FMRI GLM Analysis7/15/2014 TuesdayYingying Wanghttp://neurobrain.net/2014STBCH/index.html

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Overview

Realignment Smoothing

Normalization

General linear model

Statistical parametric map (SPM)Image time-series

Parameter estimates

Design matrix

Template

Kernel

Randomfield theory

p <0.05

Statisticalinference

GLM

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Parametric statistics•one sample t-test• two sample t-test•paired t-test•Anova•AnCova•correlation• linear regression•multiple regression•F-tests•etc…

all cases of the

General Linear Model

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The General Linear Model (GLM)

T-tests, correlations and Fourier analysis work for simple designs and were common in the early days of imaging.The General Linear Model (GLM) is now available in many software packages and tends to be the analysis of choice.

Why is the GLM so great?the GLM is an overarching tool that can do anything that the simpler tests doyou can examine any combination of contrasts (e.g., intact - scrambled, scrambled - baseline) with one GLM rather than multiple correlationsthe GLM allows much greater flexibility for combining data within subjects and between subjects

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The General Linear Model (GLM)

it also makes it much easier to counterbalance orders and discard bad sections of data

the GLM allows you to model things that may account for variability in the data even though they aren’t interesting in and of themselves (e.g., head motion)

as we will see later in the course, the GLM also allows you to use more complex designs (e.g., factorial designs)

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A mass-univariate approach

Time

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General Linear Model

Equation for single voxel j:

yj = xj1 b1 + … xjL bL + ej ej ~ N(0,s2)

yj : data for scan, j = 1…J xjl : explanatory variables / covariates / regressors, l = 1…L

bl : parameters / regression slopes / fixed effects ej : residual errors, independent & identically distributed (“iid”)

(Gaussian, mean of zero and standard deviation of σ)

Equivalent matrix form:

y = Xb + e X : “design matrix” / model

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GLM

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The Design Matrix

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Estimation of the parameters

= +𝜀𝛽

𝜀 𝑁 (0 ,𝜎 2 𝐼 )

��=(𝑋𝑇 𝑋 )−1 𝑋𝑇 𝑦

i.i.d. assumptions:

OLS estimates:

��1=3.9831

��2 −7={0.6871 ,1.9598 , 1.3902 , 166.1007 , 76.4770 ,− 64.8189 }

��8=131.0040

=

�� 2= ��𝑇 ��𝑁 −𝑝�� 𝑁 (𝛽 ,𝜎2(𝑋𝑇 𝑋 )−1 )

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OLS (Ordinary Least Squares)

This is a scalar and the transpose of a scalar is a scalar

SOLUTION: OLS of the Parameters

The goal is to minimize the quadratic error between data and model

You find the extremum (max or min) of a function by taking its derivative and setting it to zero

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What are the problems?

1. BOLD responses have a delayed and dispersed form.

2. The BOLD signal includes substantial amounts of low-frequency noise.

3. The data are serially correlated (temporally auto correlated) this violates the assumptions of the noise model in the GLM

Design Error

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t

dtgftgf0

)()()(

The response of a linear time-invariant (LTI) system is the convolution of the input with the system's response to an impulse (delta function).

Problem 1: Shape of BOLD response

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Solution: Convolution model of the BOLD responseexpected BOLD

response = input function impulse response function (HRF)

HRF

t

dtgftgf0

)()()(

blue = datagreen = predicted response, taking convolved with HRFred = predicted response, NOT taking into account the HRF

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Problem 2: Low frequency noise

blue = datablack = mean + low-frequency driftgreen = predicted response, taking into account low-frequency driftred = predicted response, NOT taking into

account low-frequency drift

Linear model

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discrete cosine transform (DCT) set

discrete cosine transform (DCT) set

Solution 2: High pass filtering

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Problem 3: Serial correlations

non-identity non-independencet1 t2 t1

t2

i.i.d

n: number of scans

n

n

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Problem 3: Serial correlations

n

n

auto covariancefunction

withwithttt aee 1 ),0(~ 2 Nt

1st order autoregressive process: AR(1)

n: number of scans

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Solution 3: Pre-whitening

1. Use an enhanced noise model with multiple error covariance components, i.e. e ~ N(0,2V) instead of e ~ N(0,2I).

2. Use estimated serial correlation to specify filter matrix W for whitening the data.

WeWXWy

This is i.i.d

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How do we define W ?• Enhanced noise model

• Remember linear transform for Gaussians

• Choose W such that error covariance becomes spherical

• Conclusion: W is a simple function of V so how do we estimate V ?

WeWXWy

),0(~ 2VNe

),(~

),,(~22

2

aaNy

axyNx

2/1

2

22 ),0(~

VW

IVW

VWNWe

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Linear Model

effects estimate

error estimate

statistic

• GLM includes all known experimental effects and confounds• Convolution with a canonical HRF• High-pass filtering to account for low-frequency drifts• Estimation of multiple variance components (e.g. to account for

serial correlations)

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Contrasts in the GLM

•We can examine whether a single predictor is significant (compared to the baseline)

•We can also examine whether a single predictor is significantly greater than another predictor

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Implementation of GLM in SPM

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Contrasts A contrast selects a specific effect of

interest.

A contrast is a vector of length .

is a linear combination of regression coefficients .

[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

𝑐𝑇 �� 𝑁 (𝑐𝑇 𝛽 ,𝜎2 𝑐𝑇 (𝑋𝑇 𝑋 )−1 𝑐 )

[1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

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Hypothesis Testing - Introduction

Is the mean of a measurement different from zero?

Mean of severalmeasurements

Many experiments

𝑇=𝜇𝜎√𝑛

Ratio of effect vs. noise

t-statistic Null D

istribution of T

Null distribution

What distribution of T would we

get for m = 0?𝜎 𝜇=

𝜎1,2

√𝑛

0

m1, s1

Exp A m2, s2

Exp B

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Hypothesis Testing Null Hypothesis H0

Typically what we want to disprove (no effect). The Alternative Hypothesis HA expresses outcome of interest.

To test an hypothesis, we construct “test statistics”.

Test Statistic T The test statistic summarises

evidence about H0. Typically, test statistic is small in

magnitude when the hypothesis H0 is true and large when false.

We need to know the distribution of T under the null hypothesis. Null Distribution of T

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Hypothesis Testing

p-value: A p-value summarises evidence against H0. This is the chance of observing a value more

extreme than t under the null hypothesis.

Null Distribution of T

Significance level α: Acceptable false positive rate α. threshold uα

Threshold uα controls the false positive rate

t

p-value

Null Distribution of T

u

Conclusion about the hypothesis: We reject the null hypothesis in favour of the

alternative hypothesis if t > uα

)|( 0HuTp

𝑝 (𝑇>𝑡∨𝐻0 )

t

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cT = 1 0 0 0 0 0 0 0

T =

contrast ofestimated

parameters

varianceestimate

effect of interest > 0 ?=

amplitude > 0 ?=

b1 = cT b > 0 ?

b1 b2 b3 b4 b5 ...

T-test - one dimensional contrasts – SPM{t}

Question:

Null hypothesis: H0: cTb=0 H0: cTb=0

Test statistic:

pNTT

T

T

T

tcXXc

c

c

cT

~ˆ

ˆ

)ˆvar(

ˆ

12

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T-contrast in SPM

con_???? image

Tc

ResMS image

pN

T

ˆˆ

ˆ 2

spmT_???? image

SPM{t}

For a given contrast c:

yXXX TT 1)(ˆ beta_???? images

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T-test: a simple example

Q: activation during listening ?

Null hypothesis: b1 = 0

Passive word listening versus rest

SPMresults: Threshold T = 3.2057 {p<0.001}voxel-level

p uncorrected

T

( Zº) Mm mm mm

13.94 Inf 0.000 -63 -27 15 12.04 Inf 0.000 -48 -33 12 11.82 Inf 0.000 -66 -21 6 13.72 Inf 0.000 57 -21 12 12.29 Inf 0.000 63 -12 -3 9.89 7.83 0.000 57 -39 6 7.39 6.36 0.000 36 -30 -15 6.84 5.99 0.000 51 0 48 6.36 5.65 0.000 -63 -54 -3

𝑡= 𝑐𝑇 ��

√var (𝑐𝑇 ��)

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T-test: summary

T-test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate).

T-contrasts are simple combinations of the betas; the T-statistic does not depend on the scaling of the regressors or the scaling of the contrast.

H0: 0Tc vs HA: 0Tc

Alternative hypothesis:

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��=(𝑋𝑇 𝑋 )−1 𝑋𝑇 𝑦

Scaling issue

The T-statistic does not depend on the scaling of the regressors.

cXXc

c

c

cT

TT

T

T

T

12ˆ

ˆ

)ˆvar(

ˆ

[1 1 1 1 ]

Be careful of the interpretation of the contrasts themselves (eg, for a second level analysis):

sum ≠ average

The T-statistic does not depend on the scaling of the contrast.

/ 4

Tc

Sub

ject

1

[1 1 1 ]

Sub

ject

5

Contrast depends on scaling.Tc/ 3

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F-test - the extra-sum-of-squares principleModel comparison:

Full model ?

X1 X0

or Reduced model?

X0 Test statistic: ratio of explained variability and unexplained variability (error)

1 = rank(X) – rank(X0)2 = N – rank(X)

RSS

2ˆ fullRSS0

2ˆreduced

RSS

RSSRSSF

0

21 ,~ FRSS

ESSF

Null Hypothesis H0: True model is X0 (reduced model)

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F-test - multidimensional contrasts – SPM{F}

Test multiple linear hypotheses:

Full model ?

Null Hypothesis H0: b3 = b4 = b5 = b6 = b7 = b8 = 0

X1 (b3-8)X0 0 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

cT =

cTb = 0

Is any of b3-8 not equal 0?

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F-contrast in SPM

ResMS image

pN

T

ˆˆ

ˆ 2

spmF_???? images

SPM{F}

ess_???? images

( RSS0 - RSS )

yXXX TT 1)(ˆ beta_???? images

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F-test example: movement related effects

Design matrix

2 4 6 8

10

20

30

40

50

60

70

80

contrast(s)

Design matrix2 4 6 8

1020304050607080

contrast(s)

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F-test: summary

F-tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler (nested) model model comparison.

0000

0100

0010

0001

In testing uni-dimensional contrast with an F-test, for example b1 – b2, the result will be the same as testing b2 – b1. It will be exactly the square of the t-test, testing for both positive and negative effects.

F tests a weighted sum of squares of one or several combinations of the regression coefficients b.

In practice, we don’t have to explicitly separate X into [X1X2] thanks to multidimensional contrasts.

Hypotheses:

0 : Hypothesis Null 3210 H

0 oneleast at : Hypothesis eAlternativ kAH

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Variability described by Variability described by

Orthogonal regressors

Variability in YTesting for Testing for

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Correlated regressorsV

aria

bilit

y de

scrib

ed b

y V

ariability described by

Shared variance

Variability in Y

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Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

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Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

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Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

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Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

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Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

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Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for and/or

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Design orthogonality

For each pair of columns of the design matrix, the orthogonality matrix depicts the magnitude of the cosine of the angle between them, with the range 0 to 1 mapped from white to black.

If both vectors have zero mean then the cosine of the angle between the vectors is the same as the correlation between the two variates.

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Correlated regressors: summary• We implicitly test for an additional effect only. When

testing for the first regressor, we are effectively removing the part of the signal that can be accounted for by the second regressor: implicit orthogonalisation.

• Orthogonalisation = decorrelation. Parameters and test on the non modified regressor change.Rarely solves the problem as it requires assumptions about which regressor to uniquely attribute the common variance. change regressors (i.e. design) instead, e.g. factorial designs. use F-tests to assess overall significance.

• Original regressors may not matter: it’s the contrast you are testing which should be as decorrelated as possible from the rest of the design matrix

x1

x2

x1

x2

x1

x2x^

x^

2

1

2x^ = x2 – x1.x2 x1

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Design efficiency

1122 ))(ˆ(),,ˆ( cXXcXce TT

)ˆvar(

ˆ

T

T

c

cT The aim is to minimize the standard error of a t-contrast

(i.e. the denominator of a t-statistic).

cXXcc TTT 12 )(ˆ)ˆvar( This is equivalent to maximizing the efficiency e:

Noise variance Design variance

If we assume that the noise variance is independent of the specific design:

11 ))((),( cXXcXce TT

This is a relative measure: all we can really say is that one design is more efficient than another (for a given contrast).

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Design efficiencyA B

A+B

A-B

𝑋𝑇 𝑋=( 1 − 0.9− 0.9 1 )

High correlation between regressors leads to low sensitivity to each regressor alone.

We can still estimate efficiently the difference between them.

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Types of error Reality

H1

H0

H0 H1

True negative (TN)

True positive (TP)

False positive (FP)

False negative (FN)

specificity: 1- = TN / (TN + FP)= proportion of actual negatives which are correctly identified

sensitivity (power): 1- = TP / (TP + FN)= proportion of actual positives which are correctly identified

h

h

hh

Decision

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Correction for Multiple Comparisons

•With conventional probability levels (e.g., p < .05) and a huge number of comparisons (e.g., 64 x 64 x 12 = 49,152), a lot of voxels will be significant purely by chance

•e.g., .05 * 49,152 = 2458 voxels significant due to chance

•How can we avoid this? need to do correction for multiple comparisons

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(1) Bonferroni correction• Divide desired p value by number of comparisons• Example:• desired p value: p < .05• number of voxels: 50,000• required p value: p < .05 / 50,000 p < .000001

• quite conservative• can use less stringent values• e.g., Brain Voyager can use the number of voxels in the

cortical surface• the luxury of a localizer task is that it allows you to

correct for only those voxels you are interested in (i.e., the voxels that fall within your ROI)

Source: Jody Culham’s web slides

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(2) Cluster correction

•falsely activated voxels should be randomly dispersed

•set minimum cluster size to be large enough to make it unlikely that a cluster of that size would occur by chance

•assumes that data from adjacent voxels are uncorrelated (not true)

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(3) Test-retest reliability

•Perform statistical tests on each half of the data

•The probability of a given voxel appearing in both purely by chance is the square of the p value used in each half

• e.g., .001 x .001 = .000001 •Alternatively, use the first half to select an

ROI and evaluate your hypothesis in the second half.

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(4) Consistency between subjects•Though seldom discussed, consistency

between subjects provides further encouragement that activation is not spurious. Present group and individual data.

(5) Screen saver scan

•You can test how many voxels really do light up when really nothing is happening.

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ROI Voxel Field‘volume’ resolution*

volume independence

FWE FDR

*voxels/volume

Height Extent

ROI Voxel Field

Height Extent

There is a multiple testing problem (‘voxel’ or ‘blob’ perspective).More corrections needed as ..

Detect an effect of unknown extent & location

Volume , Independence

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Book – must readStatistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier, 2007.

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Home Work

•http://www.fil.ion.ucl.ac.uk/spm/data/auditory/

•http://www.fil.ion.ucl.ac.uk/spm/doc/spm8_manual.pdf#Chap:data:auditory

•Walk through SPM manual chapter 28•Sample data can be found in the general

folder•If you have any question, ask me or

Danielle.•Bonus for those who finish the homework

by next training session.