The General Linear Model and Statistical Parametric Mapping

1
Experimental Design and Optimisation Rik
Henson With thanks to Karl Friston, Oliver
Josephs
2
Overview
1. A Taxonomy of Designs 2. Blocked vs
Randomised Designs 3. Efficient Designs
3
A taxonomy of design

• Categorical 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

4
A taxonomy of design

• Categorical 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

5
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
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A taxonomy of design

• Categorical 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

7
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)

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) 1 -1 0 0 (Object - Colour
naming) 0 0 1 -1 R,V - V P,R,V -
P,V R R R (assuming RxP 0 see later)
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Cognitive Conjunctions
9
A taxonomy of design

• Categorical 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

10
A (linear) parametric contrast
Linear effect of time
11
A taxonomy of design

• Categorical 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

12
Nonlinear parametric design matrix
E.g, F-contrast 0 1 0 on Quadratic Parameter gt
Inverted U response to increasing word
presentation rate in the DLPFC
Polynomial expansion f(x) b1 x b2 x2
... (N-1)th order for N levels
13
A taxonomy of design

• Categorical 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

14
Interactions and pure insertion

• Presence of an interaction can show a failure of
  pure insertion (using earlier example)

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
(Object Colour) x (Viewing Naming) 1 -1 0
0 - 0 0 1 -1 1 -1 ? 1 -1 1 -1 -1
1 R,V - V - P,R,V,RxP - P,V R
R,RxP RxP
15
Interactions and pure insertion
16
A taxonomy of design

• Categorical 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

17
(Linear) Parametric Interaction
A (Linear) Time-by-Condition Interaction (Genera
tion strategy?)
Contrast 5 3 1 -1 -3 -5 ? -1 1
18
Nonlinear Parametric Interaction

• Factorial Design with 2 factors
• Gen/Rep (Categorical, 2 levels)
• Time (Parametric, 6 levels)
• Time effects modelled with both linear and
  quadratic components

19
A taxonomy of design

• Categorical 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

20
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)
V1 activity
time
attention
V5 activity
no attention
Attentional modulation of V1 - V5 contribution
V1 activity
21
Psycho-physiological Interaction (PPI)
0 0 1
V1 activity
time
attention
V5 activity
no attention
V1 activity
V1xAtt
22
Overview
1. A Taxonomy of Designs 2. Blocked vs
Randomised Designs 3. Efficient Designs
23
Epoch vs Events

• Epochs are periods of sustained stimulation (e.g,
  box-car functions)
• Events are impulses (delta-functions)
• In SPM99, epochs and events are distinct (eg, in
  choice of basis functions)
• In SPM2/5, all conditions are specified in terms
  of their 1) onsets and 2) durations
• events simply have zero duration
• Near-identical regressors can be created by 1)
  sustained epochs, 2) rapid series of events
  (SOAslt3s)
• i.e, designs can be blocked or randomised
  models can be epoch or event-related

24
Advantages of Event-related Models
1. Randomised (intermixed) trial
order c.f. confounds of blocked designs
(Johnson et al 1997) 2. Post hoc / subjective
classification of trials e.g, according to
subsequent memory (Wagner et al 1998) 3. Some
events can only be indicated by subject (in
time) e.g, spontaneous perceptual changes
(Kleinschmidt et al 1998) 4. Some trials cannot
be blocked e.g, oddball designs (Clark et
al., 2000) 5. More accurate models even for
blocked designs? e.g, (Price et al, 1999)
25
Disadvantages of Randomised Designs
1. Less efficient for detecting effects than
are blocked designs (see later) 2. Some
psychological processes may be better blocked
(eg task-switching, attentional instructions)
26

Mixed Designs

• Blocks of trials with varying SOAs
• Blocks are modelled as epochs (sustained or
  state effect)
• Trials are modelled as events (transient or
  item effects)
• (normally confounded in conventional blocked
  designs)
• Varying (some short, some long) SOAs between
  trials needed to decorrelate epoch and
  event-related covariates (see later)
• For example, Chawla et al (1999)
• Visual stimulus dots periodically changing in
  colour or motion
• Epochs of attention to 1) motion, or 2) colour
• Events are target stimuli differing in motion or
  colour

27

(Chawla et al 1999)
28

Mixed Designs

• Blocks of trials with varying SOAs
• Blocks are modelled as epochs (sustained or
  state effect)
• Trials are modelled as events (transient or
  item effects)
• (normally confounded in conventional blocked
  designs)
• Varying (some short, some long) SOAs between
  trials needed to decorrelate epoch and
  event-related covariates (see later)
• Allows conclusion that selective attention
  modulates BOTH
• 1) baseline activity (state-effect, additive)
• 2) evoked response (item-effect, multiplicative)
• (But note tension between maximising fMRI
  efficiency to separate item and state effects,
  and maximising efficiency for each effect alone,
  and between long SOAs and maintaining a
  cognitive set)

29
Overview
1. A Taxonomy of Designs 2. Blocked vs
Randomised Designs 3. Efficient Designs
30
General Advice

• Scan as long as subjects can accommodate (eg
  40-60mins) keep subjects as busy as possible!
• If a Group study, number of subjects more
  important than time per subject (though
  additional set-up time may encourage multiple
  experiments per subject)
• Do not contrast conditions that are far apart in
  time (because of low-freq noise)
• Randomize the order, or randomize the SOA, of
  conditions that are close in time
• http//www.mrc-cbu.cam.ac.uk/Imaging/Common/fMRI-e
  fficiency.shtml

31
Expanded Overview
1. A Taxonomy of Designs 2. Blocked vs
Randomised Designs 3. Efficient Designs 3.1
Response vs Baseline (signal-processing) 3.2
Response 1 - Response 2 (statistics) 3.3
Response 1 Response 2 (correlations)
32
Fixed SOA 16s
Stimulus (Neural)
HRF
Predicted Data
?

Not particularly efficient
33
Fixed SOA 4s
Stimulus (Neural)
HRF
Predicted Data
Very Inefficient
34
Randomised, SOAmin 4s
Stimulus (Neural)
HRF
Predicted Data
More Efficient
35
Blocked, SOAmin 4s
Stimulus (Neural)
HRF
Predicted Data
Even more Efficient
36
Blocked, epoch 20s
Stimulus (Neural)
HRF
Predicted Data
?
Blocked-epoch (with small SOA) and Time-Freq
equivalences
37
Sinusoidal modulation, f 1/33s
Stimulus (Neural)
HRF
Predicted Data
The most efficient design of all!
38
High-pass Filtering

• fMRI contains low frequency noise
• Physical (scanner drifts)
• Physiological (aliased)
• cardiac (1 Hz)
• respiratory (0.25 Hz)

39
Blocked (80s), SOAmin4s, highpass filter
1/120s
Stimulus (Neural)
HRF
Predicted Data
Dont have long (gt60s) blocks!
40
Randomised, SOAmin4s, highpass filter 1/120s
Stimulus (Neural)
HRF
Predicted Data
(Randomised design spreads power over frequencies)
41
2. How about multiple conditions?

• We have talked about detecting a basic response
  vs baseline, but how about detecting differences
  between two or more response-types (event-types)?

42
Design Efficiency

• T cTb / std(cTb)
• std(cTb) sqrt(?2cT(XTX)-1c)
• For max. T, want min. contrast variability
  (Friston et al, 1999)
• If assume that noise variance (?2) is unaffected
  by changes in X
• then want maximal efficiency, e
• e(c,X) cT (XTX)-1 c -1

43
Efficiency - Multiple Event-types

• Design parametrised by
• SOAmin Minimum SOA
• pi(h) Probability of event-type i given
  history h of last m events
• With n event-types pi(h) is a nm ?? n Transition
  Matrix
• Example Randomised AB
• A B A 0.5 0.5
• B 0.5 0.5
• gt ABBBABAABABAAA...

Josephs Henson (1999)
44
Efficiency - Multiple Event-types

• Example Alternating AB
• A B A 0 1
• B 1 0
• gt ABABABABABAB...

Josephs Henson (1999)

• Example Permuted AB
• A B
• AA 0 1
• AB 0.5 0.5
• BA 0.5 0.5
• BB 1 0
• gt ABBAABABABBA...

45
Efficiency - Multiple Event-types

• Example Null events
• A B
• A 0.33 0.33
• B 0.33 0.33
• gt AB-BAA--B---ABB...
• Efficient for differential and main effects at
  short SOA
• Equivalent to stochastic SOA (Null Event like
  third unmodelled event-type)
• Selective averaging of data (Dale Buckner 1997)

Josephs Henson (1999)
46
Interim Conclusions

• Optimal design for one contrast may not be
  optimal for another
• With randomised designs, optimal SOA for
  differential effect (A-B) is minimal SOA
  (assuming no saturation see later), whereas
  optimal SOA for main effect (AB) is 16-20s
• Inclusion of null events improves efficiency for
  main effect at short SOAs (at cost of efficiency
  for differential effects)
• If order constrained, intermediate SOAs (5-20s)
  can be optimal
• If SOA constrained, pseudorandomised designs can
  be optimal (but may introduce context-sensitivity)

47
3. How about separating responses?

• What if interested in both contrasts 1 0 and 0
  1?
• For example
• 1) Mixed designs (item-state effects)
• 2) Working Memory trials (stimulus-response)
• In the efficiency of a contrast (see earlier)
• e(c,X) cT (XTX)-1 c -1
• XTX represents covariance of regressors in
  design matrix
• High covariance increases elements of (XTX)-1
• So, when correlation between regressors,
  efficiency to detect effect of each one
  separately is reduced

48
Correlations between Regressors
1 1
1 -1
Negative correlation between two regressors means
separate (orthogonal) effect of each is estimated
poorly, though difference between regressors
estimated well
49
Eg 1 Item and State effects (see earlier)
Blocks 40s, Fixed SOA 4s
Efficiency 16 1 0 (Item Effect)
Correlation .97
Not good
50
Eg 1 Item and State effects (see earlier)
Blocks 40s, Randomised SOAmin 2s
Efficiency 54 1 0 (Item Effect)
Correlation .78
Better
51
Eg 2 Stimulus-Response Paradigms
Each trial consists of 2 successive events e.g,
Stimulus - Response Each event every
4s (Stimulus every 8s)
Efficiency 29 1 0 (Stimulus)
Correlation -.65
52
Eg 2 Stimulus-Response Paradigms
Each trial consists of 2 successive events e.g,
Stimulus - Response Solution 1 Time between
Stim- Resp events jittered from 0-8 seconds...
Efficiency 40 1 0 (Stimulus)
Correlation .33
53
Eg 2 Stimulus-Response Paradigms
Each trial consists of 2 successive events e.g,
Stimulus - Response Solution 2 Stim event every
8s, but Resp event only occurs on 50 trials...
Efficiency 47 1 0 (Stimulus)
Correlation -.24
54
Nonlinear Effects

55
The End
This talk appears as Chapter 15 in the SPM
bookhttp//www.mrc-cbu.cam.ac.uk/rh01/Henson_De
sign_SPMBook_2006_preprint.pdf
For further info on how to design an efficient
fMRI experiment, seehttp//www.mrc-cbu.cam.ac.uk
/Imaging/Common/fMRI-efficiency.shtml
56
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

57
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)

58
Note on Epoch Durations

• As duration of epochs increases from 0 to 2s,
  shape of convolved response changes little
  (mainly amplitude of response changes)
• Since it is the amplitude that is effectively
  estimated by the GLM, the results for epochs of
  constant duration lt2s will be very similar to
  those for events (at typical SNRs)
• If however the epochs vary in duration from
  trial-to-trial (e.g, to match RT), then epoch and
  event models will give different results
• However, while RT-related duration may be
  appropriate for motor regions, it may not be
  appropriate for all regions (e.g, visual)
• Thus a parametric modulation of events by RT
  may be a better model in such situations

59
Epoch vs Events
Rate 1/4s
Rate 1/2s

• Though blocks of trials can be modelled as either
  epochs (boxcars) or runs of events
• interpretation of parameters differs
• Consider an experiment presenting words at
  different rates in different blocks
• An epoch model will estimate parameter that
  increases with rate, because the parameter
  reflects response per block
• An event model may estimate parameter that
  decreases with rate, because the parameter
  reflects response per word

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Blocked
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Time
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Blocked Design
Data
Model
Epoch model
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BOLD Response Latency (Iterative)

• Four-parameter HRF, nonparametric Random Effects
  (SNPM99)
• Advantages of iterative vs linear
• Height independent of shape Canonical height
  confounded by latency (e.g, different shapes
  across subjects) no slice-timing error
• 2. Distinction of onset/peak latency
  Allowing better neural inferences?
• Disadvantages of iterative
• 1. Unreasonable fits (onset/peak tension)
  Priors on parameter distributions?
  (Bayesian estimation)
• 2. Local minima, failure of convergence?
• 3. CPU time (3 days for above)

FIR used to deconvolve data, before nonlinear
fitting over PST
66
Efficiency Detection vs Estimation

• Detection power vs Estimation efficiency
  (Liu et al, 2001)
• Detect response, or characterise shape of
  response?
• Maximal detection power in blocked designs
• Maximal estimation efficiency in randomised
  designs
• gt simply corresponds to choice of basis
  functions
• detection canonical HRF
• estimation FIR

67
Efficiency - Single Event-type

• Design parametrised by
• SOAmin Minimum SOA
• p(t) Probability of event at
  each SOAmin
• Deterministic p(t)1 iff tnT
• Stationary stochastic p(t)constant
• Dynamic stochastic
• p(t) varies (eg blocked)

Blocked designs most efficient! (with small
SOAmin)
68
1. Basic Response vs Baseline

• To detect a basic event-related response versus
  baseline
• Do not present stimuli at a fixed rate
• Varying the SOA (eg via null events), with a
  minimal shortest SOA, is more efficient
• Presenting stimuli rapidly within on/off blocks
  of 20s is even more efficient (though
  psychological downsides, eg predictability?)
• Longer blocks (gt60 seconds) can be confounded by
  low-frequency noise