SPM Course

Zurich, February 2012

Statistical InferenceGuillaume Flandin

Wellcome Trust Centre for Neuroimaging

University College London

Normalisation

Statistical Parametric MapImage time-series

Parameter estimates

General Linear ModelRealignment Smoothing

Design matrix

Anatomicalreference

Spatial filter

StatisticalInference

RFT

p <0.05

A mass-univariate approach

Time

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 )

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

𝑐𝑇 �� 𝑁 (𝑐𝑇 𝛽 ,𝜎2𝑐𝑇 (𝑋𝑇 𝑋 )−1𝑐 )

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

Hypothesis Testing

p-value:

A p-value summarises evidence against H0.

This is the chance of observing 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 )

cT = 1 0 0 0 0 0 0 0

T =

contrast ofestimated

parameters

varianceestimate

box-car amplitude > 0 ?=

b1 = cTb> 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

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

T-test: a simple example

Q: activation during listening ?

Q: activation during listening ?

cT = [ 1 0 0 0 0 0 0 0]

Null hypothesis:Null hypothesis: 01

Passive word listening versus rest

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

p uncorrectedT ( 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 6.19 5.53 0.000 -30 -33 -18 5.96 5.36 0.000 36 -27 9 5.84 5.27 0.000 -45 42 9 5.44 4.97 0.000 48 27 24 5.32 4.87 0.000 36 -27 42

1

𝑡= 𝑐𝑇 ��

√var (𝑐𝑇 ��)

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:

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 ]

[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

/ 3

Tc

Sub

ject

1S

ubje

ct 5

Contrast depends on scaling.Tc

F-test - the extra-sum-of-squares principle Model comparison:

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

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

 

 

F-test - multidimensional contrasts – SPM{F} Tests multiple linear hypotheses:

0 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1

cT =

H0: b4 = b5 = ... = b9 = 0

X1 (b4-9)X0

Full model? Reduced model?

H0: True model is X0

X0

test H0 : cTb = 0 ?

SPM{F6,322}

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

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)

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

Variability described by Variability described by

Orthogonal regressors

Variability in YTesting for Testing for

Correlated regressorsV

aria

bilit

y de

scrib

ed b

y V

ariability described by

Shared variance

Variability in Y

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for

Correlated regressors

Var

iabi

lity

desc

ribed

by

Variability described by

Variability in Y

Testing for and/or

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.

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

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).

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.

Bibliography:

Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier, 2007.

Plane Answers to Complex Questions: The Theory of Linear Models. R. Christensen, Springer, 1996.

Statistical parametric maps in functional imaging: a general linear approach. K.J. Friston et al, Human Brain Mapping, 1995.

Ambiguous results in functional neuroimaging data analysis due to covariate correlation. A. Andrade et al., NeuroImage, 1999.

Estimating efficiency a priori: a comparison of blocked and randomized designs. A. Mechelli et al., NeuroImage, 2003.

Estimability of a contrast

If X is not of full rank then we can have Xb1 = Xb2 with b1≠ b2 (different parameters).

The parameters are not therefore ‘unique’, ‘identifiable’ or ‘estimable’.

For such models, XTX is not invertible so we must resort to generalised inverses (SPM uses the pseudo-inverse).

1 0 11 0 11 0 11 0 10 1 10 1 10 1 10 1 1

One-way ANOVA(unpaired two-sample t-test)

Rank(X)=2

[1 0 0], [0 1 0], [0 0 1] are not estimable.[1 0 1], [0 1 1], [1 -1 0], [0.5 0.5 1] are estimable.

Example:

parameters

imag

es

Fact

or1

Fact

or2

Mea

n

parameter estimability(gray

® b not uniquely specified)

Three models for the two-samples t-test

1 11 11 11 10 10 10 10 1

1 01 01 01 0 0 1 0 1 0 1 0 1

1 0 11 0 11 0 11 0 10 1 10 1 10 1 10 1 1

β1=y1

β2=y2

β1+β2=y1

β2=y2

[1 0].β = y1

[0 1].β = y2

[0 -1].β = y1-y2

[.5 .5].β = mean(y1,y2)

[1 1].β = y1

[0 1].β = y2

[1 0].β = y1-y2

[.5 1].β = mean(y1,y2)

β1+β3=y1

β2+β3=y2

[1 0 1].β = y1

[0 1 1].β = y2

[1 -1 0].β = y1-y2

[.5 0.5 1].β = mean(y1,y2)

Multidimensional contrasts

Think of it as constructing 3 regressors from the 3 differences and complement this new design matrix such that data can be fitted in the same exact way (same error, same fitted data).

Example: working memory

B: Jittering time between stimuli and response.

Stimulus Response Stimulus Response Stimulus ResponseA B C

Tim

e (s)

Tim

e (s)

Tim

e (s)

Correlation = -.65Efficiency ([1 0]) = 29

Correlation = +.33Efficiency ([1 0]) = 40

Correlation = -.24Efficiency ([1 0]) = 47

C: Requiring a response on a randomly half of trials.