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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. Motivations. Data Modeling Characterize Signal

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data modeling general linear model statistical inference
Data ModelingGeneral 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

motivations
Motivations
  • Data Modeling
    • Characterize Signal
    • Characterize Noise
  • Statistical Inference
    • Detect signal
    • Localization (Where’s the blob?)
outline
Outline
  • Data Modeling
    • General Linear Model
    • Linear Model Predictors
    • Temporal Autocorrelation
    • Random Effects Models
  • Statistical Inference
    • Statistic Images & Hypothesis Testing
    • Multiple Testing Problem
basic fmri example
Basic fMRI Example
  • Data at one voxel
    • Rest vs.passive word listening
  • Is there an effect?
a linear model
A Linear Model
  • “Linear” in parameters 1&2

error

=

+

+

b1

b2

Time

e

x1

x2

Intensity

in matrix form
=

+

Y

… in matrix form.

N: Number of scans, p: Number of regressors

linear model predictors
Linear Model Predictors
  • Signal Predictors
    • Block designs
    • Event-related responses
  • Nuisance Predictors
    • Drift
    • Regression parameters
signal predictors
Signal Predictors
  • Linear Time-Invariant system
  • LTI specified solely by
    • Stimulus function ofexperiment
    • Hemodynamic ResponseFunction (HRF)
      • Response to instantaneousimpulse

Blocks

Events

convolution examples
Block Design

Event-Related

Convolution Examples

Experimental Stimulus Function

Hemodynamic Response Function

Predicted Response

hrf models
SPM’s HRFHRF Models
  • Canonical HRF
    • Most sensitive if it is correct
    • If wrong, leads to bias and/or poor fit
      • E.g. True responsemay be faster/slower
      • E.g. True response may have smaller/bigger undershoot
hrf models1
HRF Models
  • Smooth Basis HRFs
    • More flexible
    • Less interpretable
      • No one parameter explains the response
    • Less sensitive relativeto canonical (only if canonical is correct)

Gamma Basis

Fourier Basis

hrf models2
HRF Models
  • Deconvolution
    • Most flexible
      • Allows any shape
      • Even bizarre, non-sensical ones
    • Least sensitive relativeto canonical (again, ifcanonical is correct)

Deconvolution Basis

drift models
Drift Models
  • Drift
    • Slowly varying
    • Nuisance variability
  • Models
    • Linear, quadratic
    • Discrete Cosine Transform

Discrete Cosine Transform Basis

general linear model recap
General Linear ModelRecap
  • Fits data Y as linear combination of predictor columns of X
  • Very “General”
    • Correlation, ANOVA, ANCOVA, …
  • Only as good as your X matrix
temporal autocorrelation
Temporal Autocorrelation
  • Standard statistical methods assume independent errors
    • Error i tells you nothing about j i  j
  • fMRI errors not independent
    • Autocorrelation due to
    • Physiological effects
    • Scanner instability
temporal autocorrelation in brief
Temporal AutocorrelationIn Brief
  • Independence
  • Precoloring
  • Prewhitening
autocorrelation independence model
Autocorrelation: Independence Model
  • Ignore autocorrelation
  • Leads to
    • Under-estimation of variance
    • Over-estimation of significance
    • Too many false positives
autocorrelation precoloring
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

[Friston, et al., “To smooth or not to smooth…” NI 12:196-208 2000]

autocorrelation prewhitening
Autocorrelation:Prewhitening
  • Statistically optimal solution
  • If know true autocorrelation exactly, canundo 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
autocorrelation models
Autocorrelation: Models
  • Autoregressive
    • Error is fraction of previous error plus “new” error
    • AR(1): i = i-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( i, i-k )
      • Software: FSL’s FEAT
statistic images hypothesis testing
Statistic Images &Hypothesis Testing
  • For each voxel
    • Fit GLM, estimate betas
      • Write b for estimate of 
    • But usually not interested in all betas
      • Recall  is a length-p vector
building statistic images
Building Statistic Images

Predictor of interest

b1

b2

b3

b4

b5

b6

b7

b8

b9

=

+

´

=

+

Y

X

b

e

building statistic images1
c’ = 1 0 0 0 0 0 0 0

b1b2b3b4b5....

contrast ofestimatedparameters

c’b

T =

T =

varianceestimate

s2c’(X’X)+c

Building Statistic Images
  • Contrast
    • A linear combination of parameters
    • c’
hypothesis test
Hypothesis Test
  • So now have a value T for our statistic
  • How big is big
    • Is T=2 big? T=20?
hypothesis testing
P-val Hypothesis Testing
  • Assume Null Hypothesis of no signal
  • Given that there is nosignal, how likely is our measured T?
  • P-value measures this
    • Probability of obtaining Tas large or larger
  •  level
    • Acceptable false positive rate

T

random effects models
Random Effects Models
  • GLM has only one source of randomness
    • Residual error
  • But people are another source of error
    • Everyone activates somewhat differently…
fixed vs random effects
Distribution of each subject’s effectFixed vs.RandomEffects

Subj. 1

Subj. 2

  • Fixed Effects
    • Intra-subject variation suggests all these subjects different from zero
  • Random Effects
    • Intersubject variation suggests population not very different from zero

Subj. 3

Subj. 4

Subj. 5

Subj. 6

0

random effects for fmri
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
random effects for fmri random vs fixed
Random Effects for fMRIRandom 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
multiple testing problem
t > 2.5

t > 4.5

t > 0.5

t > 1.5

t > 3.5

t > 5.5

t > 6.5

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!
mcp solutions measuring false positives
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)
fwer mcp solutions
FWER MCP Solutions
  • Bonferroni
  • Maximum Distribution Methods
    • Random Field Theory
    • Permutation
fwer mcp solutions1
FWER MCP Solutions
  • Bonferroni
  • Maximum Distribution Methods
    • Random Field Theory
    • Permutation
fwer mcp solutions controlling fwer w max
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

fwer mcp solutions random field theory
FWER MCP Solutions:Random Field Theory
  • Euler Characteristic u
    • Topological Measure
      • #blobs - #holes
    • At high thresholds,just counts blobs
    • FWER = P(Max voxel u | Ho) = P(One or more blobs | Ho) P(u  1 | Ho) E(u| Ho)

Threshold

Random Field

Suprathreshold Sets

controlling fwer permutation test
5%

Parametric Null Max Distribution

5%

Nonparametric Null Max Distribution

Controlling FWER: Permutation Test
  • Parametric methods
    • Assume distribution ofmax statistic under nullhypothesis
  • Nonparametric methods
    • Use data to find distribution of max statisticunder null hypothesis
    • Any max statistic!
measuring false positives
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)
slide43
11.3%

11.3%

12.5%

10.8%

11.5%

10.0%

10.7%

11.2%

10.2%

9.5%

6.7%

10.5%

12.2%

8.7%

10.4%

14.9%

9.3%

16.2%

13.8%

14.0%

Control of Per Comparison Rate at 10%

Percentage of Null Pixels that are False Positives

Control of Familywise Error Rate at 10%

FWE

Occurrence of Familywise Error

Control of False Discovery Rate at 10%

Percentage of Activated Pixels that are False Positives

controlling fdr benjamini hochberg
p(i) i/V qControlling FDR:Benjamini & Hochberg
  • Select desired limit q on E(FDR)
  • Order p-values, p(1)p(2) ...  p(V)
  • Let r be largest i such that
  • Reject all hypotheses corresponding top(1), ... , p(r).

1

p(i)

p-value

i/V q

0

0

1

i/V

conclusions
Conclusions
  • Analyzing fMRI Data
    • Need linear regression basics
    • Lots of disk space, and time
    • Watch for MTP (no fishing!)
thanks
Thanks
  • Slide help
    • Stefan Keibel, Rik Henson, JB Poline, Andrew Holmes
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