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FMRI Data Analysis:
I. Basic Analyses and the General Linear Model
FMRI Graduate Course (NBIO 381, PSY 362)
Dr. Scott Huettel, Course Director
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
When do we not need statistical analysis?
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Inter-ocular Trauma Test (Lockhead, personal communication)
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Why use statistical analyses?
• Replaces simple subtractive methods
– Signal highly corrupted by noise
• Typical SNRs: 0.2 – 0.5
– Sources of noise
• Thermal variation (unstructured)
• Physiological, task variability (structured)
• Assesses quality of data
– How reliable is an effect?
– Allows distinction of weak, true effects from strong,
noisy effects
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
What do our analyses generate?
• Statistical Parametric Maps
• Brain maps of statistical quality of
measurement
– Examples: correlation, regression approaches
– Displays likelihood that the effect observed
is due to chance factors
– Typically expressed in probability (e.g., p <
0.001), or via t or z statistics
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
What are our statistics for?
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Key Concepts
• Within-subjects analyses
– Simple non-GLM approaches (older)
– General Linear Model (GLM)
• Across-subjects analyses
– Fixed vs. Random effects
• Correction for Multiple Comparisons
• Displaying Data
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Simple Hypothesis-Driven Analyses
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t-test across conditions
Time point analysis (i.e., t-test)
Correlation
Fourier analysis
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Correlation Approaches (old-school)
• How well does our data match an
expected hemodynamic response?
• Special case of General Linear Model
• Limited by choice of HDR
– Assumes particular correlation template
– Does not model task-unrelated variability
– Does not model interactions between events
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Fourier Analysis
• Fourier transform: converts information in time domain
to frequency domain
– Used to change a raw time course to a power spectrum
– Hypothesis: any repetitive/blocked task should have power at
the task frequency
• BIAC function: FFTMR
– Calculates frequency and phase plots for time series data.
• Equivalent to correlation in frequency domain
• Subset of general linear model
– Same as if used sine and cosine as regressors
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Power
12s on, 12s off
FMRI – Week 9 – Analysis I
Frequency (Hz)
Scott Huettel, Duke University
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0.005 0.047 0.089 0.130 0.172 0.214 0.255 0.297
Frequency
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FMRI – Week 9 – Analysis I
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Phase Angle (Degrees)
Scott Huettel, Duke University
The General Linear Model (GLM)
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Basic Concepts of the GLM
• GLM treats the data as a linear combination of
model functions plus noise
– Model functions have known shapes
– Amplitude of functions are unknown
– Assumes linearity of HDR; nonlinearities can be
modeled explicitly
• GLM analysis determines set of amplitude
values that best account for data
– Usual cost function: least-squares deviance of
residual after modeling (noise)
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Signal, noise, and the General
Linear Model
Y  M  
Amplitude (solve for)
Measured Data
Noise
Design Model
Cf. Boynton et al., 1996
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Form of the GLM
Model
Model Functions
*
Amplitudes
+
Noise
=
N Time Points
Data
N Time Points
Model Functions
Y  M  
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Design Matrices
Images
Model Parameters
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Regressors
(How much of the
variance in the data
does each explain?)
Contrasts
(Does one regressor
explain more variance
than another?)
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Task and Nuisance Regressors
Task Regressors
FMRI – Week 9 – Analysis I
Nuisance (Motion)
Regressors
Scott Huettel, Duke University
Hemodynamic and Basis Functions
Double Gamma
Gaussian
Gamma
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
The optimal relation between regressors
depends on our research question
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Suppose that we have two
correlated regressors.
R1: Motor?
R2: Visual?
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Because of their correlation,
the design is inefficient at
distinguishing the
contributions of R1 and R2 to
the activation of a voxel.
Good Contrast
R1
R2
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Bad Contrast
R1
R2
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Value of R2 (at each point in time)
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Value of R1 (at each point in time)
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Let’s now make the
regressors anti-correlated .
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Now, the design allows us to
separate the contributions of
each regressor, but cannot
look at their common effect.
Good Contrast
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Bad Contrast
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Value of R2 (at each point in time)
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Value of R1 (at each point in time)
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
We can shift our block design
in time, so that the regressors
are off-set.
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This makes the activation
uncorrelated, but doesn’t
efficiently use the space.
Value of R2 (at each point in time)
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Value of R1 (at each point in time)
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
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Now, we get more of a
“cloud” arrangement of the
time points.
(Squareness and lack of
evenness is caused by my
simulation approach)
Good Contrast
R1
R2
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FMRI – Week 9 – Analysis I
Good Contrast
R1
R2
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Value of R2 (at each point in time)
And, we can make the
regressors uncorrelated with
each other through
randomization.
Value of R1 (at each point in time)
Scott Huettel, Duke University
Orthogonalization of Regressors
NonOrthogonal
Cue
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Cue
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Orthogonal
FMRI – Week 9 – Analysis I
Target
Regressor
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(Orthogonalized)
Scott Huettel, Duke University
Setting up Parametric Effects
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Parametric Effect
Constant Effect
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FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Fixed and Random Effects
Comparisons
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Fixed Effects
• Fixed-effects Model
– Assumes that effect is constant (“fixed”) in the population
– Uses data from all subjects to construct statistical test
– Examples
• Averaging across subjects before a t-test
• Taking all subjects’ data and then doing an ANOVA
– Allows inference to subject sample
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Random Effects
• Random-effects Model
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Assumes that effect varies across the population
Accounts for inter-subject variance in analyses
Allows inferences to population from which subjects are drawn
Especially important for group comparisons
Required by many reviewers/journals
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Key Concepts of Random Effects
• Assumes that activation parameters may vary across
subjects
– Since subjects are randomly chosen, activation parameters may
vary within group
– (Fixed-effects models assume that parameters are constant
across individuals)
• Calculates descriptive statistic for each subject
– i.e., parameter estimate from regression model
• Uses all subjects’ statistics in a higher-level analysis
– i.e., group significance based on the distribution of subjects’
values.
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
The Problem of Multiple Comparisons
P < 0.05 (1682 voxels)
FMRI – Week 9 – Analysis I
P < 0.01 (364 voxels)
P < 0.001 (32 voxels)
Scott Huettel, Duke University
A
t = 2.10, p < 0.05 (uncorrected)
FMRI – Week 9 – Analysis I
B
C
t = 3.60, p < 0.001 (uncorrected)
t = 7.15, p < 0.05,
Bonferroni Corrected
Scott Huettel, Duke University
Options for Multiple Comparisons
• Statistical Correction (e.g., Bonferroni)
– Family-wise Error Rate
– False Discovery Rate (FDR)
• Cluster Analyses
• ROI Approaches
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Statistical Corrections
• If more than one test is made, then the
collective alpha value is greater than the
single-test alpha
– That is, overall Type I error increases
• One option is to adjust the alpha value of the
individual tests to maintain an overall alpha
value at an acceptable level
– This procedure controls for overall Type I error
– Known as Bonferroni Correction
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
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Probability of Type I Error
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FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Bonferroni Correction
• Very severe correction
– Results in very strict significance values
– Typical brain may have up to ~30,000 functional voxels
• P(Type I error) ~ 1.0 ; Corrected alpha ~ 0.000003
• Greatly increases Type II error rate
• Is not appropriate for correlated data
– If data set contains correlated data points, then the effective
number of statistical tests may be greatly reduced
– Most fMRI data has significant correlation
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Gaussian Field Theory
• Approach developed by Worsley and colleagues
to account for multiple comparisons
• Provides false positive rate for fMRI data based
upon the smoothness of the data
– If data are very smooth, then the chance of noise
points passing threshold is reduced
• Recommendation: Use a combination of voxel
and cluster correction methods
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Cluster Analyses
• Assumptions
– Assumption I: Areas of true fMRI activity will
typically extend over multiple voxels
– Assumption II: The probability of observing an
activation of a given voxel extent can be calculated
• Cluster size thresholds can be used to reject
false positive activity
– Forman et al., Mag. Res. Med. (1995)
– Xiong et al., Hum. Brain Map. (1995)
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
How many foci of activation?
Data from motor/visual event-related task (used in laboratory)
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
How large should clusters be?
• At typical alpha values, even small cluster sizes provide
good correction
– Spatially Uncorrelated Voxels
• At alpha = 0.001, cluster size 3
• Type 1 rate to << 0.00001 per voxel
– Highly correlated Voxels
• Smoothing (FW = 0.5 voxels)
• Increases needed cluster size to 7 or more voxels
• Efficacy of cluster analysis depends upon shape and
size of fMRI activity
– Not as effective for non-convex regions
– Power drops off rapidly if cluster size > activation size
Data from Forman et al., 1995
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
False Discovery Rate
• Controls the expected proportion of false
positive values among suprathreshold values
– Genovese, Lazar, and Nichols (2002, NeuroImage)
– Does not control for chance of any face positives
• FDR threshold determined based upon observed
distribution of activity
– So, sensitivity increases because metric becomes
more lenient as voxels become significant
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
(sum)
Genovese, et al., 2002
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
ROI Comparisons
• Changes basis of statistical tests
– Voxels: ~16,000
– ROIs : ~ 1 – 100
• Each ROI can be thought of as a very large
volume element (e.g., voxel)
– Anatomically-based ROIs do not introduce bias
• Potential problems with using functional ROIs
– Functional ROIs result from statistical tests
– Therefore, they cannot be used (in themselves) to
reduce the number of comparisons
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Voxel and ROI analyses are similar, in concept
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Summary of Multiple Comparison Correction
• Basic statistical corrections are often too severe for
fMRI data
• What are the relative consequences of different error
types?
– Correction decreases Type I rate: fewer false positives
– Correction increases Type II rate: more misses
• Alternate approaches may be more appropriate for fMRI
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Cluster analyses
Region of interest approaches
Smoothing and Gaussian Field Theory
False Discovery Rate
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Displaying Data
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Never
Mask!
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University
Summary of Basic Analysis Methods
• Simple experimental designs
– Blocked: t-test, Fourier analysis
– Event-related: correlation, t-test at time points
• Complex experimental designs
– Regression approaches (GLM)
• Critical problem: Minimization of Type I Error
– Strict Bonferroni correction is too severe
– Cluster analyses improve
– Accounting for smoothness of data also helps
• Use random-effects analyses to allow
generalization to the population
FMRI – Week 9 – Analysis I
Scott Huettel, Duke University