (M/F)
Another Made-up, Non-fMRI Example
Females
Males
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A Non-fMRI Example
Outcome Data Design matrix Model parameters Residuals
= X +
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• Coefficients (betas) for individual predictors test for variance uniquely explained by that predictor
• When predictors are intercorrelated, interpreting the betas can become very tricky! • Changes in sign of one beta when you add others• Changes in significance of one beta when you add others
• This is because the predictors are attempting to explain the SAME variance
• This is called multicollinearity.
Multicollinearity
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Implications for fMRI
• You have at least some control over the design matrix, X, because you manipulate the stimulation.
• Avoid multicollinearity and complex issues with good experimental design!
• Do not use a design with many or highly correlated predictors and expect the modeling to sort everything out
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Lecture Outline
1. Hypothesis Testing (covered last night)Null hypothesis vs. alternative hypothesis; Testing hypotheses about population based on a sample; Sampling distributions & Central Limit Theorem; t-statistic, t-distribution, t-tests, p-values; Interpreting results, Type I error, Type II error; One-tailed vs. two-tailed tests; Multiple comparisons
2. General Linear ModelRegression, multiple regression, model fitting, matrix notation, design matrix, example, issues
QUESTIONS?
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Lecture Outline
1. Hypothesis Testing (covered last night)Null hypothesis vs. alternative hypothesis; Testing hypotheses about population based on a sample; Sampling distributions & Central Limit Theorem; t-statistic, t-distribution, t-tests, p-values; Interpreting results, Type I error, Type II error; One-tailed vs. two-tailed tests; Multiple comparisons
2. General Linear ModelRegression, multiple regression, model fitting, matrix notation, design matrix, example, issues
3. Overview of fMRI data analysisModel specification, parameter estimation, contrasts, statistical parametric maps, threshold for significance (correcting for multiple comparisons)
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A simple research question?• What are the neural correlates of positive
valuation (see recent meta analysis by Bartra et al.)
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fMRI: Example
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Time
Off
On
Off
On
X1
X2
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β1 β2
β1
β1
β1 β1
β1β1β1
β1
β1 β1
β1 β1β1
β1
β1 β1
β1 β1
β1β1β1
β1
β2β2
β2
β2
β2β2
β2
β2β2
β2β2
β2β2β2 β2β2
β2β2 β2
β2
β2
β2
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-
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The SPM way of Plotting the Variables
= +
y X e
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T-test – contrasts – SPM{t}
cT = 1 -1 0 0 0 0 0 0
T =
contrast ofestimated
parameters
varianceestimate
Difference between seeing loved one and random face > 0 ?
=b1 – b2 = cTb> 0 ?
b1 b2 b3 b4 b5 ...
Question:
Null hypothesis: H0: cTb=0
Test statistic:
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T-test: Another Simple Example
Q: activation during attribution?
cT = [ 1 0 ]
Null hypothesis: 1=0
Why versus rest
Design matrix
0.5 1 1.5 2 2.5
10
20
30
40
50
60
70
80
1
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More specifically…
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• Tasks are typically designed to isolate mental processes of interest by comparing parameter estimates (betas)– Viewing hot vs. neutral faces– Why vs. how
• The comparison involves a linear combination of the parameter estimates– Note: investigating a single parameter estimate is
also a contrast and sometimes termed a contrast against the implicit baseline
Isolating Mental Processes
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Voxel-wise Time Series Analysis
Time
BOLD signal
Tim
e
single voxeltime series
Modelspecification
Parameterestimation
Hypothesis
Statistic
SPM
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fMRI Example: One Voxel
Temporal series fMRI voxel time course
amplitude
time
Sou
rce:
J-B
. P
olin
e
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Model Specification:Building The Design Matrix
fMRI Data Design matrix
Model parameters
Residuals
= X +
Predicted task response
intercept
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Parameter Estimation/Model Fitting
y
ERROR
=
+
+
0
1
Find values that producebest fit to observed data
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The SPM Way of Plotting the Variables
= +
y X e
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• beta_####.img– Parameter estimates (β)
• T-contrast– spm_con_####.img
• Contrast image• If you take the
appropriate beta_####.img’s and linearly combine them, it will be identical to the spm_con_####.img
– spm_T_####.img• T-statistic image
• F-contrast– spm_ess_####.img
• Extra sum-of-squares image
– spm_F_####.img• F-statistic image
• ResMS.img– Residual sum-of-squares
(σ 2)
What SPM Computes
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Statistical Parametric Maps
con_???? image
beta_???? images
spmT_???? image
SPM{t}
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T-test – One Dimensional Contrasts – SPM{t}
cT = 1 0 0 0 0 0 0 0
T =
contrast ofestimated
parameters
varianceestimate
box-car amplitude > 0 ?=
b1 = cTb> 0 ?
b1 b2 b3 b4 b5 ...
Question:
Null hypothesis: H0: cTb=0
Test statistic:
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Model Specification:Building The Design Matrix, Adding Predictors
fMRI Data Design matrix Model parameters
Residuals
= X +
Predicted task response
intercept
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The SPM Way of Plotting the Variables
= +
y X e
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T-test – One Dimensional Contrasts – SPM{t}
cT = 1 -1 0 0 0 0 0 0
T =
contrast ofestimated
parameters
varianceestimate
box-car amplitude > 0 ?=
b1 = cTb> 0 ?
b1 b2 b3 b4 b5 ...
Question:
Null hypothesis: H0: cTb=0
Test statistic:
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T-test: A Simple Example
Q: Greater activation during why than
how?cT = [ 1
-1 ]
Null hypothesis: 1-2=0
Why vs. How
Design matrix
0.5 1 1.5 2 2.5
10
20
30
40
50
60
70
80
1
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T-test: A slightly more complicated example
What happens in the brain during attribution?
A
B
C
D
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Another slightly more complicated design
• Do peer ratings change underlying assessments of the attractiveness of faces or merely public reporting (see work by Klucharev et al. and Zaki et al.)
• Manipulate peer ratings (higher, lower, same, no rating)
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• We can estimate the full design matrix, and then apply contrasts to betas after model fitting to estimate effects of interest and make inferences.
t-contrasts
Time
Design Matrix (X) A B C D Covariates
fit apply C
Courtesy of Tor Wager
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t-contrasts
C’b is effect magnitude estimate for (A + B) – (C + D), equivalent to mean(A,B) - mean(C,D) or WHY> HOW
C’b is beta for A only, or WHY_faces > rest
C’b is sum (average) beta for A and B, or WHY > rest
• Statistical contrasts typically sum to zero so that E(C’b) = 0. Makes testing easy.
• Scaling doesn’t matter for most inference, but good to get into habit of using fractions if you need to average.
– [1 -1] and [.5 -.5] give same statistical result (though can affect other calculations like percent signal change).
Time
Design Matrix (X) A B C D
-1
-1
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• T-contrast– Test of a specific combination of predictors
• Linear combination differs from zero (e.g., why – how)?
– Signed: effect can be positive or negative• Some areas might be higher for why and some higher for how
• F-contrast– Test on a set of predictors
• Variance explained by a set of predictors greater than zero?
– Unsigned: amount of variance explained can only be positive– Any t-contrast can be specified as a F-contrast (without the
sign)
Two Kinds of Contrasts
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• Test of set of linear contrasts
– 1) Does a set of regressors explain a significant amount of variance?
• E.g., model with and without temporal derivatives
– 2) Set of differences• E.g., an overall effect of time (T1-T2; T2-T3)
F-Contrast
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• Contrasts can be a linear combination of parameter estimates (t-contrast) or a set of linear combinations of parameter estimates (F-contrast)
• Inference on these contrasts can be performed at the single subject level where degrees of freedom are dictated by number of scans and number parameters modeled
Summary
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Questions?
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Note: Multiple Runs
• Each run is estimated by a separate GLM– Run: one set of scans (typically 5-10
min)– SPM calls this a “Session” although in
typical parlance, a “Session” refers everything that occurs in-between putting a subject in the scanner and taking them out
• Due to– Mean signal varies on a run-by-run
basis• Different intercept term for each run• SPM scales images on a per run basis so
that mean signal across a run is 100
– Low frequency drift and temporal autocorrelation are defined within a run
Put participant in scanner Localizer Functional Run 1 Block 1 Trial 1 Trial 2 … Trial n Block 2 … Block n Functional Run 2 … Functional Run n StructuralTake participant out of scanner
BlockRunSession
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Design Matrix in SPM: One Run
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Design Matrix in SPM: Multiple Runs
Run 1Predictors of interest
Intercept
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Design Matrix in SPM: Multiple Runs
Run 2Predictors of interestIntercept
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Note re: contrasts and multiple runs
• If your runs all have the same conditions in the same order (aim for this):– You can specify the contrast weights for one run
and SPM will assume same for other runs• If your runs don’t conform this way:
– You can specify custom contrast across all run/conditions
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Example
Input: 0 0 1Input: 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1
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Questions?
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Parametric modulation
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What if you want to model a continuous variable?
e.g., making mental state attributions for people rated more or less similar to the participant
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• Modeling changes in brain activity as a function of performance or other continuous variables across multiple levels can provide strong inference– Load, confidence, RT, etc.
Parametric Modulation
Goal (purchase) value of an item
Courtesy of Tor Wager
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Examples
• Effect of attribution (Why > How) modulated by:– Identification with person in the photo– Attractiveness of person in the photo– Accuracy of attribution (i.e., consensus with other
raters)
• Parametric effect on face viewing of:– Ppt. attractiveness rating of face– Peer rating of face– Peer rating of face, controlling for initial rating of face
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Parametric Psychological Events
Principle Predictor: Captures Average Response
Idealized Data
Fitted Response
Parametric Modulator: Captures Modulations
Courtesy of Tor Wager
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• Modulators are orthogonalized w.r.t. principle regressor and each other– Order matters– Modulators will capture residual activation not
captured by preceding regressors
• Example:– Why, Why*Similarity, Why*Attractiveness– Why, Why*Attractiveness, Why*Similarity
Parametric Modulation: Details
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• Modulators model differences in amplitude of response only– Not duration
• Suggested to vary duration of event rather than use parametric modulation in these cases (“variable epoch” model)
Parametric Modulation: Details
Average response
Parametric modulator
(parametric modulator framework)
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Visual cortex: Contrast-modulation (red) vs. duration-modulation (blue)
Grinband et al., 2008
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Parametric Modulation: Contrasts with PMs
= +
y X e
Contrast: 1 0 = main effect of condition AContrast: 0 1 = effect of pm
AA*pm
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• Include modulator(s) can be a good idea to capture parametric effects
• Understand that modulator(s) are orthogonalized to principle regressor and each other when making inferences– Can be adjusted in SPM code by removing calls to
“spm_orth”
• Amplitude modulators may not capture appropriate variance if modulation affects both amplitude and duration of response– Including “variable epochs” and orthogonalizing them w.r.t.
principle regressor may be more appropriate
Parametric Modulation: Summary
Covariates/ Nuisances
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• fMRI data contains large contributions from factors of non-interest (nuisances)• E.g. Scanner instability
• A few such nuisances are:• Low frequency drift• Temporal autocorrelation• Motion
Nuisances in fMRI
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Low Frequency Drift
fMRI Noise: Time domain Frequency domain
Unf
ilter
ed
Lot’s of low frequency power!85
Effect of Low Frequency Drift
blue = datablack = mean + low-frequency driftgreen = predicted response, taking into account
low-frequency driftred = predicted response, not taking into account low-frequency drift
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Accounting for Low Frequency Drift
High-pass filter: High frequency elements are allowed to pass
Default in SPM is 128
In SPM, filter is applied to both data and design
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High-Pass Filtering
fMRI Noise: Time domain Frequency domain
60 s HP filter
1/60 = .016 Hz
Unf
ilter
edF
ilter
ed
Courtesy of Tor Wager88
Low Frequency Drift Considerations
• Avoid long epochs or conditions that repeat at a very low frequency
• Check if filter removes power from predictors of interest and adjust if necessary
• Rule of thumb: 2 x the longest periodicity in the design
Predictor
Power
Filtered frequencies
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• fMRI data contains large contributions from factors of non-interest (nuisances)• E.g. Scanner instability
• A few such nuisances are:• Low frequency drift• Temporal autocorrelation• Motion
Nuisances in fMRI
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Temporal Autocorrelation
• Nearby time-points exhibit positive correlation• Physiological and
scanner instability
Lag 0
r = 1
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Temporal Autocorrelation
• Nearby time-points exhibit positive correlation• Physiological and
scanner instability
Lag 1
r = 0.6
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Temporal Autocorrelation
• Nearby time-points exhibit positive correlation• Physiological and
scanner instability
Lag 2
r = 0.3
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Temporal Autocorrelation
• Nearby time-points exhibit positive correlation• Physiological and
scanner instability
Lag 3
r = 0.1
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• Nearby time-points exhibit positive correlation• Physiological and scanner instability
• GLM assumes that errors are independent• Correlated errors violate this assumption (changes
degrees of freedom)
Temporal Autocorrelation
Autocorrelation function
Random errorsAutocorrelated
Errors 95
• Model correlation structure, V– Model data ignoring temporal autocorrelation– Use residuals to estimate temporal autocorrelation
• Assume autoregressive (AR) model– Signal at time t depends on signal at previous time points plus noise
• Use estimated autocorrelation to filter the data and design matrix
Temporal Autocorrelation: Solution
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Autocorrelation: Case Study
“Model serial correlations: none”
“Model serial correlations: AR(1)”
Courtesy of Tor Wager97
• Understand that autocorrelation exists and that there are algorithms to correct it– Specific algorithm will vary from package to package
• Understand that it impacts inference on individuals (1st level) if not accounted for– Residual variance will be biased
• Parameter estimates will be unbiased if autocorrelation is not taken into account
• Parameter estimates will be less precise if autocorrelation is not taken into account
Temporal Autocorrelation: Take Home
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• fMRI data contains large contributions from factors of non-interest (nuisances)• E.g. Scanner instability
• A few such nuisances are:• Low frequency drift• Temporal autocorrelation• Motion
Nuisances in fMRI
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• Realignment spatially corrects images for motion• However, motion has profound affects on signal that
realignment does not correct
Motion Confounds
Motion
Signal Spikes
Power et al., 2012, NeuroImage100
• Realignment procedures output motion parameters– 6 per scan (x, y, z,
pitch, roll, yaw)– Including these
parameters as nuisance regressors can capture motion-related signal changes and improve model fits
Motion Regression
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Motion Regression Improvements
• Motion parameters capture absolute displacement over time– i.e. motion drift
• Motion-related signal artifacts are typically reflected in scan-to-scan changes– i.e. displacement relative to last scan
• Good idea to include differential motion to capture scan-to-scan changes– But remove sign
• Recommended approach is to include 24 total motion regressors (Lund et al.,
2005, NeuroImage):– Displacement, Differential, Squared Displacement, Squared Differential
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• Motion can sometimes be synched to task– Button press– Shifting during ITI– Motion regressors can be confounded with
task-related regressors in these cases
• May be a good idea to set a motion tolerance criterion and exclude motion regressors from low motion subjects
Motion Regression: Caveats
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• GLM is fit to each voxel, estimating linear contribution of each predictor
• Parameter estimates (β’s) estimate the amplitude of each predictor
• Estimates are affected by presence and absence of other predictors, so it is important to model the data as accurately as possible
• Nuisance confounds are addressed in both the data and design– High-pass filter: removed low frequency drift– Pre-whiten: remove temporal autocorrelation
• Nuisance confounds and methods to address them limit the kinds of predictors that can be effectively included in the GLM
• Multiple runs (called “Session” in SPM) are each estimated by a separate GLM
Summary
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Additional notes
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• Power of predictors depends on the appropriateness of the basis function
• Thus far, assumed a “canonical” hemodynamic response function (HRF)– Double-gamma function– Peak 5s– Undershoot 15s (1/6 height of peak)– ~20s to return to baseline
• True HRF varies between individuals AND regions
Hemodynamic Response
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HRFs Vary Across Brain Regions and TasksProblem for Canonical Model!
Co
urt
esy
of T
or
Wa
ge
r
Aversive picture, n = 30 Aversive anticipation
Stimulus On• HRF shape depends on:
• Vasculature
• Time course of neural activity
See Aguirre et al. 1998, NeuroImage
Checkerboard, n = 10 Thermal pain, n = 23
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• Typically, we are interested in the amplitude of the HRF
• If true HRF peaks earlier or later than expected, amplitude estimates will be systematically biased– True HRFs often peak earlier than canonical HRF (c.f.
Handwerker et al., 2004, NeuroImage; Steffener et al., 2010, NeuroImage)
• May be especially problematic for closely spaced events– Unaccounted for variance can be shifted to earlier or later
events
HRF and Bias
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• One popular approach is to use derivatives of the canonical HRF to capture timing and dispersion related variance
Basis Sets: HRF + Derivatives
1st derivative of the HRF is the “temporal” derivative: captures peaks that occur earlier or later than expected
2nd derivative of the HRF is the “dispersion” derivative: captures narrower or wider responses than expected
Canonical only
Canonical + derivs
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Basis Sets: HRF + Derivatives
Adding derivative can account for delay
Note, including derivative, but basing tests only on the non-deriv does not help!
Calhoun et al., 2004, NeuroImageCalhoun et al., 2004, NeuroImage
110
Basis Sets: Finite Impulse Response
• Another approach is to make no assumptions about the shape of the HRF
• Instead, a set of shifted delta functions is used to capture any response: Finite Impulse Response (FIR) set
111
• Fit is sometimes better with multiple basis sets, inference is harder
• Most common approach is to use the SPM canonical HRF
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• Providing a better model fit– Basis functions
• Modeling continuous variation– Parametric modulators
• Improved confound reduction– Motion regression
Outline
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• Model fits can be made more accurate with the use of basis sets– Can help if canonical HRF is not appropriate– BUT, cost in more complicated inference or methods to extract
amplitude estimates from set of fitted parameters
• Parametric modulators can capture changes across several levels– Can lead to stronger inferences– BUT, note that duration modulation may require a variable epoch
approach
• Motion regression can substantially reduce variability– Can remove severe signal spikes– BUT, may be a good idea to set a motion tolerance criterion
Take-Home
114
• The SPM listserv is a great, active resource for questions about SPM and SPM-related analyses
• Subscribe– Search “spm listserv”, click on first link (SPM LIST
at JISCMAIL.AC.UK), then click on subscribe
• If you have a question– First search the archives to see if it’s already been
asked and answered– If it hasn’t, email your question to
SPM Listserv
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Lecture Outline
1. Hypothesis Testing (covered last night)Null hypothesis vs. alternative hypothesis; Testing hypotheses about population based on a sample; Sampling distributions & Central Limit Theorem; t-statistic, t-distribution, t-tests, p-values; Interpreting results, Type I error, Type II error; One-tailed vs. two-tailed tests; Multiple comparisons
2. General Linear ModelRegression, multiple regression, model fitting, matrix notation, design matrix, example, issues
3. Overview of fMRI data analysisModel specification, parameter estimation, contrasts, statistical parametric maps, threshold for significance (correcting for multiple comparisons)
QUESTIONS?