1. Classical inference and design efficiency Zurich SPM Course 2014
Jakob Heinzle
Translational Neuromodeling Unit (TNU)
Institute for Biomedical Engineering (IBT)
University and ETH Zürich
Many thanks to K. E. Stephan, G. Flandin and others for material
Translational Neuromodeling Unit

2. Statistical Inference
Overview
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
Classical Inference and Design Efficiency

3. A mass-univariate approach
Time
Classical Inference and Design Efficiency

4. Estimation of the parameters
i.i.d. assumptions:
𝜀~𝑁(0, 𝜎 2 𝐼)
𝛽 = ( 𝑋 𝑇 𝑋) −1 𝑋 𝑇 𝑦
OLS estimates:
𝛽 1 =3.9831
𝛽 2−7 ={0.6871, , , , , − } 𝛽
𝛽 8 =
𝑦 = +𝜀
Ways to reduce the noise.
𝜀 =
𝜎 2 = 𝜀 𝑇 𝜀 𝑁−𝑝
𝛽 ~𝑁 𝛽, 𝜎 2 ( 𝑋 𝑇 𝑋) −1
Classical Inference and Design Efficiency

5. Contrasts 𝑐 𝑇 𝛽 ~𝑁 𝑐 𝑇 𝛽, 𝜎 2 𝑐 𝑇 ( 𝑋 𝑇 𝑋) −1 𝑐
A contrast selects a specific effect of interest.
A contrast 𝑐 is a vector of length 𝑝.
𝑐 𝑇 𝛽 is a linear combination of regression coefficients 𝛽.
[ ]
[ ]
𝑐= [ …] 𝑇
𝑐 𝑇 𝛽=𝟏× 𝛽 1 +𝟎× 𝛽 2 +𝟎× 𝛽 3 +𝟎× 𝛽 4 +…
= 𝜷 𝟏
𝑐= [1 0 0 −1 0 …] 𝑇
𝑐 𝑇 𝛽=𝟏× 𝛽 1 +𝟎× 𝛽 2 +𝟎× 𝛽 3 +−𝟏× 𝛽 4 +…
= 𝜷 𝟏 − 𝜷 𝟒
𝑐 𝑇 𝛽 ~𝑁 𝑐 𝑇 𝛽, 𝜎 2 𝑐 𝑇 ( 𝑋 𝑇 𝑋) −1 𝑐
Classical Inference and Design Efficiency

6. Hypothesis Testing - Introduction
Is the mean of a measurement different from zero?
Mean of several
measurements
Many experiments
Ratio of
effect vs. noise
 t-statistic
Null Distribution of T
Null distribution
What distribution of T
would we
get for m = 0?
m1, s1
Exp A
m2, s2
Exp B
𝑇= 𝜇 𝜎 𝑛
𝜎 𝜇 = 𝜎 1,2 𝑛
Classical Inference and Design Efficiency

7. Hypothesis Testing
To test an hypothesis, we construct “test statistics”.
Null Hypothesis H0
Typically what we want to disprove (no effect).
 The Alternative Hypothesis HA expresses outcome of interest.
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
Animation how the null distribution is generated. Based on classical statistics^.
Classical Inference and Design Efficiency

8. Hypothesis Testing u t t 𝑝 𝑇>𝑡| 𝐻 0 Significance level α:
Acceptable false positive rate α.
 threshold uα
Threshold uα controls the false positive rate t 
Null Distribution of T
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 a value more extreme than t under the null hypothesis.
p-value
Null Distribution of T
𝑝 𝑇>𝑡| 𝐻 0
Classical Inference and Design Efficiency

9. T-test - one dimensional contrasts – SPM{t}
effect of interest > 0 ? =
amplitude > 0 ?
b1 = cTb > 0 ?
cT =
Question:
b1 b2 b3 b4 b5 ...
Null hypothesis:
H0: cTb=0
T =
contrast of estimated parameters
variance estimate
Test statistic:
Classical Inference and Design Efficiency

10. T-contrast in SPM For a given contrast c: ResMS image beta_???? images
# beta images = p = # columns of X
con_???? image
spmT_???? image
SPM{t}
Classical Inference and Design Efficiency

11. T-test: a simple example
Passive word listening versus rest
Q: activation during listening ?
Null hypothesis: b1 = 0
SPMresults:
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
𝑡= 𝑐 𝑇 𝛽 var 𝑐 𝑇 𝛽
Classical Inference and Design Efficiency

12. T-test: summary
T-test is a signal-to-noise measure (ratio of estimate to standard deviation of estimate).
H0:
vs HA:
Alternative hypothesis:
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.
Classical Inference and Design Efficiency

13. Scaling issue 𝛽 = ( 𝑋 𝑇 𝑋) −1 𝑋 𝑇 𝑦
[ ]
/ 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.
Contrast depends on scaling.
[ ]
Subject 5
/ 3
Be careful of the interpretation of the contrasts themselves (eg, for a second level analysis): sum ≠ average
𝛽 = ( 𝑋 𝑇 𝑋) −1 𝑋 𝑇 𝑦
Classical Inference and Design Efficiency

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

15. F-test - multidimensional contrasts – SPM{F}
Test multiple linear hypotheses:
Null Hypothesis H0: b3 = b4 = b5 = b6 = b7 = b8 = 0
cTb = 0 X0
X1 (b3-8)
cT =
Bessere Designmatrix.
Full model ?
Is any of b3-8 not equal 0?
Classical Inference and Design Efficiency

16. F-contrast in SPM ( RSS0 - RSS ) ResMS image beta_???? images
ess_???? images
spmF_???? images
( RSS0 - RSS )
SPM{F}
Classical Inference and Design Efficiency

17. 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)
Classical Inference and Design Efficiency

18. 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.
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.
Classical Inference and Design Efficiency

19. Orthogonal regressors
Variability described by 𝑋 1
Variability described by 𝑋 2
Testing for 𝑋 1
Testing for 𝑋 2
Variability in Y
Classical Inference and Design Efficiency

20. Correlated regressors
Shared variance
Variability described by 𝑋 1
Variability described by 𝑋 2
Variability in Y
Classical Inference and Design Efficiency

21. Correlated regressors
Testing for 𝑋 1
Variability described by 𝑋 1
Variability described by 𝑋 2
Variability in Y
Classical Inference and Design Efficiency

22. Correlated regressors
Testing for 𝑋 2
Variability described by 𝑋 1
Variability described by 𝑋 2
Variability in Y
Classical Inference and Design Efficiency

23. Correlated regressors
Variability described by 𝑋 1
Variability described by 𝑋 2
Variability in Y
Classical Inference and Design Efficiency

24. Correlated regressors
Testing for 𝑋 1
Variability described by 𝑋 1
Variability described by 𝑋 2
Variability in Y
Classical Inference and Design Efficiency

25. Correlated regressors
Testing for 𝑋 2
Variability described by 𝑋 1
Variability described by 𝑋 2
Variability in Y
Classical Inference and Design Efficiency

26. Correlated regressors
Testing for 𝑋 1 and/or 𝑋 2
Variability described by 𝑋 1
Variability described by 𝑋 2
Variability in Y
Classical Inference and Design Efficiency

27. 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.
Classical Inference and Design Efficiency

28. 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 x2 x2 x2 x^ 2
x^ = x2 – x1.x2 x1 x1 2 x1 x1 x^ 1
Classical Inference and Design Efficiency

29. 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).
Classical Inference and Design Efficiency

30. Design efficiency A B A+B A-B
𝑐= [1 0] 𝑇 : 𝑒 𝑐,𝑋 =18.1
𝑐= [ ] 𝑇 : 𝑒 𝑐,𝑋 =19.0 𝑐= [1 −1] 𝑇 : 𝑒 𝑐,𝑋 =95.2
𝑋 𝑇 𝑋= 1 −0.9 −0.9 1
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.
Classical Inference and Design Efficiency

31. 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.
Classical Inference and Design Efficiency