Transcript Data Modeling, General Linear Model & Statistical Inference

Data Modeling
General Linear Model &
Statistical Inference
Thomas Nichols, Ph.D.
Assistant Professor
Department of Biostatistics
http://www.sph.umich.edu/~nichols
Brain Function and fMRI
ISMRM Educational Course
July 11, 2002
1
Motivations
• Data Modeling
– Characterize Signal
– Characterize Noise
• Statistical Inference
– Detect signal
– Localization (Where’s the blob?)
2
Outline
• Data Modeling
–
–
–
–
General Linear Model
Linear Model Predictors
Temporal Autocorrelation
Random Effects Models
• Statistical Inference
– Statistic Images & Hypothesis Testing
– Multiple Testing Problem
3
Basic fMRI Example
• Data at one
voxel
– Rest vs.
passive
word
listening
• Is there an
effect?
4
Time
• “Linear” in
parameters
b1 & b2
= b1
Intensity
+ b2
x1
error
A Linear Model
+
e
x2
5
Linear model, in image form…
= b1
Y

b1 x1
+ b2
+
 b2 x2

e
6
Linear model, in image form…
Estimated
= bˆ1
Y

bˆ1 x1
+ bˆ2
+
ˆ
b

2 x2

eˆ
7
… in image matrix form…
 bˆ1 
ˆ 
 b 2 
=
Y

X

bˆ
+

eˆ
8
… in matrix form.
p
1
1
1
Y  Xb  e
b
Y
N
=
N
X
p
+
e
N
N: Number of scans, p: Number of regressors
9
Linear Model Predictors
• Signal Predictors
– Block designs
– Event-related responses
• Nuisance Predictors
– Drift
– Regression parameters
10
Signal Predictors
• Linear Time-Invariant system
• LTI specified solely by
– Stimulus function of
experiment
Blocks
Events
– Hemodynamic Response
Function (HRF)
• Response to instantaneous
impulse
11
Convolution
Examples
Block Design
Event-Related
Experimental
Stimulus
Function
Hemodynamic
Response
Function
Predicted
Response
12
HRF Models
• Canonical HRF
– Most sensitive
if it is correct
– If wrong, leads to
bias and/or poor fit
• E.g. True response
may be faster/slower
• E.g. True response
may have smaller/
bigger undershoot
SPM’s HRF
13
HRF Models
• Smooth Basis HRFs
– More flexible
– Less interpretable
• No one parameter
explains the response
Gamma Basis
– Less sensitive relative
to canonical (only
if canonical is correct)
Fourier Basis
14
HRF Models
• Deconvolution
– Most flexible
• Allows any shape
• Even bizarre,
non-sensical ones
– Least sensitive relative
to canonical (again, if
canonical is correct)
Deconvolution Basis
15
Drift Models
• Drift
– Slowly varying
– Nuisance variability
• Models
– Linear, quadratic
– Discrete Cosine Transform
Discrete Cosine 16
Transform Basis
General Linear Model
Recap
• Fits data Y as linear combination of
predictor columns of X
Y  Xb  e
• Very “General”
– Correlation, ANOVA, ANCOVA, …
• Only as good as your X matrix
17
Temporal Autocorrelation
• Standard statistical methods assume
independent errors
– Error ei tells you nothing about ej i  j
• fMRI errors not independent
– Autocorrelation due to
– Physiological effects
– Scanner instability
18
Temporal Autocorrelation
In Brief
• Independence
• Precoloring
• Prewhitening
19
Autocorrelation:
Independence Model
• Ignore autocorrelation
• Leads to
– Under-estimation of variance
– Over-estimation of significance
– Too many false positives
20
Autocorrelation:
Precoloring
• Temporally blur, smooth your data
– This induces more dependence!
– But we exactly know the form of the
dependence induced
– Assume that intrinsic autocorrelation is
negligible relative to smoothing
• Then we know autocorrelation exactly
• Correct GLM inferences based on “known”
autocorrelation
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[Friston, et al., “To smooth or not to smooth…” NI 12:196-208 2000]
Autocorrelation:
Prewhitening
• Statistically optimal solution
• If know true autocorrelation exactly, can
undo the dependence
– De-correlate your data, your model
– Then proceed as with independent data
• Problem is obtaining accurate estimates of
autocorrelation
– Some sort of regularization is required
• Spatial smoothing of some sort
22
Autocorrelation Redux
Advantage
Disadvantage
Software
Indep.
Simple
Inflated
significance
All
Precoloring
Avoids
Statistically
autocorr. est. inefficient
Whitening
Statistically
optimal
SPM99
Requires precise FSL,
autocorr. est.
SPM2
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Autocorrelation: Models
• Autoregressive
– Error is fraction of previous error plus
“new” error
– AR(1): ei = ei-1 + I
• Software: fmristat, SPM99
• AR + White Noise or ARMA(1,1)
– AR plus an independent WN series
• Software: SPM2
• Arbitrary autocorrelation function
– k = corr( ei, ei-k )
• Software: FSL’s FEAT
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Statistic Images &
Hypothesis Testing
• For each voxel
Y  Xb  e
– Fit GLM, estimate betas
• Write b for estimate of b
– But usually not interested in all betas
• Recall b is a length-p vector
25
Building Statistic Images
Predictor of interest
b1
b2
b3
b4
=
b5
+
b6
b7
b8
b9
Y
=
X

b
+
e
26
Building Statistic Images
c’ = 1 0 0 0 0 0 0 0
• Contrast
– A linear combination
of parameters
– c’b
contrast of
estimated
parameters
T=
c’b
T=
variance
estimate
b1 b2 b3 b4 b5 ....
s2c’(X’X)+c
27
Hypothesis Test
• So now have a value T for our statistic
• How big is big
– Is T=2 big? T=20?
28
Hypothesis Testing
• Assume Null Hypothesis of no signal
• Given that there is no
signal, how likely
is our measured T?
• P-value measures this
T
P-val
– Probability of obtaining T
as large or larger
•  level
– Acceptable false positive rate
29
Random Effects Models
• GLM has only one source of randomness
Y  Xb  e
– Residual error
• But people are another source of error
– Everyone activates somewhat differently…
30
Fixed vs.
Random
Effects
• Fixed Effects
– Intra-subject
variation suggests
all these subjects
different from zero
• Random Effects
– Intersubject
variation suggests
population not
very different from
zero
Distribution of
each subject’s
effect
Subj. 1
Subj. 2
Subj. 3
Subj. 4
Subj. 5
Subj. 6
0
31
Random Effects for fMRI
• Summary Statistic Approach
– Easy
• Create contrast images for each subject
• Analyze contrast images with one-sample t
– Limited
• Only allows one scan per subject
• Assumes balanced designs and homogeneous meas. error.
• Full Mixed Effects Analysis
– Hard
• Requires iterative fitting
• REML to estimate inter- and intra subject variance
– SPM2 & FSL implement this, very differently
– Very flexible
32
Random Effects for fMRI
Random vs. Fixed
• Fixed isn’t “wrong”, just usually isn’t of interest
• If it is sufficient to say
“I can see this effect in this cohort”
then fixed effects are OK
• If need to say
“If I were to sample a new cohort from the
population I would get the same result”
then random effects are needed
33
Multiple Testing Problem
• Inference on statistic images
– Fit GLM at each voxel
– Create statistic images of effect
• Which of 100,000 voxels are significant?
– =0.05  5,000 false positives!
t > 0.5
t > 1.5
t > 2.5
t > 3.5
t > 4.5
t > 5.5
t > 6.5
34
MCP Solutions:
Measuring False Positives
• Familywise Error Rate (FWER)
– Familywise Error
• Existence of one or more false positives
– FWER is probability of familywise error
• False Discovery Rate (FDR)
– R voxels declared active, V falsely so
• Observed false discovery rate: V/R
– FDR = E(V/R)
35
FWER MCP Solutions
• Bonferroni
• Maximum Distribution Methods
– Random Field Theory
– Permutation
36
FWER MCP Solutions
• Bonferroni
• Maximum Distribution Methods
– Random Field Theory
– Permutation
37
FWER MCP Solutions:
Controlling FWER w/ Max
• FWER & distribution of maximum
FWER = P(FWE)
= P(One or more voxels  u | Ho)
= P(Max voxel  u | Ho)
• 100(1-)%ile of max distn controls FWER
FWER = P(Max voxel  u | Ho)  

u
38
FWER MCP Solutions:
Random Field Theory
• Euler Characteristic u
– Topological Measure
• #blobs - #holes
Threshold
– At high thresholds,
Random Field
just counts blobs
– FWER = P(Max voxel  u | Ho)
= P(One or more blobs | Ho)
 P(u  1 | Ho)
 E(u | Ho)
39 Sets
Suprathreshold
Controlling FWER:
Permutation Test
• Parametric methods
– Assume distribution of
max statistic under null
hypothesis
• Nonparametric methods
5%
Parametric Null Max Distribution
– Use data to find
distribution of max statistic
5%
under null hypothesis
Nonparametric Null Max Distribution
– Any max statistic!
40
Measuring False Positives
• Familywise Error Rate (FWER)
– Familywise Error
• Existence of one or more false positives
– FWER is probability of familywise error
• False Discovery Rate (FDR)
– R voxels declared active, V falsely so
• Observed false discovery rate: V/R
– FDR = E(V/R)
41
Measuring False Positives
FWER vs FDR
Noise
Signal
Signal+Noise
42
Control of Per Comparison Rate at 10%
11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2%
Percentage of Null Pixels that are False Positives
9.5%
Control of Familywise Error Rate at 10%
Occurrence of Familywise Error
FWE
Control of False Discovery Rate at 10%
6.7%
10.4% 14.9% 9.3% 16.2% 13.8% 14.0% 10.5% 12.2%
Percentage of Activated Pixels that are False Positives
8.7%
43
Controlling FDR:
Benjamini & Hochberg
1
• Select desired limit q on E(FDR)
• Order p-values, p(1)  p(2)  ...  p(V)
• Let r be largest i such that
p(i)  i/V  q
p-value
i/V  q
0
• Reject all hypotheses
corresponding to
p(1), ... , p(r).
p(i)
0
1
i/V
44
Conclusions
• Analyzing fMRI Data
– Need linear regression basics
– Lots of disk space, and time
– Watch for MTP (no fishing!)
45
Thanks
• Slide help
– Stefan Keibel, Rik Henson, JB Poline, Andrew
Holmes
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