Brain Mapping Unit
The General Linear Model
A Basic Introduction
Roger Tait (rt337@cam.ac.uk)
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Overview
 What is imaging data
 How is data pre-processed
 Hypothesis testing
 GLM: simple linear regression
 Analysis software
 How to process results
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What is imaging data?
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Data
A stack of numbers
Structural
fMRI
Functional
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Multiple Data
subjID
voxel1
voxel2
voxel 3
voxel 4
…….. voxel n
1
1227.308541 1472.770249 1417.745632 1701.294758
1288.742729
2
1612.461523 1934.953827 1677.661927 2013.194312
1465.051592
3
1466.264739 1759.517687 1559.769586 1871.723503
1827.678127
4
1499.70072 1799.640864 1842.474418 2210.969302
1316.392368
5
1598.121692 1917.746031 1510.850757 1813.020909
1740.286976
6
1408.066243 1689.679492 1399.393815 1679.272578
1534.459154
7
1555.951487 1867.141784 1588.529211 1588.529211
1516.464089
8
1397.721831 1677.266197 1523.825912 1523.825912
1340.814881
9
1333.659118 1600.390941 1384.217926 1384.217926
1461.281399
10
1453.14966 1743.779592 1558.603977 1558.603977
1406.575083
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Reorientation
Native
Reoriented
MNI152
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Basic pre-processing (fmri)
omprage.nii
obrain.nii
omrest.nii
nomrest.nii
worest.nii
wnomrest.nii
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Basic pre-processing (structural)
omprage.nii
gmomprage.nii
wgmomprage.nii
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How does standard space data
help?
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Hypothesis testing
Statistical inference is commonly done with a test
statistic (t, F, c2…) which has a distribution under
H0 mathematically derived.
For example
^b1  ^b0
t=
t
^1  ^b0)
SE(b
5%
Parametric Null Distribution
NB: this assumes that the errors
are independent and normally
distributed.
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Introducing The GLM
Y = Xb + e
DATA = MODEL + ERROR
DATA = KNOWN * UNKNOWN + ERROR
 Encapsulates: t-test (paired, un-paired), F-test, ANOVA
(one-way, two-way, main effects, factorial) MANOVA,
ANCOVA, MANCOVA, simple regression, linear
regression, multiple regression, multivariate
regression……
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GLM definition
Y = Xb + e
 Where Y is a matrix with a series of observed
measurements
 Where X is a matrix that might be a design matrix
 Where b is a matrix containing parameters to be
estimated
 And e is a matrix containing error or noise
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GLM: Simple Linear Regression
Y = b0 + X1b1 +
e
b0: is the Y axis intercept
Y
b1: is the gradient of slope
Y: the black circles
e: diff between
X
predicted Y and
observed Y
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GLM: Simple Linear Regression
Y = b0 + X1b1 + e
^
^
 This is done by choosing b0 and b1 so that the sum of
the squares of the estimated errors S ei2 is as small
as possible.
 This is called the Method of Least Squares.
 S ei2 is called the Residual Sum of Squares (RSS)
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GLM example
DATA = KNOWN * UNKNOWN + ERROR
= mean reaction time + GENDER + AGE
Y = b0 + X1b1 + X2b2+ X3b3+ X4b4+ e
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Dummy Variables
 Continuous variables
 measurements on a continuous scale (age,
mRT)
(-4.01, -0.47, 6.35, -7.06, -7.69, -14.24)
 Dummy Variables
 Code for group membership (disease, gender)
controls = 0, patients = 1
females = 1, males = -1
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Usage
 Hypothesis tests with GLM can be multivariate or
several independent univariate tests
 In multivariate tests the columns of Y are tested
together
 In univariate tests the columns of Y are tested
independently (multiple univariate tests with the
same design matrix)
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fMRI model specification
silent naming task
The model
BOLD signal
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Actual retrieved data
30
20
10
Model
0
+ve activation
0
-10
-20
-30
50
100
150
200
250
300
350
400
450
500
-ve activation
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fmri analysis with FSL
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Structural analysis with CamBA
sex
group
weight
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Structural analysis output
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Where are my clusters?
here is a big
cluster
here is a big
cluster
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Where is the cluster I am
interested in?
position mouse
cursor here
cluster
location
information
shown here
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How do my clusters help me?
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Statistical Testing
 Convert cluster into a binary mask
 Overlay mask on subject data
 Extract voxel intensities
 Do some statistical analysis to get more information
from your data
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Correlation with behaviour
for cluster Pos_002
p>0.05
close but cluster
Pos_001 does not
significantly
correlate with
behaviour HIT1
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Other Analyses
different from 0
one-sample t-test
Difference between means
two-sample t-test
Linear relationship between 2 variables
simple regression
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What else can I do to find out
more about my data?
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Other types of analyses
 Factorial designs
 Permits analysis of multiple time data
 Shows
 Main effects of Factor 1 (time)
 Main effects of Factor 2 (group)
 Interaction between Factor 1 and Factor 2
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Useful software package
 CamBA – Cambridge
 http://www-bmu.psychiatry.cam.ac.uk/software/
 FSL Randomise – Oxford
 http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/Randomise
 SPM8 – UCL
 http://www.fil.ion.ucl.ac.uk/spm/software/spm8/
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In summary
 The GLM allows us to summarize a wide variety of
research outcomes by specifying the exact equation
that best summarizes the data for a study. If the
model is wrongly specified, the estimates of the
coefficients (the beta values) are likely to be biased
(i.e. wrong) and the resulting equation will not
describe the data accurately.
 In complex situations (e.g. cognitive fMRI paradigms),
this model specification problem can be a serious and
difficult one
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Any questions?