An Introduction to Functional MRI

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Transcript An Introduction to Functional MRI

FMRI Data Analysis:
I. Basic Analyses and the General Linear Model
FMRI Undergraduate Course (PSY 181F)
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
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What are our statistics for?
FMRI – Week 9 – Analysis I
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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
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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
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Power
12s on, 12s off
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Frequency (Hz)
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Scott Huettel, Duke University
The General Linear Model (GLM)
FMRI – Week 9 – Analysis I
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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
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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
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Amplitudes
+
Noise
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N Time Points
Data
N Time Points
Model Functions
Y  M  
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Design Matrices
Images
Model Parameters
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Regressors
(How much of the
variance in the data
does each explain?)
Contrasts
(Does one regressor
explain more variance
than another?)
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The optimal relation between regressors
depends on our research question
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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
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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
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)
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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
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Good Contrast
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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)
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Fixed and Random Effects
Comparisons
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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
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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
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The Problem of Multiple Comparisons
P < 0.05 (1682 voxels)
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P < 0.01 (364 voxels)
P < 0.001 (32 voxels)
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t = 2.10, p < 0.05 (uncorrected)
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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
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Probability of Type I Error
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FMRI – Week 9 – Analysis I
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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
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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
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How many foci of activation?
Data from motor/visual event-related task (used in laboratory)
FMRI – Week 9 – Analysis I
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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
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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
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(sum)
Genovese, et al., 2002
FMRI – Week 9 – Analysis I
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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
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Voxel and ROI analyses are similar, in concept
FMRI – Week 9 – Analysis I
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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
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Displaying Data
FMRI – Week 9 – Analysis I
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Never
Mask!
FMRI – Week 9 – Analysis I
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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
Midterm Results
• Undergraduate
– Mean score: ~78/100
– Mean grade: B
• See instructor or TAs for
questions about answers
• See instructor for grading
questions/concerns
• Reminder: Projects count
as much as midterm!
FMRI – Week 9 – Analysis I
• Graduate
– Mean score: ~81/100
– Mean grade: B
• See instructor for any
examination questions
• Graduate grades can be
raised by meeting with
instructor
Scott Huettel, Duke University