Statistical InferenceGuillaume Flandin
Wellcome Trust Centre for Neuroimaging
University College London
SPM Course
London, May 2010
NormalisationNormalisation
Statistical Parametric MapStatistical Parametric MapImage time-seriesImage time-series
Parameter estimatesParameter estimates
General Linear ModelGeneral Linear ModelRealignmentRealignment SmoothingSmoothing
Design matrix
AnatomicalAnatomicalreferencereference
Spatial filterSpatial filter
StatisticalStatisticalInferenceInference
RFTRFT
p <0.05p <0.05
Time
BOLD signal
Tim
e
single voxeltime series
single voxeltime series
Voxel-wise time series analysis
Modelspecification
Modelspecification
Parameterestimation
Parameterestimation
HypothesisHypothesis
StatisticStatistic
SPMSPM
Overview
Model specification and parameters estimation Hypothesis testing Contrasts
T-tests F-tests
Contrast estimability Correlation between regressors
Example(s) Design efficiency
Model Specification: The General Linear Model
Sphericity assumption: Independent and identically distributed (i.i.d.) error terms
N: number of scans, p: number of regressors
=
+yy X
N
1
N N
1 1p
p
Parameter Estimation: Ordinary Least Squares
Find that minimises TXy 2 TXy 2
yXXX TT 1)(ˆ yXXX TT 1)(ˆ
The Ordinary Least Estimates are:
Under i.i.d. assumptions, the Ordinary Least Squares estimates are Maximum Likelihood.
),0(~ 2IN ),(~ 2IXNY
))(,(~ˆ 12 XXN TpN
T
ˆˆ
ˆ 2
pN
T
ˆˆ
ˆ 2
Hypothesis Testing
The 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”.
The 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.
Observation of test statistic t, a realisation of T Null Distribution of T
)|( 0HtTp )|( 0HtTp
Significance level α:
Acceptable false positive rate α.
threshold uα
Threshold uα controls the false positive rate
t
P-val
Null Distribution of T
u
The conclusion about the hypothesis:
We reject the null hypothesis in favour of the alternative hypothesis if t > uα
)|( 0HuTp
Contrasts
A contrast selects a specific effect of interest: a contrast c is a vector of length p.
cTβ is a linear combination of regression coefficients β.
))(,(~ˆ 12 cXXccNc TTTT ))(,(~ˆ 12 cXXccNc TTTT
Under i.i.d assumptions:
We are usually not interested in the whole β vector.
cTβ = 1x1 + 0x2 + 0x3 + 0x4 + 0x5 + . . .
cT = [1 0 0 0 0 …]
cTβ = 0x1 + -1x2 + 1x3 + 0x4 + 0x5 + . . .
cT = [0 -1 1 0 0 …]
cT = 1 0 0 0 0 0 0 0
T =
contrast ofestimated
parameters
varianceestimate
box-car amplitude > 0 ?=
1 = cT> 0 ?
1 2 3 4 5 ...
T-test - one dimensional contrasts – SPM{t}
Question:
Null hypothesis: H0: cT=0 H0: cT=0
Test statistic:
pNTT
T
T
T
tcXXc
c
c
cT
~ˆ
ˆ
)ˆvar(
ˆ
12
pNTT
T
T
T
tcXXc
c
c
cT
~ˆ
ˆ
)ˆvar(
ˆ
12
T-contrast in SPM
ResMS image
yXXX TT 1)(ˆ
con_???? image
Tc
pN
T
ˆˆ
ˆ 2
beta_???? images
spmT_???? image
SPM{t}
For a given contrast c:
T-test: a simple example
Q: activation during listening ?
Q: activation during listening ?
cT = [ 1 0 ]
Null hypothesis:Null hypothesis: 01
)ˆ(
ˆ
T
T
cStd
ct
)ˆ(
ˆ
T
T
cStd
ct
Passive word listening versus rest
SPMresults:Height threshold T = 3.2057 {p<0.001}
Statistics: p-values adjusted for search volume
set-levelc p
cluster-levelp corrected p uncorrectedk E
voxel-levelp FWE-corr p FDR-corr p uncorrectedT (Z
)
mm mm mm
0.000 10 0.000 520 0.000 0.000 0.000 13.94 Inf 0.000 -63 -27 150.000 0.000 12.04 Inf 0.000 -48 -33 120.000 0.000 11.82 Inf 0.000 -66 -21 6
0.000 426 0.000 0.000 0.000 13.72 Inf 0.000 57 -21 120.000 0.000 12.29 Inf 0.000 63 -12 -30.000 0.000 9.89 7.83 0.000 57 -39 6
0.000 35 0.000 0.000 0.000 7.39 6.36 0.000 36 -30 -150.000 9 0.000 0.000 0.000 6.84 5.99 0.000 51 0 480.002 3 0.024 0.001 0.000 6.36 5.65 0.000 -63 -54 -30.000 8 0.001 0.001 0.000 6.19 5.53 0.000 -30 -33 -180.000 9 0.000 0.003 0.000 5.96 5.36 0.000 36 -27 90.005 2 0.058 0.004 0.000 5.84 5.27 0.000 -45 42 90.015 1 0.166 0.022 0.000 5.44 4.97 0.000 48 27 240.015 1 0.166 0.036 0.000 5.32 4.87 0.000 36 -27 42
Design matrix
0.5 1 1.5 2 2.5
10
20
30
40
50
60
70
80
1
X voxel-levelp uncorrectedT ( Z) 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
T-test: a few remarks
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 Unilateral test:
cXXc
c
c
cT
TT
T
T
T
12ˆ
ˆ
)ˆvar(
ˆ
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: 4 = 5 = = 9 = 0
X1 (4-9)X0
Full model? Reduced model?
H0: True model is X0
X0
test H0 : cT = 0 ?
SPM{F6,322}
F-contrast in SPM
ResMS image
yXXX TT 1)(ˆ pN
T
ˆˆ
ˆ 2
beta_???? images
spmF_???? images
SPM{F}
ess_???? images
( RSS0 - RSS )
For a given contrast c:
F-test example: movement-related effects
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).
F-test: a few remarks
F-tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler (nestednested) model Model comparison.Model comparison.
0000
0100
0010
0001
In testing uni-dimensional contrast with an F-test, for example 1 – 2, the result will be the same as testing 2 – 1. It will be exactly the square of the t-test, testing for both positive and negative effects.
F tests a weighted sum of squaressum of squares of one or several combinations of the regression coefficients .
In practice, we don’t have to explicitly separate X into [X1X2] thanks to multidimensional contrastsmultidimensional contrasts.
Hypotheses:
0 : Hypothesis Null 3210 H
0 oneleast at : Hypothesis eAlternativ kAH
Estimability of a contrast
If X is not of full rank then we can have X1 = X2 with 1≠ 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 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.
parameters
imag
es
Fact
or1
Fact
or2
Mea
n
parameter estimability(gray
not uniquely specified)
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.
The cosine of the angle between two vectors a and b is obtained by:
ba
bacos
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.
Shared variance
Orthogonal regressors.Orthogonal regressors.
Shared variance
Correlated regressors, for example:green: subject ageyellow: subject score
Testing for the Testing for the greengreen::
Shared variance
Correlated regressors.
Testing for the Testing for the redred::
Shared variance
Highly correlated.Entirely correlated non estimable
Testing for the Testing for the greengreen::
Shared variance
If significant, can be GG and/or YY
Testing for the Testing for the greengreen and and yellowyellow
Examples
A few remarks
We implicitly test for an additional effect only, be careful if there is correlation
- Orthogonalisation = decorrelation : not generally needed- Parameters and test on the non modified regressor change
It is always simpler to have orthogonal regressors and therefore designs.
In case of correlation, use F-tests to see the overall significance. There is generally no way to decide to which regressor the « common » part should be attributed to.
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
Design efficiency
1122 ))(ˆ(),,ˆ( cXXcXce TT 1122 ))(ˆ(),,ˆ( cXXcXce TT
)ˆvar(
ˆ
T
T
c
cT
)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( 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 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 efficiency The efficiency of an estimator is a measure of how reliable it is and
depends on error variance (the variance not modeled by explanatory variables in the design matrix) and the design variance (a function of the explanatory variables and the contrast tested).
XTX represents covariance of regressors in design matrix; high covariance increases elements of (XTX)-1.
High correlation between regressors leads to low sensitivity to each regressor alone.
cXXc TT 1)( cXXc TT 1)(
19.0
9.01cT=[1 0]: 5.26
cT=[1 1]: 20
cT=[1 -1]: 1.05
Bibliography:
Statistical Parametric Mapping: The Analysis of Functional Brain Images. Elsevier, 2007.
With many thanks to J.-B. Poline, Tom Nichols, S. Kiebel, R. Henson for slides.
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.
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