Zurich February 2012 Statistical Inference Guillaume Flandin Wellcome Trust Centre for Neuroimaging University College London Normalisation Statistical Parametric Map Image timeseries Parameter estimates ID: 276556
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Slide1
SPM CourseZurich, February 2012
Statistical Inference
Guillaume Flandin
Wellcome Trust Centre for Neuroimaging
University College LondonSlide2
Normalisation
Statistical Parametric Map
Image time-series
Parameter estimates
General Linear Model
Realignment
Smoothing
Design matrix
Anatomical
reference
Spatial filter
Statistical
Inference
RFT
p <0.05Slide3
A mass-univariate approach
TimeSlide4
Estimation of the parameters
=
i.i.d. assumptions:
OLS estimates:
=
Slide5
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]
Slide6
Hypothesis TestingNull 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 H
0
.
Typically, test statistic is small in magnitude when the hypothesis H
0
is true and large when false.
We need to know the distribution of T under the null hypothesis.
Null Distribution of TSlide7
Hypothesis Testingp-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
Slide8
c
T
=
1
0 0 0 0 0 0 0
T
=
contrast
of
estimated
parameters
varianceestimate
box-car amplitude > 0 ?
=
b
1 = cT
b> 0 ?
b1 b2 b3 b4 b5
...T-test - one dimensional contrasts – SPM{t}
Question:Null hypothesis:
H0: c
Tb=0
Test statistic:Slide9
T-contrast in SPM
con_???? image
ResMS image
spmT_???? image
SPM{
t
}
For a given contrast
c
:
beta_???? imagesSlide10
T-test: a simple example
Q: activation during listening ?
cT = [ 1 0 0 0 0 0 0 0]
Null hypothesis:
Passive word listening versus rest
SPMresults:
Height threshold T = 3.2057 {p<0.001}
voxel-level
p
uncorrected
T
(
Z
º
)
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
Slide11
T-test: summaryT-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:
vs H
A
:
Alternative hypothesis:Slide12
Scaling issueThe T-statistic does not depend on the scaling of the regressors.
[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
Subject 1
Subject 5
Contrast depends on scaling.Slide13
F-test - the extra-sum-of-squares principle
Model comparison:
Null Hypothesis H0: True model is X0 (reduced model)
Full model ?
X
1
X
0
or Reduced model?
X
0
Test statistic:
ratio of explained variability and unexplained variability (error)
1
= rank(X) – rank(X
0
)
2
= N – rank(X)
RSS
RSS
0
Slide14
F
-test - multidimensional contrasts – SPM{F
}Tests multiple linear hypotheses:
0 0 0
1
0 0 0 0 0
0 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 = ... = b
9 = 0
X
1
(
b4-9
)
X0
Full model?
Reduced model?
H
0
:
True model is
X
0
X
0
test H0
: cTb = 0 ?
SPM{F
6,322
}Slide15
F-contrast in SPM
ResMS image
spmF_???? images
SPM{F}
ess_???? images
( RSS
0
- RSS
)
beta_???? imagesSlide16
F-test example: movement related effects
Design matrix
2
4
6
8
10
20
30
40
50
60
70
80
contrast(s)
Design matrix
2
4
6
8
10
20
30
40
50
60
70
80
contrast(s)Slide17
F-test: summaryF-tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler (nested) model model comparison
.
In testing uni-dimensional contrast with an
F
-test, for example
b
1
–
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 [X
1X2] thanks to multidimensional contrasts.
Hypotheses:Slide18
Variability described by
Variability described by
Orthogonal regressors
Variability in
Y
Testing for
Testing for
Slide19
Correlated regressors
Variability described by
Variability described by
Shared variance
Variability in
YSlide20
Correlated regressors
Variability described by
Variability described by
Variability in
Y
Testing for
Slide21
Correlated regressors
Variability described by
Variability described by
Variability in
Y
Testing for
Slide22
Correlated regressors
Variability described by
Variability described by
Variability in
YSlide23
Correlated regressors
Variability described by
Variability described by
Variability in
Y
Testing for
Slide24
Correlated regressors
Variability described by
Variability described by
Variability in
Y
Testing for
Slide25
Correlated regressors
Variability described by
Variability described by
Variability in
Y
Testing for
and/or
Slide26
Design orthogonalityFor 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.Slide27
Correlated regressors: summaryWe 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
x
1
x
2
x
1
x
2
x
1
x
2
x
^
x
^
2
1
2
x
^
= x
2
– x
1
.x
2 x1Slide28
Design efficiency
The aim is to minimize the standard error of a
t
-contrast (i.e. the denominator of a t-statistic).
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:
This is a relative measure: all we can really say is that one design is more efficient than another (for a given contrast).Slide29
Design efficiency
A
B
A+B
A-B
High
correlation between regressors leads to low sensitivity to each regressor alone
.
We can
still estimate efficiently the difference between
them.Slide30
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.Slide31
Estimability of a contrastIf X is not of full rank then we can have Xb1 = Xb2 with b1≠ b
2 (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 1
1 0 1
1 0 1
1 0 1
0 1 1
0 1 1
0 1 1
0 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
images
Factor
1
Factor
2
Mean
parameter estimability
(gray
®
b
not uniquely specified)Slide32
Three models for the two-samples t-test
1 1
1 1
1 1
1 1
0 1
0 1
0 1
0 1
1 0
1 0
1 0
1 0
0 1
0 1
0 1
0 1
1 0 1
1 0 1
1 0 1
1 0 1
0 1 1
0 1 1
0 1 1
0 1 1
β
1
=y
1
β
2
=y
2
β
1
+
β
2
=y
1
β
2
=y
2
[1 0].
β
= y
1
[0 1].
β
= y
2
[0 -1].
β
= y
1
-y
2
[.5 .5].
β
= mean(y
1
,y
2
)
[1 1].
β
= y
1
[0 1].
β
= y
2
[1 0].
β
= y
1
-y
2
[.5 1].
β
= mean(y
1
,y
2
)
β
1
+
β
3
=y
1
β
2
+
β
3
=y
2
[1 0 1].
β
= y
1
[0 1 1].
β
= y
2
[1 -1 0].
β
= y
1
-y
2
[.5 0.5 1].
β
= mean(y
1
,y
2
)Slide33
Multidimensional contrastsThink 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).Slide34
Example: working memoryB: Jittering time between stimuli and response.
Stimulus
Response
Stimulus
Response
Stimulus
Response
A
B
C
Time (s)
Time (s)
Time (s)
Correlation = -.65
Efficiency ([1 0]) = 29
Correlation = +.33
Efficiency ([1 0]) = 40
Correlation = -.24
Efficiency ([1 0]) = 47
C: Requiring a response on a randomly half of trials.