1. Karl Friston, Andrew Holmes, Stefan Kiebel, Will Penny
Statistical Analysis
Rik Henson
With thanks to:
Karl Friston, Andrew Holmes, Stefan Kiebel, Will Penny

2. Overview fMRI time-series Statistical Parametric Map
General Linear Model
Design matrix
Parameter Estimates
Smoothing
kernel
Motion
correction
Spatial
normalisation
Standard
template

3. Some Terminology
SPM (“Statistical Parametric Mapping”) is a massively univariate approach - meaning that a statistic (e.g., T-value) is calculated for every voxel - using the “General Linear Model”
Experimental manipulations are specified in a model (“design matrix”) which is fit to each voxel to estimate the size of the experimental effects (“parameter estimates”) in that voxel…
… on which one or more hypotheses (“contrasts”) are tested to make statistical inferences (“p-values”), correcting for multiple comparisons across voxels (using “Gaussian Field Theory”)
The parametric statistics assume continuous-valued data and additive noise that conforms to a “normal” distribution (“nonparametric” versions of SPM eschew such assumptions)

4. Some Terminology
SPM usually focused on “functional specialisation” - i.e. localising different functions to different regions in the brain
One might also be interested in “functional integration” - how different regions (voxels) interact
Multivariate approaches work on whole images and can identify spatial/temporal patterns over voxels, without necessarily specifying a design matrix (PCA, ICA)...
… or with an experimental design matrix (PLS, CVA), or with an explicit anatomical model of connectivity between regions - “effective connectivity” - eg using Dynamic Causal Modelling

5. Overview 1. General Linear Model Design Matrix 2. fMRI timeseries
Global normalisation
2. fMRI timeseries
Highpass filtering
HRF convolution
Temporal autocorrelation
3. Statistical Inference Gaussian Field Theory
4. Random Effects
5. Experimental Designs
6. Effective Connectivity

6. Overview 1. General Linear Model Design Matrix 2. fMRI timeseries
Global normalisation
2. fMRI timeseries
Highpass filtering
HRF convolution
Temporal autocorrelation
3. Statistical Inference Gaussian Field Theory
4. Random Effects
5. Experimental Designs
6. Effective Connectivity

7. General Linear Model… Parametric statistics one sample t-test
two sample t-test
paired t-test
Anova
AnCova
correlation
linear regression
multiple regression
F-tests
etc…
all cases of the General Linear Model

8. General Linear Model y = Xb + e Equation for single (and all) voxels:
yj = xj1 b1 + … + xjP bP + ej ej ~ N(0,s2)
yj : data for scan, j = 1…N
xjp : explanatory variables / covariates / regressors, p = 1…P
bp : parameters / regression slopes / fixed effects
ej : residual errors, independent & identically (normally) distributed
Equivalent matrix form:
y = Xb + e
X : “design matrix” / model

9. Matrix Formulation Equation for scan j Simultaneous equations for
scans 1..N(J)
…that can be solved
for parameters b1..P(L)
Regressors
Scans

10. General Linear Model (Estimation)
Residual errors and estimated error variance are:
r = y - Y 2 = rTr / df
where df are the degrees of freedom (assuming iid):
df = N - rank(X) (=N-P if X full rank)
( R = I - XX+ r = Ry df = trace(R) )
Estimate parameters from least squares fit to data, y:
b = (XTX)-1XTy = X+y (OLS estimates)
Fitted response is:
Y = Xb ^ ^ ^

11. General Linear Model (Inference)
Specify contrast (hypothesis), c, a linear combination of parameter estimates, cT b
c = [ ] T ^
Calculate T-stastistic for that contrast:
T(N-p) = cTb / var(cTb) = cTb / sqrt(2cT(XTX)-1c)
(c is a vector), or an F-statistic:
F(p-p0,N-p) = [(r0Tr0 – rTr) / (p-p0)] / [rTr / (N-P)]
where r0 and p0 are parameters of the reduced model specified by c (which is a matrix)
Prob. of falsely rejecting Null hypothesis, H0: cTb=0 (“p-value”) ^
c = [ ] F

12. Example PET experiment
12 scans, 3 conditions (1-way ANOVA)
yj = x1j b1 + x2j b2 + x3j b3 + x4j b4 + ej
where (dummy) variables:
x1j = [0,1] = condition A (first 4 scans)
x2j = [0,1] = condition B (second 4 scans)
x3j = [0,1] = condition C (third 4 scans)
x4j = [1] = grand mean
T-contrast : [ ] tests whether A>B
[ ] tests whether B>A
F-contrast: [ ] tests main effect of A,B,C
rank(X)=3 11 9 12 8 21 19 22 18 31 29 32 28 =
-10 10 20 + 1 -1 2 -2
c=[ ], T=10/sqrt(3.3*8)
df=12-3=9, T(9)=1.94, p<.05

13. Global Effects May be variation in PET tracer dose from scan to scan
Such “global” changes in image intensity (gCBF) confound local / regional (rCBF) changes of experiment
Adjust for global effects by:
- AnCova (Additive Model) - PET?
- Proportional Scaling fMRI?
Can improve statistics when orthogonal to effects of interest (as here)…
…but can also worsen when effects of interest correlated with global (as next)
global
global
AnCova
global
Scaling

14. Global Effects (AnCova)
12 scans, 3 conditions, 1 confounding covariate
yj = x1j b1 + x2j b2 + x3j b3 + x4j b4 + x5j b5 + ej
where (dummy) variables:
x1j = [0,1] = condition A (first 4 scans)
x2j = [0,1] = condition B (second 4 scans)
x3j = [0,1] = condition C (third 4 scans)
x4j = grand mean
x5j = global signal (mean over all voxels)
(further mean-corrected over all scans)
Global correlated here with conditions (and time)
Global estimate can be scaled to, eg, 50ml/min/dl
b1 b b b b5
rank(X)=4 11 9 12 8 21 19 22 18 31 29 32 28 =
1.7
5.0
8.3 15
6.7 + 1 -1 2 -2
c=[ ], T=3.3/sqrt(3.8*8)
df=12-4=8, T(8)=0.61, p>.05

15. Global Effects (fMRI)
Two types of scaling: Grand Mean scaling and Global scaling
Grand Mean scaling is automatic, global scaling is optional
Grand Mean scales by 100/mean over all voxels and ALL scans (i.e, single number per session)
Global scaling scales by 100/mean over all voxels for EACH scan (i.e, a different scaling factor every scan)
Problem with global scaling is that TRUE global is not (normally) known…
…we only estimate it by the mean over voxels
So if there is a large signal change over many voxels, the global estimate will be confounded by local changes
This can produce artifactual deactivations in other regions after global scaling
Since most sources of global variability in fMRI are low frequency (drift), high-pass filtering may be sufficient, and many people to not use global scaling

16. A word on correlation/estimability
If any column of X is a linear combination of any others (X is rank deficient), some parameters cannot be estimated uniquely (inestimable)
… which means some contrasts cannot be tested (eg, only if sum to zero)
This has implications for whether “baseline” (constant term) is explicitly or implicitly modelled
rank(X)=2
A B A+B 
cm = [ ]
cd = [ ] 
A B
“implicit”
cm = [1 0]
cd = [1 -1] 
b1 = 1.6
b2 = 0.7
cd*b = [1 -1]*b = 0.9
A A+B
“explicit”
b1 = 0.9
b2 = 0.7
cd = [1 0] 
cd*b = [1 0]*b = 0.9

17. A word on correlation/estimability
If any column of X is a linear combination of any others (X is rank deficient), some parameters cannot be estimated uniquely (inestimable)
… which means some contrasts cannot be tested (eg, only if sum to zero)
This has implications for whether “baseline” (constant term) is explicitly or implicitly modelled
(rank deficiency might be thought of as perfect correlation…)
rank(X)=2
A B A+B 
cm = [ ]
cd = [ ] 
A B
“implicit”
A A+B
“explicit”
T = 1 1
0 1
X(1) * T = X(2)
c(1) * T = c(2)
[ 1 -1 ] * = [ 1 0 ]
1 1
0 1

18. A word on correlation/estimability
When there is high (but not perfect) correlation between regressors, parameters can be estimated…
…but the estimates will be inefficient estimated (ie highly variable)
…meaning some contrasts will not lead to very powerful tests
A B A+B 
cm = [ ]
cd = [ ] 
A B A+B
convolved with HRF!
cm = [ ]
cd = [ ] 
()
SPM shows pairwise correlation between regressors, but this will NOT tell you that, eg, X1+X2 is highly correlated with X3…
… so some contrasts can still be inefficient, even though pairwise correlations are low

19. A word on orthogonalisation
To remove correlation between two regressors, you can explicitly orthogonalise one (X1) with respect to the other (X2):
X1^ = X1 – (X2X2+)X1 (Gram-Schmidt)
Paradoxically, this will NOT change the parameter estimate for X1, but will for X2
In other words, the parameter estimate for the orthogonalised regressor is unchanged!
This reflects fact that parameter estimates automatically reflect orthogonal component of each regressor…
…so no need to orthogonalise, UNLESS you have a priori reason for assigning common variance to the other regressor Y X2 X1
b2^
X1^ b2 b1

20. A word on orthogonalisation
X X2
b1 = 0.9
b2 = 0.7
X1 X2^
Orthogonalise X2
(Model M1)
Orthogonalise X1
(Model M2)
X1^ X2
b1(M2) = 0.9
b2(M2) = 1.15
= b1(M1) – b2(M1)
= ( b1(M1) + b2(M1) )/2
b1(M1) = 1.6
b2(M1) = 0.7
T =
-0.5 1

21. Overview 1. General Linear Model Design Matrix 2. fMRI timeseries
Global normalisation
2. fMRI timeseries
Highpass filtering
HRF convolution
Temporal autocorrelation
3. Statistical Inference Gaussian Field Theory
4. Random Effects
5. Experimental Designs
6. Effective Connectivity

22. fMRI Analysis Scans are treated as a timeseries…
… and can be filtered to remove low-frequency (1/f) noise
Effects of interest are convolved with haemodynamic (BOLD) response function (HRF), to capture sluggish nature of response
Scans can no longer be treated as independent observations…
… they are typically temporally autocorrelated (for TRs<8s)

23. fMRI Analysis Scans are treated as a timeseries…
… and can be filtered to remove low-frequency (1/f) noise
Effects of interest are convolved with haemodynamic (BOLD) response function (HRF), to capture sluggish nature of response
Scans can no longer be treated as independent observations…
… they are typically temporally autocorrelated (for TRs<8s)

24. (Epoch) fMRI example… = b1 b2 + + (t) voxel timeseries
box-car function = b1 b2 +
baseline (mean)
+ (t)
(box-car
unconvolved)

25. (Epoch) fMRI example… y b1 b2   +  = X (voxel time series)
data vector b1 b2 
parameters  + 
error vector = X
design matrix

26. Low frequency noise Low frequency noise: Physical (scanner drifts)
aliasing
Low frequency noise:
Physical (scanner drifts)
Physiological (aliased)
cardiac (~1 Hz)
respiratory (~0.25 Hz)
power spectrum
highpass
filter
power spectrum
noise
signal
(eg infinite 30s on-off)

27. (Epoch) fMRI example… ...with highpass filter1 2 3 4 5 6 7 8 9
parameters   +
error vector  =
design matrix X
data vector y

28. (Epoch) fMRI example… …fitted and adjusted data
Raw fMRI timeseries
Adjusted data
fitted box-car
highpass filtered (and scaled)
fitted high-pass filter
Residuals

29. fMRI Analysis Scans are treated as a timeseries…
… and can be filtered to remove low-frequency (1/f) noise
Effects of interest are convolved with haemodynamic (BOLD) response function (HRF), to capture sluggish nature of response
Scans can no longer be treated as independent observations…
… they are typically temporally autocorrelated (for TRs<8s)

30. Convolution with HRF  = Unconvolved fit Residuals Convolved fit
hæmodynamic response 
convolved with HRF =
Boxcar function
Convolved fit
Residuals (less structure)

31. fMRI Analysis Scans are treated as a timeseries…
… and can be filtered to remove low-frequency (1/f) noise
Effects of interest are convolved with haemodynamic (BOLD) response function (HRF), to capture sluggish nature of response
Scans can no longer be treated as independent observations…
… they are typically temporally autocorrelated (for TRs<8s)

32. Temporal autocorrelation…
Because the data are typically correlated from one scan to the next, one cannot assume the degrees of freedom (dfs) are simply the number of scans minus the dfs used in the model – need “effective degrees of freedom”
In other words, the residual errors are not independent:
Y = Xb + e e ~ N(0,s2V) V  I, V=AA'
where A is the intrinsic autocorrelation
Generalised least squares:
KY = KXb + Ke Ke ~ N(0, s2V) V = KAA'K'
(autocorrelation is a special case of “nonsphericity”…)

33. Temporal autocorrelation (History)
KY = KXb + Ke Ke ~ N(0, s2V) V = KAA'K'
One method is to estimate A, using, for example, an AR(p) model, then:
K = A-1 V = I (allows OLS)
This “pre-whitening” is sensitive, but can be biased if K mis-estimated
Another method (SPM99) is to smooth the data with a known autocorrelation that swamps any intrinsic autocorrelation:
K = S V = SAA'S’ ~ SS' (use GLS)
Effective degrees of freedom calculated with Satterthwaite approximation ( df = trace(RV)2/trace(RVRV) )
This is more robust (providing the temporal smoothing is sufficient, eg 4s FWHM Gaussian), but less sensitive
Most recent method (SPM2) is to restrict K to highpass filter, and estimate residual autocorrelation A using voxel-wide, one-step ReML…

34. Nonsphericity and ReML (SPM2)
New in
SPM2
Nonsphericity and ReML (SPM2)
Scans
cov(e)
spherical
Nonsphericity means (kind of) that:
Ce = cov(e)  s2I
Nonsphericity can be modelled by set of variance components:
Ce = 1Q1 + 2Q2 + 3Q3 ...
(i are hyper-parameters)
- Non-identical (inhomogeneous): (e.g, two groups of subjects)
- Non-independent (autocorrelated): (e.g, white noise + AR(1))

35. Nonsphericity and ReML (SPM2)
New in
SPM2
Nonsphericity and ReML (SPM2)
Joint estimation of parameters and hyper-parameters requires ReML
ReML gives (Restricted) Maximum Likelihood (ML) estimates of (hyper)parameters, rather than Ordinary Least Square (OLS) estimates
ML estimates are more efficient, entail exact dfs (no Satterthwaite approx)…
…but computationally expensive: ReML is iterative (unless only one hyper-parameter)
To speed up:
Correlation of errors (V) estimated by pooling over voxels
Covariance of errors (s2V) estimated by single, voxel-specific scaling hyperparameter
Ce = ReML( yyT, X, Q ) ^
bOLS = (XTX)-1XTy (= X+y)
bML = (XTCe-1X)-1XTCe-1y
V = ReML(  yjyjT, X, Q )

36. Nonsphericity and ReML (SPM2)
New in
SPM2
Nonsphericity and ReML (SPM2)
Voxels to be pooled collected by first-pass through data (OLS)
(biased if correlation structure
not stationary across voxels?)
Correlation structure V estimated iteratively using ReML once, pooling over all voxels
Remaining hyper-parameter estimated using V and ReML noniteratively, for each voxel
Estimation of nonsphericity is used to pre-whiten the data and design matrix, W=V-1/2 (or by KW, if highpass filter K present)
(which is why design matrices in SPM2 can differ from those in SPM99 after estimation) X B W WX

37. The Full-Monty T-test (SPM2)
New in
SPM2
The Full-Monty T-test (SPM2)

38. Overview 1. General Linear Model Design Matrix 2. fMRI timeseries
Global normalisation
2. fMRI timeseries
Highpass filtering
HRF convolution
Temporal autocorrelation
3. Statistical Inference Gaussian Field Theory
4. Random Effects
5. Experimental Designs
6. Effective Connectivity

39. Multiple comparisons…
If n=100,000 voxels tested with pu=0.05 of falsely rejecting Ho...
…then approx n  pu (eg 5,000) will do so by chance (false positives, or “type I” errors)
Therefore need to “correct” p-values for number of comparisons
A severe correction would be a Bonferroni, where pc = pu /n…
…but this is only appropriate when the n tests independent…
… SPMs are smooth, meaning that nearby voxels are correlated
=> Gaussian Field Theory...
SPM{t}
Eg random noise
Gaussian 10mm FWHM
(2mm pixels)
pu = 0.05

40. Gaussian Field Theory
Consider SPM as lattice representation of continuous random field
“Euler characteristic” - topological measure of “excursion set” (e.g, # components - # “holes”)
Smoothness estimated by covariance of partial derivatives of residuals (expressed as “resels” or FWHM)
Assumes: 1) residuals are multivariate normal 2) smoothness » voxel size (practically, FWHM  3  VoxDim)
Not necessarily stationary: smoothness estimated locally as resels-per-voxel

41. E[(WAu)] = S Rd (W) rd (u)
Generalised Form
General form for expected Euler characteristic for D dimensions:
E[(WAu)] = S Rd (W) rd (u)
Rd (W): d-dimensional Minkowski
– function of dimension, d, space W and smoothness:
R0(W) = (W) Euler characteristic of W
R1(W) = resel diameter
R2(W) = resel surface area
R3(W) = resel volume
rd (W): d-dimensional EC density of Z(x)
– function of dimension, d, threshold, u, and statistic, e.g. Z-statistic:
r0(u) = 1- (u)
r1(u) = (4 ln2)1/2 exp(-u2/2) / (2p)
r2(u) = (4 ln2) exp(-u2/2) / (2p)3/2
r3(u) = (4 ln2)3/2 (u2 -1) exp(-u2/2) / (2p)2
r4(u) = (4 ln2)2 (u3 -3u) exp(-u2/2) / (2p)5/2

42. set-level: P(c  3, n  k, t  u) = 0.019
Levels of Inference
Three levels of inference:
extreme voxel values
voxel-level inference
big suprathreshold clusters
cluster-level inference
many suprathreshold clusters
set-level inference
Omnibus: P(c  7, t  u) = 0.031
voxel-level: P(t  4.37) = .048
n=82
n=32
n=12
set-level: P(c  3, n  k, t  u) = 0.019
cluster-level: P(n  82, t  u) = 0.029
Parameters:
“Height” threshold, u - t > 3.09
“Extent” threshold, k voxels
Dimension, D
Volume, S voxels
Smoothness, FWHM voxels

43. (Spatial) Specificity vs. Sensitivity

44. Small-volume correction
If have an a priori region of interest, no need to correct for whole- brain!
But can use GFT to correct for a Small Volume (SVC)
Volume can be based on:
An anatomically-defined region
A geometric approximation to the above (eg rhomboid/sphere)
A functionally-defined mask (based on an ORTHOGONAL contrast!)
Extent of correction can be APPROXIMATED by a Bonferonni correction for the number of resels…
..but correction also depends on shape (surface area) as well as size (volume) of region (may want to smooth volume if rough)

45. Example SPM window

46. Overview 1. General Linear Model Design Matrix 2. fMRI timeseries
Global normalisation
2. fMRI timeseries
Highpass filtering
HRF convolution
Temporal autocorrelation
3. Statistical Inference Gaussian Field Theory
4. Random Effects
5. Experimental Designs
6. Effective Connectivity

47. Fixed vs. Random Effects
Subjects can be Fixed or Random variables
If subjects are a Fixed variable in a single design matrix (SPM “sessions”), the error term conflates within- and between-subject variance
In PET, this is not such a problem because the within-subject (between-scan) variance can be as great as the between-subject variance; but in fMRI the between-scan variance is normally much smaller than the between-subject variance
If one wishes to make an inference from a subject sample to the population, one needs to treat subjects as a Random variable, and needs a proper mixture of within- and between-subject variance
In SPM, this is achieved by a two-stage procedure:
1) (Contrasts of) parameters are estimated from a (Fixed Effect) model for each subject
2) Images of these contrasts become the data for a second design matrix (usually simple t-test or ANOVA)
Multi-subject Fixed Effect model
Subject 1
error df ~ 300
Subject 2
Subject 3
Subject 4
Subject 5
Subject 6

48. Two-stage “Summary Statistic” approach
1st-level (within-subject)
2nd-level (between-subject) b1 ^ b2 b3 b4 b5 b6
contrast images of cbi 
N=6 subjects
(error df =5)
One-sample
t-test
1 ^
2
3
4
5
6
w = within-subject error
p < (uncorrected)
SPM{t} ^
bpop 
WHEN special case of n independent observations per subject:
var(bpop) = 2b / N + 2w / Nn

49. Limitations of 2-stage approach
New in
SPM2
Limitations of 2-stage approach
Summary statistic approach is a special case, valid only when each subject’s design matrix is identical (“balanced designs”)
In practice, the approach is reasonably robust to unbalanced designs (Penny, 2004)
More generally, exact solutions to any hierarchical GLM can be obtained using ReML
This is computationally expensive to perform at every voxel (so not implemented in SPM2)
Plus modelling of nonsphericity at 2nd-level can minimise potential bias of unbalanced designs…

50. Inhomogeneous variance
New in
SPM2
Nonsphericity again!
When tests at 2nd-level are more complicated than 1/2-sample t-tests, errors can be non i.i.d
For example, two groups (e.g, patients and controls) may have different variances (non-identically distributed; inhomogeniety of variance)
Or when taking more than one parameter per subject (repeated measures, e.g, multiple basis functions in event-related fMRI), errors may be non-independent
(If nonsphericity correction selected, inhomogeniety assumed, and further option for repeated measures)
Same method of variance component estimation with ReML (that used for autocorrelation) is used
(Greenhouse-Geisser correction for repeated- measures ANOVAs is a special case approximation)
Inhomogeneous variance
(3 groups of 4 subjects) Q 1 2 3
Repeated measures
(3 groups of 4 subjects) Q

51. Hierarchical Models y = X(1) (1) + e(1) (1) = X(2) (2) + e(2) …
New in
SPM2
Hierarchical Models
Two-stage approach is special case of Hierarchical GLM
In a Bayesian framework, parameters of one level can be made priors on distribution of parameters at lower level: “Parametric Empirical Bayes” (Friston et al, 2002)
The parameters and hyperparameters at each level can be estimated using EM algorithm (generalisation of ReML)
Note parameters and hyperparameters at final level do not differ from classical framework
Second-level could be subjects; it could also be voxels…
y = X(1) (1) + e(1)
(1) = X(2) (2) + e(2) …
(n-1) = X(n) (n) + e(n)
Ce(i) =  k(i) Qk(i)

52. Parametric Empirical Bayes & PPMs
New in
SPM2
Parametric Empirical Bayes & PPMs
Bayes rule:
p(|y) = p(y|) p()
Posterior Likelihood Prior
(PPM) (SPM)
What are the priors?
In “classical” SPM, no (flat) priors
In “full” Bayes, priors might be from theoretical arguments, or from independent data
In “empirical” Bayes, priors derive from same data, assuming a hierarchical model for generation of that data

53. Parametric Empirical Bayes & PPMs
New in
SPM2
Parametric Empirical Bayes & PPMs
Bayes rule:
p(|y) = p(y|) p()
Classical T-test
Posterior Likelihood Prior
(PPM) (SPM)
For PPMs in SPM2, priors come from distribution over voxels
If remove mean over voxels, prior mean can be set to zero (a “shrinkage” prior)
One can threshold posteriors for a given probability of a parameter estimate greater than some value …
…to give a posterior probability map (PPM)
Bayesian test

54. Parametric Empirical Bayes & PPMs
New in
SPM2
Parametric Empirical Bayes & PPMs
Activations greater than certain amount Voxels with non-zero activations
Can infer no responses Cannot “prove the null hypothesis”
No fallacy of inference Fallacy of inference (large df)
Inference independent of search volume Correct for search volume
Computationally expensive Computationally faster

55. Overview 1. General Linear Model Design Matrix 2. fMRI timeseries
Global normalisation
2. fMRI timeseries
Highpass filtering
HRF convolution
Temporal autocorrelation
3. Statistical Inference Gaussian Field Theory
4. Random Effects
5. Experimental Designs
6. Effective Connectivity

56. A taxonomy of design Categorical designs Parametric designs
Subtraction - Additive factors and pure insertion
Conjunction - Testing multiple hypotheses
Parametric designs
Linear - Cognitive components and dimensions
Nonlinear - Polynomial expansions
Factorial designs
Categorical - Interactions and pure insertion
- Adaptation, modulation and dual-task inference
Parametric - Linear and nonlinear interactions
- Psychophysiological Interactions

57. A taxonomy of design Categorical designs Parametric designs
Subtraction - Additive factors and pure insertion
Conjunction - Testing multiple hypotheses
Parametric designs
Linear - Cognitive components and dimensions
Nonlinear - Polynomial expansions
Factorial designs
Categorical - Interactions and pure insertion
- Adaptation, modulation and dual-task inference
Parametric - Linear and nonlinear interactions
- Psychophysiological Interactions

58. A categorical analysis
Experimental design
Word generation G
Word repetition R
R G R G R G R G R G R G
G - R = Intrinsic word generation
…under assumption of pure insertion,
ie, that G and R do not differ in other ways

59. A taxonomy of design Categorical designs Parametric designs
Subtraction - Additive factors and pure insertion
Conjunction - Testing multiple hypotheses
Parametric designs
Linear - Cognitive components and dimensions
Nonlinear - Polynomial expansions
Factorial designs
Categorical - Interactions and pure insertion
- Adaptation, modulation and dual-task inference
Parametric - Linear and nonlinear interactions
- Psychophysiological Interactions

60. Cognitive Conjunctions
One way to minimise problem of pure insertion is to isolate same process in several different ways (ie, multiple subtractions of different conditions)
Task (1/2)
Viewing Naming
Stimuli (A/B)
Objects Colours A1 A2 B2 B1
Visual Processing V
Object Recognition R
Phonological Retrieval P
Object viewing R,V
Colour viewing V
Object naming P,R,V
Colour naming P,V
(Object - Colour viewing) [ ]
&
(Object - Colour naming) [ ]
[ R,V - V ] & [ P,R,V - P,V ] = R & R = R
(assuming RxP = 0; see later)
Common object
recognition response (R)
Price et al, 1997

61. Cognitive Conjunctions
New in
SPM2
Cognitive Conjunctions
Original (SPM97) definition of conjunctions entailed sum of two simple effects (A1-A2 + B1-B2) plus exclusive masking with interaction (A1-A2) - (B1-B2)
Ie, “effects significant and of similar size”
(Difference between conjunctions and masking is that conjunction p-values reflect the conjoint probabilities of the contrasts)
SPM2 defintion of conjunctions uses advances in Gaussian Field Theory (e.g, T2 fields), allowing corrected p-values
However, the logic has changed slightly, in that voxels can survive a conjunction even though they show an interaction
A1-A2
B1-B2
p((A1-A2)=
(B1-B2))>P2
p(A1=A2+B1=B2)<P1 +
p(A1=A2)<p
A1-A2
B1-B2
p(B1=B2)<p

62. A taxonomy of design Categorical designs Parametric designs
Subtraction - Additive factors and pure insertion
Conjunction - Testing multiple hypotheses
Parametric designs
Linear - Cognitive components and dimensions
Nonlinear - Polynomial expansions
Factorial designs
Categorical - Interactions and pure insertion
- Adaptation, modulation and dual-task inference
Parametric - Linear and nonlinear interactions
- Psychophysiological Interactions

63. Nonlinear parametric responses
SPM{F}
E.g, F-contrast [0 1 0] on
Quadratic Parameter =>
Quadratic
Linear
Inverted ‘U’ response to
increasing word presentation
rate in the DLPFC
Polynomial expansion:
f(x) ~ b1 x + b2 x
…(N-1)th order for N levels

64. A taxonomy of design Categorical designs Parametric designs
Subtraction - Additive factors and pure insertion
Conjunction - Testing multiple hypotheses
Parametric designs
Linear - Cognitive components and dimensions
Nonlinear - Polynomial expansions
Factorial designs
Categorical - Interactions and pure insertion
- Adaptation, modulation and dual-task inference
Parametric - Linear and nonlinear interactions
- Psychophysiological Interactions

65. Interactions and pure insertion
Presence of an interaction can show a failure of pure insertion (using earlier example)… A1 A2 B2 B1
Task (1/2)
Viewing Naming
Stimuli (A/B)
Objects Colours
Visual Processing V
Object Recognition R
Phonological Retrieval P
Object viewing R,V
Colour viewing V
Object naming P,R,V,RxP
Colour naming P,V
Naming-specific
object recognition
viewing naming
Object - Colour
(Object – Colour) x (Viewing – Naming)
[ ] - [ ] = [1 -1]  [1 -1] = [ ]
[ R,V - V ] - [ P,R,V,RxP - P,V ] = R – R,RxP = RxP

66. A taxonomy of design Categorical designs Parametric designs
Subtraction - Additive factors and pure insertion
Conjunction - Testing multiple hypotheses
Parametric designs
Linear - Cognitive components and dimensions
Nonlinear - Polynomial expansions
Factorial designs
Categorical - Interactions and pure insertion
- Adaptation, modulation and dual-task inference
Parametric - Linear and nonlinear interactions
- Psychophysiological Interactions

67. Psycho-physiological Interaction (PPI)
Parametric, factorial design, in which one factor is psychological (eg attention)
...and other is physiological (viz. activity extracted from a brain region of interest)
SPM{Z}
V1 activity
Attention
time V1
attention V5
V5 activity
no attention
Attentional modulation of
V1 - V5 contribution
V1 activity

68. Psycho-physiological Interaction (PPI)
New in
SPM2
Psycho-physiological Interaction (PPI)
PPIs tested by a GLM with form:
y = (V1A).b1 + V1.b2 + A.b3 + e c = [1 0 0]
However, the interaction term of interest, V1A, is the product of V1 activity and Attention block AFTER convolution with HRF
We are really interested in interaction at neural level, but:
(HRF  V1)  (HRF  A)  HRF  (V1  A)
(unless A low frequency, eg, blocked; so problem for event-related PPIs)
SPM2 can effect a deconvolution of physiological regressors (V1), before calculating interaction term and reconvolving with the HRF
Deconvolution is ill-constrained, so regularised using smoothness priors (using ReML)

69. Overview 1. General Linear Model Design Matrix 2. fMRI timeseries
Global normalisation
2. fMRI timeseries
Highpass filtering
HRF convolution
Temporal autocorrelation
3. Statistical Inference Gaussian Field Theory
4. Random Effects
5. Experimental Designs
6. Effective Connectivity

70. Effective vs. functional connectivity
Correlations:
A B C 1
0.49 1
No connection between B and C,
yet B and C correlated because of common input from A, eg:
A = V1 fMRI time-series
B = 0.5 * A + e1
C = 0.3 * A + e2
Functional connectivity A B C
0.49
0.31
Effective connectivity
-0.02
2=0.5, ns.
Model
Correlations (Normalised ts)
Analysis of correlation structure
proper effcon (SEM)
fallacy of wrong models

71. Dynamic Causal Modelling
New in
SPM2
Dynamic Causal Modelling
PPIs allow a simple (restricted) test of effective connectivity
Structural Equation Modelling is more powerful (Buchel & Friston, 1997)
However in SPM2, Dynamic Causal Modelling (DCM) is preferred
DCMs are dynamic models specified at the neural level
The neural dynamics are transformed into predicted BOLD signals using a realistic biological haemodynamic forward model (HDM)
The neural dynamics comprise a deterministic state-space model and a bilinear approximation to model interactions between variables

72. Dynamic Causal Modelling
New in
SPM2
Dynamic Causal Modelling
The variables consist of:
connections between regions
self-connections
direct inputs (eg, visual stimulations)
contextual inputs (eg, attention)
Connections can be bidirectional
Variables estimated using EM algorithm
Priors are:
empirical (for haemodynamic model)
principled (dynamics to be convergent)
shrinkage (zero-mean, for connections)
Inference using posterior probabilities
Methods for Bayesian model comparison
direct inputs - u1
(e.g. visual stimuli)
contextual inputs - u2
(e.g. attention)
z1 V1
z2 V5
z3 SPC y1 y2 y3
z = f(z,u,z)  Az + uBz + Cu
y = h(z,h) + e
z = state vector
u = inputs
 = parameters (connection/haemodynamic) .

73. Dynamic Causal Modelling
New in
SPM2
Dynamic Causal Modelling
stimuli u1
context u2 u1  - + - Z1 u2 + z1 + Z2 - z2 - 

74. Dynamic Causal Modelling
New in
SPM2
Dynamic Causal Modelling V1
IFG V5
SPC
Motion
Photic
Attention
.82
(100%)
.42
.37
(90%)
.69 (100%)
.47
.65 (100%)
.52 (98%)
.56
(99%)
Friston et al. (2003)
Effects
Photic – dots vs fixation
Motion – moving vs static
Attenton – detect changes
Büchel & Friston (1997)
Attention modulates the backward-connections IFG→SPC and SPC→V5
The intrinsic connection V1→V5 is insignificant in the absence of motion

76. Some References
Friston KJ, Holmes AP, Worsley KJ, Poline J-B, Frith CD, Frackowiak RSJ (1995) Statistical parametric maps in functional imaging: A general linear approach” Human Brain Mapping 2:
Worsley KJ & Friston KJ (1995) Analysis of fMRI time series revisited — again” NeuroImage 2:
Friston KJ, Josephs O, Zarahn E, Holmes AP, Poline J-B (2000) “To smooth or not to smooth” NeuroImage
Zarahn E, Aguirre GK, D'Esposito M (1997) “Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5:
Holmes AP, Friston KJ (1998) “Generalisability, Random Effects & Population Inference” NeuroImage 7(4-2/3):S754
Worsley KJ, Marrett S, Neelin P, Evans AC (1992) “A three-dimensional statistical analysis for CBF activation studies in human brain”Journal of Cerebral Blood Flow and Metabolism 12:
Worsley KJ, Marrett S, Neelin P, Vandal AC, Friston KJ, Evans AC (1995) “A unified statistical approach for determining significant signals in images of cerebral activation” Human Brain Mapping 4:58-73
Friston KJ, Worsley KJ, Frackowiak RSJ, Mazziotta JC, Evans AC (1994) Assessing the Significance of Focal Activations Using their Spatial Extent” Human Brain Mapping 1:
Cao J (1999) The size of the connected components of excursion sets of 2, t and F fields” Advances in Applied Probability (in press)
Worsley KJ, Marrett S, Neelin P, Evans AC (1995) Searching scale space for activation in PET images” Human Brain Mapping 4:74-90
Worsley KJ, Poline J-B, Vandal AC, Friston KJ (1995) Tests for distributed, non-focal brain activations” NeuroImage 2:
Friston KJ, Holmes AP, Poline J-B, Price CJ, Frith CD (1996) Detecting Activations in PET and fMRI: Levels of Inference and Power” Neuroimage 4:

77. PCA/SVD and Eigenimages
A time-series of 1D images
128 scans of 32 “voxels”
Expression of 1st 3 “eigenimages”
Eigenvalues and spatial “modes”
The time-series ‘reconstituted’

78. PCA/SVD and Eigenimages
APPROX.
OF Y U1 = s1 V1
APPROX.
OF Y
+ s2 U2 V2
+ s3
APPROX.
OF Y U3 V3
voxels
Y (DATA)
+ ...
time
Y = USVT = s1U1V1T s2U2V2T

79. Time x Condition interaction
Time x condition interactions (i.e. adaptation)
assessed with the SPM{T}

80. Structural Equation Modelling (SEM)
Minimise the difference between the observed (S) and implied () covariances by adjusting the path coefficients (B)
The implied covariance structure:
x = x.B + z x = z.(I - B)-1
x : matrix of time-series of Regions 1-3
B: matrix of unidirectional path coefficients
Variance-covariance structure:
xT . x =  = (I-B)-T. C.(I-B)-1
where C = zT z
xT.x is the implied variance covariance structure 
C contains the residual variances (u,v,w) and covariances
The free parameters are estimated by minimising a [maximum likelihood] function of S and  1 3 2 z
SEM
Key sentence
BUT how can we calculate an implied var-cov matrix -->Equ
Minimise the diff

81. Attention - No attention
0.43
0.75
0.47
0.76
No attention
Attention
Changes in “effective connectivity”

82. Second-order InteractionsPP
2 =11, p<0.01
0.14 V5 V1 =
V1xPP V5
The information that att is required does not come from V1, therefore top-down mechanisms: PP modulating V1 --> V5 OR PP changing the sensitivity of V5 for other inputs (ie V1)
Previous presentations V5 --> PP mod by PFC: Same principle
Modulatory influence of parietal cortex on V1 to V5