The General Linear Model and Statistical Parametric Mapping

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Experimental Design
Course Methods and models for fMRI data
analysis in neuroeconomics Christian
Ruff Laboratory for Social and Neural Systems
Research IEW, University of Zurich ICN FIL,
University College London With thanks to Rik
Henson Daniel Glaser
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Statistical parametric map (SPM)
Design matrix
Image time-series
Kernel
Realignment
Smoothing
General linear model
Gaussian field theory
Statistical inference
Normalisation
p lt0.05
Template
Parameter estimates
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Overview

• Categorical designs
• Subtraction - Pure insertion, evoked /
  differential responses
• Conjunction - Testing multiple hypotheses
• Parametric designs
• Linear - Adaptation, cognitive dimensions
• Nonlinear - Polynomial expansions, neurometric
  functions
• Factorial designs
• Categorical - Interactions and pure insertion
• Parametric - Linear and nonlinear interactions
• - Psychophysiological Interactions

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Cognitive Subtraction

• Aim
• Neural structures computing process P?
• Procedure
• Contrast Task with P control task without
  P P
• the critical assumption of pure insertion

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Cognitive Subtraction Baseline-problems

• Distant stimuli

- ? Several components differ !

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Differential event-related fMRI
Evoked responses
SPMF testing for evoked responses
Parahippocampal responses to words

• Baseline here corresponds to session mean
  (and thus processing during rest)
• Null events or long SOAs essential for
  estimation
• Cognitive interpretation hardly possible,
  but useful to define regions generally involved
  in task

BOLD EPI fMRI at 2T, TR 3.2sec. Words presented
every 16 secs (i) studied words or (ii) new words
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Differential event-related fMRI
Differential responses
SPMF testing for evoked responses
Parahippocampal responses to words
SPMF testing for differences
studied words
new words
BOLD EPI fMRI at 2T, TR 3.2sec. Words presented
every 16 secs (i) studied words or (ii) new words
Peri-stimulus time secs
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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
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Overview

• Categorical designs
• Subtraction - Pure insertion, evoked /
  differential responses
• Conjunction - Testing multiple hypotheses
• Parametric designs
• Linear - Adaptation, cognitive dimensions
• Nonlinear - Polynomial expansions, neurometric
  functions
• Factorial designs
• Categorical - Interactions and pure insertion
• Parametric - Linear and nonlinear interactions
• - Psychophysiological Interactions

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Conjunctions

• One way to minimise the baseline/pure insertion
  problem is to isolate the same process by two or
  more separate comparisons, and inspect the
  resulting simple effects for commonalities
• A test for such activation common to several
  independent contrasts is called Conjunction
• Conjunctions can be conducted across a whole
  variety of different contexts
• tasks
• stimuli
• senses (vision, audition)
• etc.
• But the contrasts entering a conjunction have to
  be truly independent!

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Conjunctions
Example Which neural structures support object
recognition, independent of task (naming vs
viewing)?
Visual Processing V Object Recognition
R Phonological Retrieval P (Object - Colour
viewing) (Object - Colour naming) ? 1 -1 0
0 0 0 1 -1 ? R,V - V P,R,V -
P,V R R R (assuming no
interaction RxP see later)
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Conjunctions
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Two flavours of inference about conjunctions

• SPM8 offers two general ways to test the
  significance of conjunctions
• Test of global null hypothesis Significant set
  of consistent effects
• ? which voxels show effects of similar
  direction (but not necessarily individual
  significance) across contrasts?
• Test of conjunction null hypothesis Set of
  consistently significant effects
• ? which voxels show, for each specified
  contrast, effects gt threshold?
• Choice of test depends on hypothesis and
  congruence of contrasts the global null test is
  more sensitive (i.e., when direction of effects
  hypothesised)

Friston et al. (2005). Neuroimage,
25661-7. Nichols et al. (2005). Neuroimage,
25653-60.
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Overview

• Categorical designs
• Subtraction - Pure insertion, evoked /
  differential responses
• Conjunction - Testing multiple hypotheses
• Parametric designs
• Linear - Adaptation, cognitive dimensions
• Nonlinear - Polynomial expansions, neurometric
  functions
• Factorial designs
• Categorical - Interactions and pure insertion
• Parametric - Linear and nonlinear interactions
• - Psychophysiological Interactions

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Parametric Designs General Approach

• Parametric designs approach the baseline problem
  by
• Varying a stimulus-parameter of interest on a
  continuum, in multiple (ngt2) steps...
• ... and relating blood-flow to this parameter
• Flexible choice of tests for such relations
• Linear
• Nonlinear Quadratic/cubic/etc.
• Data-driven (e.g., neurometric functions)
• Model-based

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A linear parametric contrast
Linear effect of time
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A nonlinear parametric contrast
The nonlinear effect of time assessed with the
SPMT
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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
... up to (N-1)th order for N levels
(SPM8 GUI offers polynomial expansion as option
during creation of parametric modulation
regressors)
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Parametric Designs Neurometric functions
versus
Rees, G., et al. (1997). Neuroimage, 6 27-78
Inverted U response to increasing word
presentation rate in the DLPFC
Rees, G., et al. (1997). Neuroimage, 6 27-78
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Parametric Designs Neurometric functions

• Coding of tactile stimuli in Anterior Cingulate
  Cortex
• Stimulus (a) presence, (b) intensity, and (c)
  pain intensity
• Variation of intensity of a heat stimulus applied
  to the right hand
• (300, 400, 500, and 600 mJ)

Büchel et al. (2002). The Journal of
Neuroscience, 22 970-6
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Parametric Designs Neurometric functions
Büchel et al. (2002). The Journal of
Neuroscience, 22 970-6
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Parametric Designs Model-based regressors
Seymour, ODoherty, et al. (2004). Nature.
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Overview

• Categorical designs
• Subtraction - Pure insertion, evoked /
  differential responses
• Conjunction - Testing multiple hypotheses
• Parametric designs
• Linear - Adaptation, cognitive dimensions
• Nonlinear - Polynomial expansions, neurometric
  functions
• Factorial designs
• Categorical - Interactions and pure insertion
• Parametric - Linear and nonlinear interactions
• - Psychophysiological Interactions

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Factorial designs Main effects and Interactions

• Main effect of task (A1 B1) (A2 B2)
• Main effect of stimuli (A1 A2) (B1 B2)
• Interaction of task and stimuli Can show a
  failure of pure insertion
• (A1 B1) (A2 B2)

interaction effect (Task x Stimuli)
Colours Objects
Colours Objects
Viewing
Naming
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Interactions and pure insertion
Components Visual processing V Object
recognition R Phonological retrieval P Interactio
n RxP Interaction (name object - colour) -
(view object - colour) ? 1 -1 0 0 - 0 0 1
-1 P,R,V RxP - P,V - R,V - V
RxP

Object-naming-specific activations
adjusted rCBF
Context no naming naming
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Interactions and pure insertion
Interactions cross-over and simple We
can selectively inspect our data for one or the
other by masking during inference
A1 A2 B1 B2
A1 A2 B1 B2
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Overview

• Categorical designs
• Subtraction - Pure insertion, evoked /
  differential responses
• Conjunction - Testing multiple hypotheses
• Parametric designs
• Linear - Adaptation, cognitive dimensions
• Nonlinear - Polynomial expansions, neurometric
  functions
• Factorial designs
• Categorical - Interactions and pure insertion
• Parametric - Linear and nonlinear interactions
• - Psychophysiological Interactions

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

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Overview

• Categorical designs
• Subtraction - Pure insertion, evoked /
  differential responses
• Conjunction - Testing multiple hypotheses
• Parametric designs
• Linear - Adaptation, cognitive dimensions
• Nonlinear - Polynomial expansions, neurometric
  functions
• Factorial designs
• Categorical - Interactions and pure insertion
• Parametric - Linear and nonlinear interactions
• - Psychophysiological Interactions

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Psycho-physiological Interaction (PPI)
Parametric, factorial design, in which one factor
is a psychological context ...and the other is
a physiological source (activity extracted from
a brain region of interest)
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Psycho-physiological Interaction (PPI)
Parametric, factorial design, in which one factor
is a psychological context ...and the other is
a physiological source (activity extracted from
a brain region of interest)
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Psycho-physiological Interaction (PPI)
V1 activity
time
attention
V5 activity
no attention
Attentional modulation of V1 - V5 contribution
V1 activity
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Psycho-physiological Interaction (PPI)
0 0 1
V1 activity
time
attention
V5 activity
no attention
V1 activity
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Psycho-physiological Interaction (PPI)
SPMZ
Stimuli Faces or objects
Faces
PPC
IT
adjusted rCBF
Objects
medial parietal activity
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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, e.g., blocked mainly
  problem for event-related PPIs)
• SPM5 can effect a deconvolution of physiological
  regressors (V1), before calculating interaction
  term and reconvolving with the HRF the PPI
  button

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Overview

• Categorical designs
• Subtraction - Pure insertion, evoked /
  differential responses
• Conjunction - Testing multiple hypotheses
• Parametric designs
• Linear - Adaptation, cognitive dimensions
• Nonlinear - Polynomial expansions, neurometric
  functions
• Factorial designs
• Categorical - Interactions and pure insertion
• Parametric - Linear and nonlinear interactions
• - Psychophysiological Interactions

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Mixed Designs

• Simultaneously measuring effects that are
• transient (item- or event-related)
• sustained (state- or epoch-related)
• What is the best design to estimate both?

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A bit more formallyEfficiency

• Sensitivity, or efficiency, e
• e(c,X) cT (XTX)-1 c -1
• XTX represents covariance of regressors in design
  matrix
• High covariance increases elements of (XTX)-1
• gt So, when correlation between regressors is
  high,
• sensitivity to each regressor alone is low

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Item effect only
Blocks 40s, Fixed SOA 4s
Efficiency 565 (Item Effect)
OK
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Item and state effects
Blocks 40s, Fixed SOA 4s
Efficiency 16 (Item Effect)
Correlation .97
Not good
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Item and state effects
Blocks 40s, Randomised SOAmin 2s
Efficiency 54 (Item Effect)
Correlation .78
Better!
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Mixed design 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
• Randomised, long SOAs between events (targets) to
  decorrelate epoch and event-related covariates
• Attention modulates BOTH
• 1) baseline activity (state-effect, additive)
• 2) evoked response (item-effect, multiplicative)

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Mixed design example Chawla et al. (1999)
Mixed Designs (Chawla et al 1999)