Understanding fMRI 1st Level Analysis: Basis Functions and GLM Assumptions

 
fMRI 1st Level Analysis:
Basis functions, parametric
modulation and correlated
regression
 
 
MfD 04/12/18
Alice Accorroni – Elena Amoruso
Overview
Normalisation
Normalisation
Statistical Parametric Map
Statistical Parametric Map
Parameter estimates
Parameter estimates
General Linear Model
General Linear Model
Realignment
Realignment
Smoothing
Smoothing
Design matrix
Anatomical
Anatomical
reference
reference
Spatial filter
Spatial filter
Statistical
Statistical
Inference
Inference
RFT
RFT
p <0.05
p <0.05
Preprocessing
Data Analysis
 
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The GLM and its assumptions
 
Neural activity
function is correct
 
HRF
is correct
 
Linear time-invariant
system
 
The GLM and its assumptions
 
HRF
is correct
The GLM and its assumptions
 
Gamma functions
 
2 Gamma functions added
 
The GLM and its assumptions
 
HRF
is correct
The GLM and its assumptions
Neural activity
function is correct
HRF
is correct
Linear time-invariant
system
Aguirre, Zarahn and D’Esposito, 1998; Buckner, 2003; Wager, Hernandez, Jonides and Lindquist, 2007a;
 
Brain region differences in BOLD
 
+
 
Brain region differences in BOLD
 
+
 
Aversive Stimulus
 
Brain region differences in BOLD
 
Sommerfield et al 2006
 
Temporal basis functions
 
Temporal basis functions
 
Temporal basis functions: FIR
 
Temporal basis functions
 
Temporal basis functions
 
Temporal basis functions
 
Canonical
HRF
 
HRF +
derivatives
 
Finite
Impulse
Response
 
Temporal basis functions
 
Canonical
HRF
 
HRF +
derivatives
 
Finite
Impulse
Response
 
Temporal basis functions
 
What is the best basis set?
What is the best basis set?
 
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if the model fits the data well:
-
R
2
  is high (reflects the proportion of variance in Y explained by the
regressor X)
-
the corresponding 
p
 value 
will be low
 
Regression analysis
 examines the
relation of a dependent variable Y to a
specified independent variables X:
 
 
Y = a + bX
 
M
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R
e
g
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s
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n
 
Multiple regression characterises the relationship between several
independent variables (or 
regressors
), X1, X2, X3 etc, and a single
dependent variable
, Y:
 
Y = 
β
1
X
1 
+ 
β
2
X
2
 +…..+ 
β
L
X
L
 + 
ε
 
The X variables are combined linearly and each has its own regression
coefficient β (
weight
)
βs reflect the independent contribution of each regressor, X, to the
value of the dependent variable, Y
i.e. the proportion of the variance in Y accounted for by each regressor
after all other regressors are accounted for
 
Multiple regression results are sometimes difficult to interpret:
 the overall 
p
 value of a fitted model is very 
low
i.e. the model fits the data well
but individual 
p
 values for the regressors are 
high
i.e. none of the X variables has a significant impact on predicting
Y.
How is this possible?
Caused when two (or more)  regressors are highly correlated: problem
known as
 
multicollinearity
 
M
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a
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y
 
Are correlated regressors a problem?
No
when you want to predict Y from X1 and X2
Because 
R
2
 and 
p
 will be correct
Yes
when you want assess impact of individual regressors
Because 
individual 
p
 values can be misleading: a p value can
be 
high
, even though the variable is important
 
 
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-
 
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a
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p
l
e
 
Question: how can the 
perceived clarity 
of a auditory stimulus be
predicted from the 
loudness and frequency 
of that stimulus?
Perception experiment in which subjects had to judge the clarity of an
auditory stimulus.
Model to be fit:
  
Y = β1X1 + β2X2 + ε
  
Y = judged clarity of stimulus
  
X1 = loudness
  
X2 = frequency
 
Regression analysis: multicollinearity example
 
What happens when X1 (frequency) and X2 (loudness) are
collinear, i.e., strongly correlated?
 
 
Correlation loudness
& frequency : 
0.945
(p<0.000)
 
High loudness values
correspond to high
frequency values
FREQUENCY
 
Regression analysis: multicollinearity example
 
Contribution of individual predictors (
simple regression
):
 
 
X2 (frequency) entered as sole predictor:
Y = 0.824X
1 
+ 26.94
R
2
 = 0.68 (68% explained variance in Y)
p < 0.000
 
X1 (loudness) is entered as sole predictor:
Y = 0.859X
1 
+ 24.41
R
2
 = 0.74 (74% explained variance in Y)
p < 0.000
 
Collinear regressors X1 and X2 entered together (
multiple
regression
):
Resulting model:
Y = 0.756X1 + 0.551X2 + 26.94
R2 = 0.74 (74% explained variance in Y)
p < 0.000
 
Individual regressors:
X1 (loudness): 
 
R2 = 0.850  , p < 0.000
X2 (frequency):
 
R2 = 0.555, 
p < 0.594
 
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R
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The 
General Linear Model 
can be seen as an extension of multiple
regression (or multiple regression is just a simple form of the General
Linear Model):
Multiple Regression only looks at one Y variable
GLM allows you to analyse 
several Y variables 
in a linear combination
(time series in voxel)
ANOVA, t-test, F-test, etc. are also forms of the GLM
 
General Linear Model and fMRI
 
     
     
Y
            
=           X
         
.        
β
        
+          
ε
 
Observed data
Y is the BOLD signal
at various time
points at a single
voxel
 
Design matrix
Several components
which explain the
observed data Y:
-
Different stimuli
-
Movement regressors
 
Parameters
Define the
contribution of
each component
of the design
matrix to the value
of Y
 
Error/residual
Difference between
the observed data, Y,
and that predicted
by the model, X
β
.
 
Experiment
:
Which areas of the brain are active in visual movement processing?
Subjects press a button when a shape on the screen suddenly moves
Model
 to be fit:
  
Y = 
β
1
X
1 
+ 
β
2
X
2
 + 
ε
   
Y = BOLD response
   
X1 = visual component
   
X2 = motor response
 
 
 
 
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M
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C
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a
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t
y
 
 If the regressors are linearly dependent the results of the 
GLM
 are not easy
to interpret
 
H
o
w
 
d
o
 
I
 
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e
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w
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i
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?
 
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a
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x
1
 
x
2
 
x
2
*
 
y
 
y = 
1
X
1 
+
 
2
*X
2 
*
 
1
 = 1.5
2 
*
 = 1
 
Carefully
 
design
 
your experiment
!
When sequential scheme of
predictors (stimulus – response) is
inevitable:
 inject jittered delay (see B)
use a probabilistic R
1
-R
2
sequence (see C))
 
Orthogonalizing might lead to self-
fulfilling prophecies
 
(MRC CBU Cambridge,
http://imaging.mrc-cbu.cam.ac.uk/imaging/DesignEfficiency)
 
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D
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g
n
 
Parametric Modulations
 
Types of experimental design
 
1.
Categorical
 - comparing the activity between stimulus
types
2.
Factorial
 - combining two or more factors within a
task and looking at the effect of one factor on the
response to other factor
3.
Parametric
 
- exploring systematic changes in BOLD
signal according to some performance attributes of
the task (
difficulty levels, increasing sensory input,
drug doses, etc)
 
 
 
Complex stimuli with a number of stimulus dimensions can be
modelled by a set of 
parametric modulators 
tied to the
presentation of each stimulus.
 
This means that:
Can look at the contribution of each 
stimulus dimension
independently
Can test predictions about the 
direction and scaling of BOLD
responses due to these different dimensions (e.g., linear or
non linear activation).
 
Parametric Design
Parametric Modulation
Example
: 
 
Very simple motor task  - Subject presses a button then rests.
  
Repeats this four times, with an increasing level of force.
Hypothesis
: 
 
We will see a linear increase in activation in motor cortex as the
  
force increases
Model: 
  
Parametric
 
Linear effect of force
 
C
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:
 
 
 
 
 
 
 
 
 
 
0
 
 
 
 
 
 
 
 
 
 
 
1
 
 
 
 
 
 
 
 
 
 
 
 
0
 
C
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s
t
:
 
 
 
 
 
 
 
 
 
 
0
 
 
 
 
 
 
 
 
0
 
 
 
 
 
 
 
 
1
 
 
 
 
 
 
 
0
 
Quadratic effect of force
 
Time (scans)
 
Example
: 
 
Very simple motor task  - Subject presses a button then rests.
  
Repeats this four times, with an increasing level of force.
 
Hypothesis
: 
 
We will see a linear increase in activation in motor cortex as the
  
force increases
 
Model: 
  
Parametric
 
 
 
Parametric Modulation
 
Thanks to…
 
Rik Henson’s slides:
www.mrc-cbu.cam.ac.uk/Imaging/Common/rikSPM-GLM.ppt
 
Previous years’ presenters’ slides
 
Guillaume Flandin
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Explore the exciting world of fMRI 1st level analysis focusing on basis functions, parametric modulation, correlated regression, GLM assumptions, group analysis, and more. Dive into brain region differences in BOLD signals with various stimuli and learn about temporal basis functions in neuroimaging research. Visualize the complexities of statistical parametric mapping, estimation, and group-level analyses in functional MRI studies.


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  1. fMRI 1st Level Analysis: Basis functions, parametric modulation and correlated regression MfD 04/12/18 Alice Accorroni Elena Amoruso

  2. Overview Preprocessing Data Analysis Statistical Parametric Map Design matrix Spatial filter Realignment General Linear Model Smoothing Statistical Inference RFT Normalisation p <0.05 Anatomical reference Parameter estimates

  3. Estimation Estimation (1 (1st level) ) Group Analysis Group Analysis (2 (2nd level) ) st level nd level

  4. The GLM and its assumptions Neural activity function is correct HRF is correct Linear time-invariant system

  5. The GLM and its assumptions HRF is correct

  6. The GLM and its assumptions Gamma functions 2 Gamma functions added

  7. The GLM and its assumptions HRF is correct

  8. The GLM and its assumptions Neural activity function is correct HRF is correct Linear time-invariant system Aguirre, Zarahnand D Esposito, 1998; Buckner, 2003; Wager, Hernandez, Jonides and Lindquist, 2007a;

  9. Brain region differences in BOLD +

  10. Brain region differences in BOLD + Aversive Stimulus

  11. Brain region differences in BOLD Sommerfield et al 2006

  12. Temporal basis functions

  13. Temporal basis functions

  14. Temporal basis functions: FIR

  15. Temporal basis functions

  16. Temporal basis functions

  17. Temporal basis functions Canonical HRF HRF + derivatives Finite Impulse Response

  18. Temporal basis functions Canonical HRF HRF + derivatives Finite Impulse Response

  19. Temporal basis functions

  20. What is the best basis set?

  21. What is the best basis set?

  22. Correlated Regressors Correlated Regressors

  23. Regression analysis Regression analysis 56 Regression analysis examines the 55 relation of a dependent variable Y to a 54 specified independent variables X: 53 52 LOUDNESS Y = a + bX 51 50 110 120 130 140 150 160 170 PITCH - if the model fits the data well: R2 is high (reflects the proportion of variance in Y explained by the regressor X) the corresponding p value will be low -

  24. Multiple Regression Multiple Regression Multiple regression characterises the relationship between several independent variables (or regressors), X1, X2, X3 etc, and a single dependent variable, Y: Y = 1X1 + 2X2+ ..+ LXL + The X variables are combined linearly and each has its own regression coefficient (weight) s reflect the independent contribution of each regressor, X, to the value of the dependent variable, Y i.e. the proportion of the variance in Y accounted for by each regressor after all other regressors are accounted for

  25. Multicollinearity Multicollinearity Multiple regression results are sometimes difficult to interpret: the overall p value of a fitted model is very low i.e. the model fits the data well but individual p values for the regressors are high i.e. none of the X variables has a significant impact on predicting Y. How is this possible? Caused when two (or more) regressors are highly correlated: problem known asmulticollinearity

  26. Multicollinearity Multicollinearity Are correlated regressors a problem? No when you want to predict Y from X1 and X2 Because R2 and p will be correct Yes when you want assess impact of individual regressors Because individual p values can be misleading: a p value can be high, even though the variable is important

  27. Multicollinearity Multicollinearity - - Example Example Question: how can the perceived clarity of a auditory stimulus be predicted from the loudness and frequency of that stimulus? Perception experiment in which subjects had to judge the clarity of an auditory stimulus. Model to be fit: Y = 1X1 + 2X2 + Y = judged clarity of stimulus X1 = loudness X2 = frequency

  28. Regression analysis: multicollinearity example What happens when X1 (frequency) and X2 (loudness) are collinear, i.e., strongly correlated? 56 55 Correlation loudness & frequency : 0.945 (p<0.000) 54 53 52 High loudness values correspond to high frequency values LOUDNESS 51 50 110 120 130 140 150 160 170 FREQUENCY PITCH

  29. Regression analysis: multicollinearity example Contribution of individual predictors (simple regression): X1 (loudness) is entered as sole predictor: Y = 0.859X1 + 24.41 R2 = 0.74 (74% explained variance in Y) p < 0.000 X2 (frequency) entered as sole predictor: Y = 0.824X1 + 26.94 R2 = 0.68 (68% explained variance in Y) p < 0.000

  30. Collinear regressors X1 and X2 entered together (multiple regression): Resulting model: Y = 0.756X1 + 0.551X2 + 26.94 R2 = 0.74 (74% explained variance in Y) p < 0.000 Individual regressors: X1 (loudness): X2 (frequency): R2 = 0.555, p < 0.594 R2 = 0.850 , p < 0.000

  31. GLM and Correlated Regressors GLM and Correlated Regressors The General Linear Model can be seen as an extension of multiple regression (or multiple regression is just a simple form of the General Linear Model): Multiple Regression only looks at one Y variable GLM allows you to analyse several Y variables in a linear combination (time series in voxel) ANOVA, t-test, F-test, etc. are also forms of the GLM

  32. General Linear Model and fMRI Y= X. + Observed data Y is the BOLD signal at various time points at a single voxel Design matrix Parameters Define the contribution of each component of the design matrix to the value of Y Error/residual Difference between the observed data, Y, and that predicted by the model, X . Several components which explain the observed data Y: -Different stimuli -Movement regressors

  33. fMRI Collinearity fMRI Collinearity If the regressors are linearly dependent the results of the GLM are not easy to interpret Experiment: Which areas of the brain are active in visual movement processing? Subjects press a button when a shape on the screen suddenly moves Model to be fit: Y = 1X1 + 2X2 + Y = BOLD response X1 = visual component X2 = motor response

  34. How do I deal with it? Ortogonalization Ortogonalization y = 1X1 + 2*X2 * y y 1 = 1.5 2 * = 1 x x2 2 x x2 2* * x x1 1

  35. How do I deal with it? Experimental Design Experimental Design Carefullydesignyour experiment! When sequential scheme of predictors (stimulus response) is inevitable: inject jittered delay (see B) use a probabilistic R1-R2 sequence (see C)) Orthogonalizing might lead to self- fulfilling prophecies (MRC CBU Cambridge, http://imaging.mrc-cbu.cam.ac.uk/imaging/DesignEfficiency)

  36. Parametric Modulations

  37. Types of experimental design 1. Categorical - comparing the activity between stimulus types 2. Factorial - combining two or more factors within a task and looking at the effect of one factor on the response to other factor 3. Parametric - exploring systematic changes in BOLD signal according to some performance attributes of the task (difficulty levels, increasing sensory input, drug doses, etc)

  38. Parametric Design Complex stimuli with a number of stimulus dimensions can be modelled by a set of parametric modulators tied to the presentation of each stimulus. This means that: Can look at the contribution of each stimulus dimension independently Can test predictions about the direction and scaling of BOLD responses due to these different dimensions (e.g., linear or non linear activation).

  39. Parametric Modulation Example: Very simple motor task - Subject presses a button then rests. Repeats this four times, with an increasing level of force. Hypothesis: We will see a linear increase in activation in motor cortex as the force increases Model: Parametric Contrast: 0 1 0 Linear effect of force Time (scans) 5 4 10 3 2 15 1 0 20 -1 -2 25 -3 -4 0 5 10 15 20 25 30 35 30 0.5 1 1.5 2 2.5 3 3.5 Regressors: press force mean

  40. Parametric Modulation Example: Very simple motor task - Subject presses a button then rests. Repeats this four times, with an increasing level of force. Hypothesis: We will see a linear increase in activation in motor cortex as the force increases Model: Parametric Quadratic effect of force Contrast: 0 0 1 0 Time (scans) 5 10 9 10 8 7 6 15 5 4 20 3 2 1 25 0 -1 0 5 10 15 20 25 30 35 30 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time (scans) Regressors: press force (force)2 mean

  41. Thanks to Rik Henson s slides: www.mrc-cbu.cam.ac.uk/Imaging/Common/rikSPM-GLM.ppt Previous years presenters slides Guillaume Flandin

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