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Data ModelingGeneral Linear Model &

Statistical InferenceThomas Nichols, Ph.D.

Assistant ProfessorDepartment of Biostatistics

http://www.sph.umich.edu/~nichols

Brain Function and fMRIISMRM Educational Course

July 11, 2002

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Motivations

• Data Modeling– Characterize Signal– Characterize Noise

• Statistical Inference– Detect signal– Localization (Where’s the blob?)

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Outline

• Data Modeling– General Linear Model – Linear Model Predictors– Temporal Autocorrelation – Random Effects Models

• Statistical Inference– Statistic Images & Hypothesis Testing– Multiple Testing Problem

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Basic fMRI Example

• Data at one voxel– Rest vs.

passive word listening

• Is there an effect?

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A Linear Model

IntensityT

ime = 1 2+ + er

ror

x1 x2

• “Linear” in parameters 1 & 2

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Linear model, in image form…

= + +1 2

Y 11x 22x

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Linear model, in image form…

= + +1̂ 2̂

Y ̂ 11̂x 22ˆ x

Estimated

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… in image matrix form…

= +

2

1

ˆ

ˆ

Y ̂ X ̂

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… in matrix form.

XY

=

+YY X

N

1

N N

1 1p

p

N: Number of scans, p: Number of regressors

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Linear Model Predictors

• Signal Predictors– Block designs– Event-related responses

• Nuisance Predictors– Drift– Regression parameters

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Signal Predictors

• Linear Time-Invariant system

• LTI specified solely by– Stimulus function of

experiment

– Hemodynamic ResponseFunction (HRF)

• Response to instantaneousimpulse

Blocks

Events

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Convolution Examples

Event-Related

Hemodynamic Response Function

Predicted Response

Block Design

Experimental Stimulus Function

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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

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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

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HRF Models

• Deconvolution– Most flexible

• Allows any shape

• Even bizarre, non-sensical ones

– Least sensitive relativeto canonical (again, if canonical is correct) Deconvolution Basis

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Drift Models

• Drift– Slowly varying– Nuisance variability

• Models– Linear, quadratic– Discrete Cosine Transform

Discrete Cosine Transform Basis

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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

XY

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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

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Temporal AutocorrelationIn Brief

• Independence

• Precoloring

• Prewhitening

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Autocorrelation: Independence Model

• Ignore autocorrelation

• Leads to – Under-estimation of variance– Over-estimation of significance– Too many false positives

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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]

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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

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Autocorrelation Redux

Advantage Disadvantage Software

Indep. Simple Inflated significance

All

Precoloring Avoids autocorr. est.

Statistically inefficient

SPM99

Whitening Statistically optimal

Requires precise autocorr. est.

FSL, SPM2

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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

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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

XY

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Building Statistic Images

=

+

= +Y X

Predictor of interest

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Building Statistic Images

• Contrast– A linear combination

of parameters– c’

T =

contrast ofestimated

parameters

varianceestimate

T =

ss22c’(X’X)c’(X’X)++cc

c’bc’b

c’ = 1 0 0 0 0 0 0 0

b b b b b ....

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Hypothesis Test

• So now have a value T for our statistic

• How big is big– Is T=2 big? T=20?

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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 T

as large or larger

level– Acceptable false positive rate

P-val

T

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Random Effects Models

• GLM has only one source of randomness

– Residual error

• But people are another source of error– Everyone activates somewhat differently…

XY

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Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

Subj. 6

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Fixed vs.RandomEffects

• 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

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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

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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

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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

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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)

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FWER MCP Solutions

• Bonferroni

• Maximum Distribution Methods– Random Field Theory– Permutation

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FWER MCP Solutions

• Bonferroni

• Maximum Distribution Methods– Random Field Theory– Permutation

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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 FWERFWER = P(Max voxel u | Ho)

u

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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)

Random Field

Suprathreshold Sets

Threshold

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Controlling FWER: Permutation Test

• Parametric methods– Assume distribution of

max statistic under nullhypothesis

• Nonparametric methods– Use data to find

distribution of max statisticunder null hypothesis

– Any max statistic!

5%

Parametric Null Max Distribution

5%

Nonparametric Null Max Distribution

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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)

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Measuring False PositivesFWER vs FDR

Signal

Signal+Noise

Noise

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FWE

6.7% 10.4% 14.9% 9.3% 16.2% 13.8% 14.0% 10.5% 12.2% 8.7%

Control of Familywise Error Rate at 10%

11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5%

Control of Per Comparison Rate at 10%

Percentage of Null Pixels that are False Positives

Control of False Discovery Rate at 10%

Occurrence of Familywise Error

Percentage of Activated Pixels that are False Positives

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Controlling 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 to p(1), ... , p(r).p(i) i/V q p(i)

i/V

i/V qp-

valu

e

0 1

01

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Conclusions

• Analyzing fMRI Data– Need linear regression basics– Lots of disk space, and time– Watch for MTP (no fishing!)

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Thanks

• Slide help– Stefan Keibel, Rik Henson, JB Poline, Andrew

Holmes