The General Linear Model

1
The General Linear Model

• A Basic Introduction
• Roger Tait (rt337_at_cam.ac.uk)

2
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
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)
worest.nii
obrain.nii
omprage.nii
omrest.nii
wnomrest.nii
nomrest.nii
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Basic pre-processing (structural)
gmomprage.nii
wgmomprage.nii
omprage.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
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 predicted Y and observed Y
X
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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
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fmri analysis with FSL
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Structural analysis with CamBA
sex
weight
group
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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
pgt0.05 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?