1. fMRI Analysis with emphasis on the general linear model
Jody Culham
Department of Psychology
University of Western Ontario
fMRI Analysis with emphasis on the general linear model
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
Slice 9, Voxel 0, 0
Even where there’s no brain, there’s noise
Slice 9, Voxel 1, 0
Slice 9, Voxel 22, 7
The signal is much higher where there is brain, but there’s still noise
Slice 9, Voxel 14, 42
Here’s a couple that sort of show the right pattern but is it “real”?
Slice 9, Voxel 18, 36
Slice 9, Voxel 9, 27
Here’s a voxel that responds well whenever there’s visual stimulation
Slice 9, Voxel 13, 41
Here’s one that responds well whenever there’s intact objects

4. Why do we need stats?
Clearly voxel surfing isn’t a viable option. We’d 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 VISUAL AREA
PREDICTED ACTIVATION IN OBJECT AREA
BOXCAR
SHIFTED
CONVOLVED
WITH HRF

6. Types of Errors Type I Error HIT Type II Error Correct Rejection
Is the region truly active?
Does our stat test indicate that the region is active?
Yes No
HIT
Type I Error
Type II Error
Correct Rejection
p value:
probability of a Type I error
e.g., p <.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 = .40 16% of variance p < .000001 r = .80

9. Complications
Not only is it hard to determine what’s 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
What’s wrong with these data?
r = .24
6% of variance
p < .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 aren’t 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 TIME Blank Screen Intact Objects Scrambled Objects
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. What’s real? A. C. B. D.

13. = + What’s real? signal noise
I created each of those time courses based by taking the predictor function and adding a variable amount of random noise
signal = +
noise

14. What’s 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 < .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. Let’s create a time course for one LO voxel

17. We’ll begin with activation
Response to Intact Objects is 4X greater than Scrambled Objects

18. Then we’ll 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 & D’Esposito, 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 let’s 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, I’ve only included parameter one in our model
Head motion can lead to other problems not predictable by these parameters

23. Now let’s throw in a pinch of linear drift
linear drift could arise from magnet noise (e.g., parts warm up) or physiological noise (e.g., subject’s head sinks)

24. and then we’ll add a dash of low frequency noise
low frequency noise can arise from magnet noise or physiological noise (e.g., subject’s cycles of alertness/drowsiness)
low frequency noise would occur over a range of frequencies but for simplicity, I’ve 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., subject’s breathing rate and heartrate)

26. When we add these all together, we get a realistic time course

27. Now let’s 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
Design Matrix x
Betas +
Residuals
“what we CAN explain”
“how much of it we CAN explain”
“what we CANNOT explain” =
“our data” x +
Statistical significance is basically a ratio of explained to unexplained variance

29. Implementation of GLM in SPM
Intact
Predictor
Scrambled
Predictor
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 1=2
fMRI Signal +
Residuals = +
when 2=0.5 =
Design Matrix x
Betas
“what we CAN explain”
“how much of it we CAN explain”
“what we CANNOT explain” =
“our data” x +
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 β = 1
x β = 0 + OR +
x β = 1
x β = 0 + +
x β = 0
x β = -1
Because the predictors are highly correlated, you can’t tell which model is best

35. Contrasts in the GLM
We can examine whether a single predictor is significant (compared to the baseline)
We can also examine whether a single predictor is significantly greater than another predictor

36. A Real Voxel
Here’s 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 can’t 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. What’s 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 < .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 < .05
number of voxels: 50,000
required p value: p < .05 / 50,000  p <
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. Test-retest reliability
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 =
Alternatively, use the first half to select an ROI and evaluate your hypothesis in the second half.

44. 5) Poor man’s 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
Jody’s rule of thumb: “If ya can’t trust the negatives, can ya trust the positives?”
Example: MT localizer data
Moving rings > stationary rings (orange)
Stationary rings > moving rings (blue)

45. 6) False discovery rate Type I Error HIT Type II Error
Genovese et al, 2002, NeuroImage
also “controls the proportion of rejected hypotheses that are falsely rejected”
standard p value (e.g., p < .01) means that a certain proportion of all voxels will be significant by chance (1%)
FDR uses q value (e.g., q < .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 hasn’t been that satisfied
Is the region truly active?
Yes No
HIT
Type I Error
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 it’s 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 didn’t 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 don’t even toy with random effects unless you’ve got 10 or more subjects (and 50+ is best)
Random effects analyses can really squash your data, especially if you don’t 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 don’t have to worry about it if you’re 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 2
SUM 10 7
Subject
Intact beta
Scram beta
Diff 1 4 3 2 -1
SUM 10 7
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 > scrambled), whereas in data set #2, only 2 of 3 subjects show a difference in that direction

51. Autocorrelation function
time
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
shift by 1 volume
shift by 2 volumes
If there’s no autocorrelation, function should drop from 1 to 0 abruptly – pink line
the points circled in yellow suggest there is some autocorrelation, especially at a shift of 1, called AR(1)

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. WARNING: UNDER CONSTRUCTION need to clarify explanation
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
You don’t necessarily need a priori hypotheses (though sometimes you can use less conservative stats if you have them)
Average all of your data together in Talairach space
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.
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?
Neuroimagers can’t even agree how to SPELL localiser/localizer!

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), it’s 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 < .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
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
0.12
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 > reaching even at moderate threshold (p < .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 < .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 Non-independence error
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 > B
Do post hoc stats showing that A is statistically higher than B
Act surprised
More subtle example of selection bias
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 would’ve predicted)
can be inspected to see if there are things happening in your data that you didn’t 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