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fMRI Analysis with emphasis on the general linear model

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Jody Culham Department of Psychology University of Western Ontario http://www.fmri4newbies.com/ fMRI Analysis with emphasis on the general linear model – PowerPoint PPT presentation

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Title: fMRI Analysis with emphasis on the general linear model


1
fMRI Analysiswith emphasis on the general linear
model
Jody Culham Department of Psychology University
of Western Ontario
http//www.fmri4newbies.com/
Last Update November 29, 2008 fMRI Course,
Louvain, Belgium
2
What data do we start with
  • 12 slices 64 voxels x 64 voxels 49,152 voxels
  • Each voxel has 136 time points
  • Therefore, for each run, we have 6.7 million data
    points
  • We often have several runs for each experiment

3
Why do we need stats?
  • We could, in principle, analyze data by voxel
    surfing move the cursor over different areas and
    see if any of the time courses look interesting

4
Why do we need stats?
  • Clearly voxel surfing isnt a viable option.
    Wed have to do it 49,152 times in this data set
    and it would require a lot of subjective
    decisions about whether activation was real
  • This is why we need statistics
  • Statistics
  • tell us where to look for activation that is
    related to our paradigm
  • help us decide how likely it is that activation
    is real

The lies and damned lies come in when you write
the manuscript
5
Predicted Responses
  • fMRI is based on the Blood Oxygenation Level
    Dependent (BOLD) response
  • It takes about 5 sec for the blood to catch up
    with the brain
  • We can model the predicted activation in one of
    two ways
  • shift the boxcar by approximately 5 seconds (2
    images x 2 seconds/image 4 sec, close enough)
  • convolve the boxcar with the hemodynamic response
    to model the shape of the true function as well
    as the delay

PREDICTED ACTIVATION IN OBJECT AREA
PREDICTED ACTIVATION IN VISUAL AREA
BOXCAR
6
Types of Errors
p value probability of a Type I error e.g., p
lt.05 There is less than a 5 probability that a
voxel our stats have declared as active is in
reality NOT active
Slide modified from Duke course
7
Statistical Approaches in a Nutshell
  • t-tests
  • compare activation levels between two conditions
  • use a time-shift to account for hemodynamic lag
  • correlations
  • model activation and see whether any areas show a
    similar pattern
  • Fourier analysis
  • Do a Fourier analysis to see if there is energy
    at your paradigm frequency

Fourier analysis images from Huettel, Song
McCarthy, 2004, Functional Magnetic Resonance
Imaging
8
Effect of Thresholds
r 0 0 of variance p lt 1
9
Complications
  • Not only is it hard to determine whats real, but
    there are all sorts of statistical problems
  • Potential problems
  • data may be contaminated by artifacts (e.g., head
    motion, breathing artifacts)
  • .05 49,152 2457 significant voxels by
    chance alone
  • many assumptions of statistics (adjacent voxels
    uncorrelated with each other adjacent time
    points uncorrelated with one another) are false

Whats wrong with these data?
r .24 6 of variance p lt .05
10
The General Linear Model
  • T-tests, correlations and Fourier analysis work
    for simple designs and were common in the early
    days of imaging
  • The General Linear Model (GLM) is now available
    in many software packages and tends to be the
    analysis of choice
  • Why is the GLM so great?
  • the GLM is an overarching tool that can do
    anything that the simpler tests do
  • you can examine any combination of contrasts
    (e.g., intact - scrambled, scrambled - baseline)
    with one GLM rather than multiple correlations
  • the GLM allows much greater flexibility for
    combining data within subjects and between
    subjects
  • it also makes it much easier to counterbalance
    orders and discard bad sections of data
  • the GLM allows you to model things that may
    account for variability in the data even though
    they arent interesting in and of themselves
    (e.g., head motion)
  • as we will see later in the course, the GLM also
    allows you to use more complex designs (e.g.,
    factorial designs)

11
A Simple Experiment
  • Lateral Occipital Complex
  • responds when subject views objects

Blank Screen
Intact Objects
Scrambled Objects
TIME
One volume (12 slices) every 2 seconds for 272
seconds (4 minutes, 32 seconds) Condition
changes every 16 seconds (8 volumes)
12
Whats real?
A.
C.
B.
D.
13
Whats real?
  • I created each of those time courses based by
    taking the predictor function and adding a
    variable amount of random noise

signal


noise
14
Whats real?
Which of the data sets below is more convincing?
15
Formal Statistics
  • Formal statistics are just doing what your
    eyeball test of significance did
  • Estimate how likely it is that the signal is real
    given how noisy the data is
  • confidence how likely is it that the results
    could occur purely due to chance?
  • p value probability value
  • If p .03, that means there is a .03/1 or 3
    chance that the results are bogus
  • By convention, if the probability that a result
    could be due to chance is less than 5 (p lt .05),
    we say that result is statistically significant
  • Significance depends on
  • signal (differences between conditions)
  • noise (other variability)
  • sample size (more time points are more
    convincing)

16
Lets create a time course for one LO voxel
17
Well begin with activation
Response to Intact Objects is 4X greater than
Scrambled Objects
18
Then well assume that our modelled activation is
off because a transient component
19
Our modelled activation could be off for other
reasons
  • All of the following could lead to inaccurate
    models
  • different shape of function
  • different width of function
  • different latency of function

20
Variability of HRF
  • Aguirre, Zarahn DEsposito, 1998
  • HRF shows considerable variability between
    subjects

different subjects
  • Within subjects, responses are more consistent,
    although there is still some variability between
    sessions

same subject, same session
same subject, different session
21
Now lets add some variability due to head motion
22
though really motion is more complex
  • Head motion can be quantified with 6 parameters
    given in any motion correction algorithm
  • x translation
  • y translation
  • z translation
  • xy rotation
  • xz rotation
  • yz rotation
  • For simplicity, Ive only included parameter one
    in our model
  • Head motion can lead to other problems not
    predictable by these parameters

23
Now lets throw in a pinch of linear drift
  • linear drift could arise from magnet noise (e.g.,
    parts warm up) or physiological noise (e.g.,
    subjects head sinks)

24
and then well add a dash of low frequency noise
  • low frequency noise can arise from magnet noise
    or physiological noise (e.g., subjects cycles of
    alertness/drowsiness)
  • low frequency noise would occur over a range of
    frequencies but for simplicity, Ive only
    included one frequency (1 cycle per run) here
  • Linear drift is really just very low frequency
    noise

25
and our last ingredient some high frequency noise
  • high frequency noise can arise from magnet noise
    or physiological noise (e.g., subjects breathing
    rate and heartrate)

26
When we add these all together, we get a
realistic time course
27
Now lets be the experimenter
  • First, we take our time course and normalize it
    using z scores
  • z (x - mean)/SD
  • normalization leads to data where
  • mean zero
  • SD 1

28
We create a GLM with 2 predictors
?1



?2

fMRI Signal
Residuals
Design Matrix

Betas
x
what we CAN explain
what we CANNOT explain
how much of it we CAN explain


x
our data
Statistical significance is basically a ratio of
explained to unexplained variance
29
Implementation of GLM in SPM
Many thanks to Øystein Bech Gadmar for creating
this figure in SPM
? Time
  • SPM represents time as going down
  • SPM represents predictors within the design
    matrix as grayscale plots (where black low,
    white high) over time
  • SPM includes a constant to take care of the
    average activation level throughout each run

30
Effect of Beta Weights
  • Adjustments to the beta weights have the effect
    of raising or lowering the height of the
    predictor while keeping the shape constant

31
The beta weight is not a correlation
  • correlations measure goodness of fit regardless
    of scale
  • beta weights are a measure of scale

32
We create a GLM with 2 predictors
when ?12


when ?20.5

Betas
x
Design Matrix
what we CAN explain
what we CANNOT explain
how much of it we CAN explain


x
our data
Statistical significance is basically a ratio of
explained to unexplained variance
33
Correlated Predictors
  • Where possible, avoid predictors that are highly
    correlated with one another
  • This is why we NEVER include a baseline predictor
  • baseline predictor is almost completely
    correlated with the sum of existing predictors


r -.53

r -.53
r -.95
Two stimulus predictors
Baseline predictor
34
Which model accounts for this data?
x ß 1
x ß 0


OR
x ß 1
x ß 0


x ß 0
x ß -1
  • Because the predictors are highly correlated, you
    cant tell which model is best

35
Contrasts in the GLM
  • We can examine whether a single predictor is
    significant (compared to the baseline)

36
A Real Voxel
  • Heres the time course from a voxel that was
    significant in the Intact -Scrambled comparison

37
Maximizing Your Power
signal


noise
  • As we saw earlier, the GLM is basically comparing
    the amount of signal to the amount of noise
  • How can we improve our stats?
  • increase signal
  • decrease noise
  • increase sample size (keep subject in longer)

38
How to Reduce Noise
  • If you cant get rid of an artifact, you can
    include it as a predictor of no interest to
    soak up variance

Example Some people include predictors from the
outcome of motion correction algorithms
Corollary Never leave out predictors for
conditions that will affect your data
39
Reducing Residuals
40
Whats this ing reviewer complaining about?!
  • Particularly if you do voxelwise stats, you have
    to be careful to follow the accepted standards of
    the field. In the past few years the following
    approaches have been recommended by the stats
    mavens
  • Correction for multiple comparisons
  • Random effects analyses
  • Correction for serial correlations

41
Correction for Multiple Comparisons
With conventional probability levels (e.g., p lt
.05) and a huge number of comparisons (e.g., 64 x
64 x 12 49,152), a lot of voxels will be
significant purely by chance e.g., .05 49,152
2458 voxels significant due to chance How can
we avoid this?
  • Bonferroni correction
  • divide desired p value by number of comparisons
  • Example
  • desired p value p lt .05
  • number of voxels 50,000
  • required p value p lt .05 / 50,000 ? p lt
    .000001
  • quite conservative
  • can use less stringent values
  • e.g., Brain Voyager can use the number of voxels
    in the cortical surface
  • small volume correction use more liberal
    thresholds in areas of the brain which you
    expected to be active

42
Correction for Multiple Comparisons
  • Gaussian field theory
  • Fundamental to SPM
  • If data are very smooth, then the chance of noise
    points passing threshold is reduced
  • Can correct for the number of resolvable
    elements (resels) rather than number of voxels

Slide modified from Duke course
43
  • 3) Cluster correction
  • falsely activated voxels should be randomly
    dispersed
  • set minimum cluster size to be large enough to
    make it unlikely that a cluster of that size
    would occur by chance
  • assumes that data from adjacent voxels are
    uncorrelated (not true)
  • Test-retest reliability
  • Perform statistical tests on each half of the
    data
  • The probability of a given voxel appearing in
    both purely by chance is the square of the p
    value used in each half
  • e.g., .001 x .001 .000001
  • Alternatively, use the first half to select an
    ROI and evaluate your hypothesis in the second
    half.

44
  • 5) Poor mans Bonferroni
  • Jack up the threshold till you get rid of the
    schmutz (especially in air, ventricles, white
    matter)
  • If you have a comparison where one condition is
    expected to produce much more activity than the
    other, turn on both tails of the comparison
  • Jodys rule of thumb If ya cant trust the
    negatives, can ya trust the positives?

Example MT localizer data Moving rings gt
stationary rings (orange) Stationary rings gt
moving rings (blue)
45
  • 6) False discovery rate
  • Genovese et al, 2002, NeuroImage
  • also controls the proportion of rejected
    hypotheses that are falsely rejected
  • standard p value (e.g., p lt .01) means that a
    certain proportion of all voxels will be
    significant by chance (1)
  • FDR uses q value (e.g., q lt .01), meaning that a
    certain proportion of the activated (colored)
    voxels will be significant by chance (1)
  • works in theory, though in practice, my lab
    hasnt been that satisfied

Is the region truly active?
Yes
No
Type I Error
HIT
Yes
Does our stat test indicate that the region is
active?
Type II Error
Correct Rejection
No
46
Correction for Temporal Correlations
Statistical methods assume that each of our time
points is independent. In the case of fMRI, this
assumption is false. Even in a screen saver
scan, activation in a voxel at one time is
correlated with its activation within 6
sec This fact can artificially inflate your
statistical significance.
47
Collapsed Fixed Effects Models
  • assume that the experimental manipulation has
    same effect in each subject
  • treats all data as one concatenated set with one
    beta per predictor (collapsed across all
    subjects)
  • e.g., Intact 2
  • Scrambled .5
  • strong effect in one subject can lead to
    significance even when others show weak or no
    effects
  • you can say that effect was significant in your
    group of subjects but cannot generalize to other
    subjects that you didnt test

48
Separate Subjects Models
  • one beta per predictor per subject
  • e.g., JC Intact 2.1
  • JC Scrambled 0.2
  • DQ Intact 1.5
  • DQ Scrambled 1.0
  • KV Intact 1.2
  • KV Scrambled 1.3
  • weights each subject equally
  • makes data less susceptible to effects of one
    rogue subject

49
Random Effects Analysis
  • Typical fMRI stats test whether the differences
    between conditions are significant in the sample
    of subjects we have tested
  • Often, we want to be able to generalize to the
    population as a whole including all potential
    subjects, not just the ones we tested
  • Random effects analyses allow you to generalize
    to the population you tested
  • Brain Voyager recommends you dont even toy with
    random effects unless youve got 10 or more
    subjects (and 50 is best)
  • Random effects analyses can really squash your
    data, especially if you dont have many subjects.
    Sometimes we refer to the random effects button
    as the make my activation go away button.
  • Reviewers are now requesting random effects
    analyses more frequently
  • You dont have to worry about it if youre using
    the ROI approach because (1) presumably the ROI
    has already been well-established across multiple
    labs and (2) posthoc analyses of results in an
    ROI approach allow you to generalize to the
    population (assuming you include individual
    variance)

underpaid graduate students in need of a few
bucks!
50
Fixed vs. Random Effects GLM
Sample Data 1
Sample Data 2
Subject Intact beta Scram beta Diff
1 4 3 1
2 2 3 -1
3 4 1 3
SUM 10 7 3
Subject Intact beta Scram beta Diff
1 4 3 1
2 2 1 1
3 4 3 1
SUM 10 7 3
  • Fixed Effects GLM cannot tell the difference
    between these data sets because (Intact sum -
    Scram sum) is the same in both cases
  • In Random Effects GLM, Data set 1 would be more
    likely to be significant because all 3 subjects
    show a trend in the same direction (intact gt
    scrambled), whereas in data set 2, only 2 of 3
    subjects show a difference in that direction

51
Autocorrelation function
To calculate the magnitude of the problem, we can
compute the autocorrelation function For a voxel
or ROI, correlate its time course with itself
shifted in time Plot these correlations by the
degree of shift
original
52
BV can correct for the autocorrelation to yield
revised (usually lower) p values
BEFORE
AFTER
53
BV Preprocessing Options
54
Temporal Smoothing of Data
  • We have the option in our software to temporally
    smooth our data (i.e., remove high temporal
    frequencies)
  • However, I recommended that you not use this
    option
  • Now do you understand why?

55
Clarification
  • correction for temporal correlations is NOT
    necessary with random effects analyses, only for
    fixed effects and individual subjects analysis

WARNING UNDER CONSTRUCTION need to clarify
explanation
56
Approach 1 Voxelwise Statistics
  1. You dont necessarily need a priori hypotheses
    (though sometimes you can use less conservative
    stats if you have them)
  2. Average all of your data together in Talairach
    space
  3. Compare two (or more) conditions using precise
    statistical procedures within every voxel of the
    brain. Any area that passes a carefully
    determined threshold is considered real.
  4. Make a list of these areas and publish it.

This is the tricky part!
57
Voxelwise Approach Example
  • Malach et al., 1995, PNAS
  • Question Are there areas of the human brain that
    are more responsive to objects than scrambled
    objects
  • You will recognize this as what we now call an LO
    localizer, but Malach was the first to identify LO

LO (red) responds more to objects, abstract
sculptures and faces than to textures, unlike
visual cortex (blue) which responds well to all
stimuli
LO activation is shown in red, behind MT
activation in green
58
The Danger of Voxelwise Approaches
  • This is one of two tables from a paper
  • Some papers publish tables of activation two
    pages long
  • How can anyone make sense of so many areas?

Source Decety et al., 1994, Nature
59
To Localize or Not to Localize?
60
Methodological Fundamentalism
The latest review I received
61
Approach 2 Region of interest (ROI) analysis
  • If you are looking at a well-established area
    (such as visual cortex, motor cortex, or the
    lateral occipital complex), its fairly easy to
    activate and identify the area
  • Do the stats and play with the threshold till you
    get something believable in the right vicinity
    based on anatomical location (e.g., sulcal
    landmarks) or functional location (e.g.,
    Talairach coordinates from prior studies)
  • Once you have found the ROI, do independent
    experiments, extract the time course information
    and determine whether activation differences
    between conditions are significant
  • Because the runs that are used to generate the
    area are independent from those used to test the
    hypothesis, liberal statistics (p lt .05) can be
    used

62
Example of ROI Approach
Culham et al., 2003, Experimental Brain
Research Does the Lateral Occipital Complex
compute object shape for grasping?
Step 1 Localize LOC
Intact Objects
Scrambled Objects
63
Example of ROI Approach
Culham et al., 2003, Experimental Brain
Research Does the Lateral Occipital Complex
compute object shape for grasping?
Step 2 Extract LOC data from experimental runs
Grasping
Reaching
NS p .35
NS p .31
64
Example of ROI Approach
Very Simple Stats
BOLD Signal Change Left Hem. LOC BOLD Signal Change Left Hem. LOC
Subject Grasping Reaching
1 0.02 0.03
2 0.19 0.08
3 0.04 0.01
4 0.10 0.32
5 1.01 -0.27
6 0.16 0.09
7 0.19 0.12
Then simply do a paired t-test to see whether the
peaks are significantly different between
conditions
Extract average peak from each subject for each
condition
NS p .35
NS p .31
  • Instead of using BOLD Signal Change, you can
    use beta weights
  • You can also do a planned contrast in Brain
    Voyager using a module called the ROI GLM

65
Utility of Doing Both Approaches
  • We also verified the result with a voxelwise
    approach

Verification of no LOC activation for grasping gt
reaching even at moderate threshold (p lt .001,
uncorrected)
66
Example The Danger of ROI Approaches
  • Example 1 LOC may be a heterogeneous area with
    subdivisions ROI analyses gloss over this
  • Example 2 Some experiments miss important areas
    (e.g., Kanwisher et al., 1997 identified one
    important face processing area -- the fusiform
    face area, FFA -- but did not report a second
    area that is a very important part of the face
    processing network -- the occipital face area,
    OFA -- because it was less robust and consistent
    than the FFA.

67
Comparing the two approaches
  • Voxelwise Analyses
  • Require no prior hypotheses about areas involved
  • Include entire brain
  • Often neglect individual differences
  • Can lose spatial resolution with intersubject
    averaging
  • Can produce meaningless laundry lists of areas
    that are difficult to interpret
  • You have to be fairly stats-savvy and include all
    the appropriate statistical corrections to be
    certain your activation is really significant
  • Popular in Europe

68
Comparing the two approaches
  • Region of Interest (ROI) Analyses
  • Extraction of ROI data can be subjected to simple
    stats (no need for multiple comparisons,
    autocorrelation or random effects corrections)
  • Gives you more statistical power (e.g., p lt .05)
  • Hypothesis-driven
  • Useful when hypotheses are motivated by other
    techniques (e.g., electrophysiology) in specific
    brain regions
  • ROI is not smeared due to intersubject averaging
  • Important for discriminating abutting areas
    (e.g., V1/V2)
  • Easy to analyze and interpret
  • Neglects other areas which may play a fundamental
    role
  • If multiple ROIs need to be considered, you can
    spend a lot of scan time collecting localizer
    data (thus limiting the time available for
    experimental runs)
  • Works best for reliable and robust areas with
    unambiguous definitions
  • Popular in North America

69
A Proposed Resolution
  • There is no reason not to do BOTH ROI analyses
    and voxelwise analyses
  • ROI analyses for well-defined key regions
  • Voxelwise analyses to see if other regions are
    also involved
  • Ideally, the conclusions will not differ
  • If the conclusions do differ, there may be
    sensible reasons
  • Effect in ROI but not voxelwise
  • perhaps region is highly variable in stereotaxic
    location between subjects
  • perhaps voxelwise approach is not powerful enough
  • Effect in voxelwise but not ROI
  • perhaps ROI is not homogenous or is
    context-specific

70
Finding the Obvious
UNDER CONSTRUCTION Need to create animation of
cards
  • Non-independence error
  • occurs when statistical tests performed are not
    independent from the means used to select the
    brain region

Arguments from Vul Kanwisher, book chapter in
press
71
Non-independence Error
  • Egregious example
  • Identify Area X with contrast of A gt B
  • Do post hoc stats showing that A is statistically
    higher than B
  • Act surprised
  • More subtle example of selection bias
  • Identify Area X with contrast of A gt B
  • Do post hoc stats showing that A is statistically
    higher than C and C is statistically greater than
    B

UNDER CONSTRUCTION Improve examples and make
graphs to illustrate
Arguments from Vul Kanwisher, book chapter in
press
72
Hypothesis- vs. Data-Driven Approaches
  • Hypothesis-driven
  • Examples t-tests, correlations, general linear
    model (GLM)
  • a priori model of activation is suggested
  • data is checked to see how closely it matches
    components of the model
  • most commonly used approach
  • Data-driven
  • Example Independent Component Analysis (ICA)
  • no prior hypotheses are necessary
  • multivariate techniques determine the patterns
    in the data that account for the most variance
    across all voxels
  • can be used to validate a model (see if the math
    comes up with the components you wouldve
    predicted)
  • can be inspected to see if there are things
    happening in your data that you didnt predict
  • can be used to identify confounds (e.g., head
    motion)

73
Example of ICA
  • hardest part is sorting the many components into
    meaningful ones vs. artifacts
  • fingerprints may help
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