Data modeling general linear model statistical inference
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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

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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


Linear model in image form

Linear model, in image form…

=

+

+


Linear model in image form1

Linear model, in image form…

Estimated

=

+

+


In image matrix form

… in image matrix form…

=

+


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 HRF

HRF 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 redux

Autocorrelation Redux


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 effect

Fixed 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)


Measuring false positives fwer vs fdr

Signal

Measuring False PositivesFWER vs FDR

Noise

Signal+Noise


Data modeling general linear model statistical inference

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 q

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 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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