1. Karl Friston, Oliver Josephs
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 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

4. 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

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

6. 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

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)
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

8. Cognitive Conjunctions

9. 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

10. A (linear) parametric contrast
Linear effect
of time

11. 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

12. Nonlinear parametric design matrix
Quadratic (2nd)
(Constant) (0th)
SPM{F}
E.g, F-contrast [0 1 0] on
Quadratic Parameter =>
Linear (1st)
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

13. 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

14. 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

15. Interactions and pure insertion

16. 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

17. (Linear) Parametric Interaction
A (Linear)
Time-by-Condition
Interaction
(“Generation strategy”?)
Contrast: [ ]  [-1 1]

18. Nonlinear Parametric Interaction
F-contrast tests for nonlinear
Generation-by-Time interaction
(including both linear and
Quadratic components)
Factorial Design with 2 factors:
Gen/Rep (Categorical, 2 levels)
Time (Parametric, 6 levels)
Time effects modelled with both linear and quadratic components…
G-R
Time
Lin
Time
Quad
G x T
Lin
G x T
Quad

19. 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

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)
SPM{Z}
V1 activity
Attention
time V1
attention V5
V5 activity
no attention
Attentional modulation of
V1 - V5 contribution
V1 activity

21. Psycho-physiological Interaction (PPI)
SPM{Z}
V1 activity
time
attention
V5 activity
no attention V1
Att
V1 x Att
V1 activity
V1xAtt

22. Overview 1. A Taxonomy of Designs 2. Blocked vs Randomised Designs
3. Efficient Designs

23. Epoch vs Events Sustained epoch Blocks of 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 (SOAs<~3s)
i.e, designs can be blocked or randomised … models can be epoch or event-related
Boxcar
function
Sustained epoch
Blocks of events
Delta
functions
Convolved
with HRF
=>

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) V5 Motion change under attention to
motion (red) or color (blue)
Item
Effect
(Evoked)
State
Effect
(Baseline)
V Color change under attention to
motion (red) or color (blue)

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

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  = Not particularly efficient… Stimulus (“Neural”)
HRF
Predicted Data  =
Not particularly efficient…

33. Fixed SOA = 4s  = Very Inefficient… Stimulus (“Neural”) HRF
Predicted Data  =
Very Inefficient…

34. Randomised, SOAmin= 4s  = More Efficient… Stimulus (“Neural”) HRF
Predicted Data  =
More Efficient…

35. Blocked, SOAmin= 4s  = Even more Efficient… 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 power spectrum highpass filter power spectrum
aliasing
fMRI contains 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)

39.  =  = Blocked (80s), SOAmin=4s, highpass filter = 1/120s
Stimulus (“Neural”)
HRF
Predicted Data 
“Effective HRF” (after highpass filtering)
(Josephs & Henson, 1999) =  =
Don’t have long (>60s) blocks!

40. Randomised, SOAmin=4s, 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 B
=> ABBBABAABABAAA...
4s smoothing; 1/60s highpass filtering
Josephs & Henson (1999)
Differential Effect (A-B)
Common Effect (A+B)

44. Efficiency - Multiple Event-types
Example: Alternating AB
A B A 0 1
B 1 0
=> ABABABABABAB...
4s smoothing; 1/60s highpass filtering
Josephs & Henson (1999)
Permuted (A-B)
Alternating (A-B)
Example: Permuted AB
A B AA AB BA BB
=> ABBAABABABBA...

45. Efficiency - Multiple Event-types
Example: Null events
A B A B
=> 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)
4s smoothing; 1/60s highpass filtering
Josephs & Henson (1999)
Null Events (A-B)
Null Events (A+B)

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 (A+B) 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
Design Matrix (X)
Efficiency = 16
[1 0] (Item Effect)
Correlation = .97
Not good…

50. Eg 1: Item and State effects (see earlier)
Blocks = 40s, Randomised SOAmin= 2s
Design Matrix (X)
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)
Stim Resp
Design Matrix (X)
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...
Stim Resp
Design Matrix (X)
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...
Stim Resp
Design Matrix (X)
Efficiency = 47
[1 0] (Stimulus)
Correlation = -.24

54. Underadditivity at short SOAs
Nonlinear Effects
Underadditivity at short SOAs
Linear
Prediction
Volterra
Implications
for Efficiency

55. The End
This talk appears as Chapter 15 in the SPM book:
For further info on how to design an efficient fMRI experiment, see:

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
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

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 <2s 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
b=3
b=5
b=11
b=9

60. Blocked Randomised Data Model O = Old Words N = New Words O1 O2 O3 N1

61. Event-Related ~4s R F R = Words Later Remembered
F = Words Later Forgotten
Event-Related
~4s
Data
Model

63. “Oddball” …
Time

64. Blocked Design “Epoch” model “Event” model
Data
“Epoch” model
Model O1 O2 O3 N1 N2 N3 N1 N2 N3
“Event” model O1 O2 O3

65. BOLD Response Latency (Iterative)
Height
Peak Delay
Onset Delay
Dispersion
Different fits
across subjects
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
Height
p<.05 (cor)
1-2
SNPM

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
=> 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 t=nT
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 (>60 seconds) can be confounded by low-frequency noise