1. Statistical Inference
Guillaume Flandin
Wellcome Trust Centre for Neuroimaging
University College London
SPM Course
Zurich, February 2009

2. Statistical Inference
Image time-series
Spatial filter
Design matrix
Statistical Parametric Map
Realignment
Smoothing
General Linear Model
Statistical Inference
RFT
Normalisation
p <0.05
Anatomical reference
Parameter estimates

3. Voxel-wise time series analysis
Model
specification
Parameter
estimation
Hypothesis
Statistic
Time
Time
BOLD signal
single voxel
time series
SPM

4. Overview
Model specification and parameters estimation
Hypothesis testing
Contrasts
T-tests
F-tests
Contrast estimability
Correlation between regressors
Example(s)
Design efficiency

5. Model Specification: The General Linear Model= + y
Sphericity assumption: Independent and identically distributed (i.i.d.) error terms
N: number of scans, p: number of regressors
The General Linear Model is an equation that expresses the observed response variable in terms of a linear combination of explanatory variables X plus a well behaved error term. Each column of the design matrix corresponds to an effect one has built into the experiment or that may confound the results.

6. Parameter Estimation: Ordinary Least Squares
Find that minimize
The Ordinary Least Estimates are:
Under i.i.d. assumptions, the Ordinary Least Squares estimates are Maximum Likelihood.

7. Hypothesis Testing
To test an hypothesis, we construct “test statistics”.
The Null Hypothesis H0
Typically what we want to disprove (no effect).
 The Alternative Hypothesis HA expresses outcome of interest.
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

8. Hypothesis Testing u t Type I Error α:
Acceptable false positive rate α.
Level  threshold uα
Threshold uα controls the false positive rate 
Observation of test statistic t, a realisation of T
Null Distribution of T
The conclusion about the hypothesis:
We reject the null hypothesis in favour of the alternative hypothesis if t > uα t
P-value:
A p-value summarises evidence against H0.
This is the chance of observing value more extreme than t under the null hypothesis.
P-val
Null Distribution of T

9. One cannot accept the null hypothesis (one can just fail to reject it)
Absence of evidence is not evidence of absence.
If we do not reject H0, then all that we are saying is that there is no strong evidence in the data to reject that H0 might be a reasonable approximation to the truth. This certainly does not mean that there is a strong evidence to accept H0.

10. cTβ = 0xb1 + -1xb2 + 1xb3 + 0xb4 + 0xb5 + . . .
Contrasts
We are usually not interested in the whole β vector.
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.
cT = [ …]
cTβ = 1xb1 + 0xb2 + 0xb3 + 0xb4 + 0xb
cT = [ …]
cTβ = 0xb1 + -1xb2 + 1xb3 + 0xb4 + 0xb
Under i.i.d assumptions:

11. T-test - one dimensional contrasts – SPM{t}
Question:
box-car amplitude > 0 ? =
b1 = cTb> 0 ?
cT =
b1 b2 b3 b4 b5 ...
Null hypothesis:
H0: cTb=0
T =
contrast of estimated parameters
variance estimate
Test statistic:

12. T-contrast in SPM For a given contrast c: ResMS image beta_???? images
con_???? image
spmT_???? image
SPM{t}

13. T-test: a simple example
Passive word listening versus rest
cT = [ ]
Q: activation during listening ?
Design matrix
0.5 1
1.5 2
2.5 10 20 30 40 50 60 70 80
Null hypothesis:
SPMresults:
Height threshold T = {p<0.001}
voxel-level p
uncorrected T ( Z )
mm mm mm
13.94
Inf
0.000
12.04
11.82
13.72
12.29
9.89
7.83
7.39
6.36
6.84
5.99
5.65
6.19
5.53
5.96
5.36
5.84
5.27
5.44
4.97
5.32
4.87
Statistics:
p-values adjusted for search volume
set-level c p
cluster-level
corrected
uncorrected k E
voxel-level
FWE-corr
FDR-corr T ( Z )
mm mm mm
0.000 10
520
13.94
Inf
12.04
11.82
426
13.72
12.29
9.89
7.83 35
7.39
6.36 9
6.84
5.99
0.002 3
0.024
0.001
5.65 8
6.19
5.53
0.003
5.96
5.36
0.005 2
0.058
0.004
5.84
5.27
0.015 1
0.166
0.022
5.44
4.97
0.036
5.32
4.87

14. 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:
vs HA:
Unilateral test: -5 -4 -3 -2 -1 1 2 3 4 5
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4 n =1 =2 =5
=10 =
Probability Density Function of Student’s t distribution

15. Scaling issue
[ ]
/ 4
Subject 1
The T-statistic does not depend on the scaling of the regressors.
The T-statistic does not depend on the scaling of the contrast.
[ ]
/ 3
However be careful of the interpretation of the contrasts themselves (eg, for a second level analysis): sum ≠ average
Subject 5

16. F-test - the extra-sum-of-squares principle
Model comparison: Full vs. Reduced model?
Null Hypothesis H0: True model is X0 (reduced model) X1 X0
RSS
Or Reduced model? X0
RSS0
Test statistic: ratio of explained variability and unexplained variability (error)
1 = rank(X) – rank(X0)
2 = N – rank(X)
Full model ?

17. F-test - multidimensional contrasts – SPM{F}
Tests multiple linear hypotheses:
H0: True model is X0
cT =
H0: b3 = b4 = ... = b9 = 0
test H0 : cTb = 0 ?
SPM{F6,322} X0
X1 (b3-9) X0
Full model?
Reduced model?

18. F-contrast in SPM ( RSS0 - RSS ) ResMS image beta_???? images
ess_???? images
spmF_???? images
( RSS0 - RSS )
SPM{F}

19. F-test example: movement related effects
To assess movement-related activation:
There is a lot of residual movement-related artifact in the data (despite spatial realignment), which tends to be concentrated near the boundaries of tissue types.
Even though we are not interested in such artifact, by including the realignment parameters in our design matrix, we “covary out” linear components of subject movement, reducing the residual error, and hence improve our statistics for the effects of interest.

20. 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).

21. F-test: a few remarks
F-tests can be viewed as testing for the additional variance explained by a larger model wrt a simpler (nested) model  Model comparison.
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:
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.

22. Estimability of a contrast
parameters
images
Factor 1 2
Mean
parameter estimability
(gray b
not uniquely specified)
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).
One-way ANOVA (unpaired two-sample t-test)
Rank(X)=2
Example:
[ ], [ ], [ ] are not estimable.
[ ], [ ], [ ], [ ] are estimable.

23. 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:
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.

24. Multicollinearity
Orthogonal regressors (=uncorrelated): By varying each separately, one can predict the combined effect of varying them jointly. x2 x1 x2
is diagonal x1
Non-orthogonal regressors (=correlated): When testing for the first regressor, we are effectively removing the part of the signal that can be accounted by the second regressor  implicit orthogonalisation. x2 x2 x2 x^ 2
x^ = x2 – x1.x2 x1 x1 2 x1 x1 x^ 1
It does not reduce the predictive power or reliability of the model as a whole.

25. Shared variance
Orthogonal regressors.

26. Shared variance Testing for the green:
Correlated regressors, for example:
green: subject age
yellow: subject score

27. Shared variance
Testing for the red:
Correlated regressors.

28. Shared variance Testing for the green: Highly correlated.
Entirely correlated non estimable

29. Shared variance Testing for the green and yellow
If significant, can be G and/or Y
Examples

30. 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

31. 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).

32. 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.
cT=[1 0]:
cT=[1 1]: 20
cT=[1 -1]:

33. 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.
With many thanks to J.-B. Poline, Tom Nichols, S. Kiebel, R. Henson for slides.

34. Example I: a first model
True signal and observed signal (--)
Model (green, pic at 6sec)
TRUE signal (blue, pic at 3sec)
Fitting (b1 = 0.2, mean = 0.11)
Residual (still contains some signal)
 Test for the green regressor not significant

35. Example I: a first model
b1 = 0.22
b2 = 0.11
Residual Var.= 0.3
p(Y| b1 = 0) 
p-value = 0.1
(t-test)
p(Y| b1 = 0)  p-value = 0.2
(F-test) = + Y
X b e

36. Example I: an “improved” model
True signal + observed signal
Model (green and red)
and true signal (blue ---)
Red regressor : temporal derivative of the green regressor
Global fit (blue)
and partial fit (green & red)
Adjusted and fitted signal
Residual (a smaller variance)
 t-test of the green regressor significant
 F-test very significant
 t-test of the red regressor very significant

37. Example I: an “improved” model
b1 = 0.22
b2 = 2.15
b3 = 0.11
Residual Var. = 0.2
p(Y| b1 = 0) 
p-value = 0.07
(t-test)
p(Y| b1 = 0, b2 = 0) 
p-value =
(F-test) = + Y
X b e
Conclusion

38. Example II: correlation between regressors
True signal
Model (green and red)
Fit (blue : global fit)
Residual

39. Example II: correlation between regressors
Residual var. = 0.3
p(Y| b1 = 0) 
p-value = 0.08
(t-test)
P(Y| b2 = 0) 
p-value = 0.07
p(Y| b1 = 0, b2 = 0) 
p-value =
(F-test) = + Y
X b e 1 2 1 2

40. Example II: correlation between regressors
True signal
Model (green and red)
red regressor has been
orthogonalised with respect to the green one
 remove everything that correlates with the green regressor
Fit
Residual

41. Example II: correlation between regressors
(0.79)
(0.85)
(0.06)
Residual var. = 0.3
p(Y| b1 = 0)
p-value =
(t-test)
p(Y| b2 = 0)
p-value = 0.07
p(Y| b1 = 0, b2 = 0)
p-value =
(F-test) = + Y
X b e 1 2 1 2
Conclusion