1. Statistics – Modelling Your Data
Chris Rorden
Modelling data:
Signal, Error and Covariates
Parametric Statistics
Thresholding Results:
Statistical power and statistical errors
The multiple comparison problem
Familywise error and Bonferroni Thresholding
Permutation Thresholding
False Discovery Rate Thresholding
Implications: null results uninterruptible

2. The fMRI signal
Last lecture: we predict areas that are involved with a task will become brighter (after a delay)
Therefore, we expect that if someone repeatedly does a task for 12 seconds, and rests for 12 seconds, our signal should look like this:

3. Calculating statistics
Does this brain area change brightness when we do the task?
Top panel: very good predictor (very little error)
Lower panel: somewhat less good predictor

4. General Linear Model
The observed data is composed of a signal that is predicted by our model and unexplained noise (Boynton et al., 1996).
Amplitude (solve for)
Design Model
Measured Data
Noise

5. What is your model? Model is predicted effect.
Consider Block desgin experiment:
Three conditions, each for 11.2sec
Press left index finger when you see ç
Press right index finger when you see è
Do nothing when you see é
Intensity
Time

6. FSL/SPM display of model
Analysis programs display model as grid.
Each column is regressor
e.g. left / right arrows.
Each row is a volume of data
for within-subject fMRI = time
Brightness of row is model’s predicted intensity.
Time
Intensity
Time

7. Statistical Contrasts
fMRI inference based on contrast.
Consider study with left arrow and right arrow as regressors
[1 0] identifies activation correlated with left arrows: we could expect visual and motor effects.
[1 –1] identifies regions that show more response to left arrows than right arrows. Visual effects should be similar, so should select contralateral motoric.
Choice of contrasts crucial to inference.

8. Statistical Contrasts
t-Test is one tailed, F-test is two-tailed.
T-test: [1 –1] mutually exclusive of [-1 1]: left>right vs right>left.
F-test: [1 –1] = [-1 1]: difference between left and right.
Choice of test crucial to inference.

9. How many regressors?
We collected data during a block design, where the participant completed 3 tasks
Left hand movement
Right hand movement
Rest
We are only interested in the brain areas involed with Left hand movement.
Should we include uninteresting right hand movement as a regressor in our statistical model?
I.E. Is a [1] analysis the same as a [1 0]?
Is a [1 0] analysis identical, better, worse or different from a [1] analysis? =?

10. Meaningful regressors decrease noise
Meaningful regressors can explain some of the variability.
Adding a meaningful regressor can reduce the unexplained noise from our contrast.

11. Correlated regressors decrease signal
If a regressor is strongly correlated with our effect, it can reduce the residual signal
Our signal is excluded as the regressor explains this variability.
Example: responses highly correlated to visual stimuli

12. Explained Variance Unexplained Variance
Single factor…
Consider a test to see how well height predicts weight.
Weight
Height
Explained Variance Unexplained Variance t=
Small t-score
height only weakly predicts weight
High t-score
height strongly predicts weight

13. Adding a second factor…
How does an additional factor influence our test?
E.G. We can add waist diameter as a regressor.
Does this regressor influence the t-test regarding how well height predicts weight?
Consider ratio of cyan to green.
Weight
Height
Waist
Increased t
Waist explains portion of weight not predicted by height.
Decreased t
Waist explains portion of weight predicted by height.

14. Regressors and statistics
Our analysis identifies three classes of variability:
Signal: Predicted effect of interest
Noise (aka Error): Unexplained variance
Covariates: Predicted effects that are not relevant.
Statistical significance is the ratio:
Covariates will
Improve sensitivity if they reduce error (explain otherwise unexplained variance).
Reduce sensitivity if they reduce signal (explain variance that is also predicted by our effect of interest).
Signal Noise t=

15. Regressors should be orthogonal
Summary
Regressors should be orthogonal
Each regressor describes independent variance.
Variance should not be explained by more than one regressor.
E.G. we will see that including temporal derivatives as regressors tend to help event related designs (temporal processing lecture).

16. Group Analysis
We typically want to make inferences about the general population
Conduct time course analysis on many people.
Identify which patterns are consistent across group.

17. Parametric Statistics
SPM and FSL conduct parametric statistics.
T-test, F-test, Correlation
These make assumptions about data.
We will not check to see if these assumptions are valid.

18. Parametric Statistics
Parameters = Assumptions
Parametric Statistics assume that data can be accurately defined by two values:
Mean = measure of central tendency
Variance = measure of noise
Means differ
Variabilities Differ

19. Parametric Statistics
Parametric Statistics are popular
Simple (complex data described by two numbers: mean and variability)
Flexible (can look at how multiple factors interact)
Powerful: very sensitive at detecting real effects
Robust: usually work even if assumptions violated
Tend to fail graciously: by becoming more conservative

20. Parametric Statistics Assume Bell-Shaped data:
Normal Distribution
Parametric Statistics Assume Bell-Shaped data:
Often, this is wrong. Mean may not be a good measure:
Positive Skew: response times, hard exam
Negative Skew: easy exam
Bimodal: some students got it

21. Rank-Order Statistics
Rank-order statistics make fewer assumptions.
Have less power (if data is normal)
Require more measurements
May fail to detect real results
Computationally slow
Classic examples:
Wilcoxon Mann-Whitney
Fligner and Policello’s robust rank order test

22. Problem with rank order statistics
While rank-order statistics are often referred to as non-parametric, most make assumptions:
WMW: assume both distributions have same shape.
FP: assume both distributions are symmetrical.
Both these tests become liberal if their assumptions are not met.
They fail catastrophically.

23. What to do? In general, use parametric tests.
In face of violations, you will simply lose power
One alternative is to use permutation testing, e.g. SnPM.
Permuation testing is only as powerful as the test statistic it uses: SnPM uses the t-test, which is sensitive to changes in mean (so it can be blind to changes in median).
Recent alternative is truly non-parametric test of Brunner and Munzel.
Can offer slightly better power than t-test if data is skewed. Rorden et al

24. Statistical Thresholding
Type I/II Errors
Power
Multiple Comparison Problem
Bonferroni Correction
Permutation Thresholding
False Discovery Rate
ROI Analysis

25. Statistics E.G. erythropoietin (EPO) doping in athletes
In endurance athletes, EPO improves performance ~ 10%
Races often won by less than 1%
Without testing, athletes forced to dope to be competitive
Dangers: Carcinogenic and can cause heart-attacks
Therefore: Measure haematocrit level to identify drug users…
haematocrit
30%
50%
If there was no noise in our measure, it would be easy to identify EPO doping:

26. The problem of noise Science tests hypotheses based on observations
We need statistics because our data is noisy
In the real world, haematocrit levels vary
This unrelated noise in our measure is called ‘error’
How to we identify dopers?
50%
In the real world, hematocrit varies between people
30%
hematocrit

27. Statistical Threshold
hematocrit
30%
50%
If we set the threshold too low, we will accuse innocent people (high rate of false alarms).
hematocrit
30%
50%
If we set the threshold too high, we will fail to detect dopers (high rate of misses).

28. Possible outcomes of drug test
Reality (unknown)
nonDoper
EPO Doper
Accuse and expel
Innocent accused (false alarm)
Type I error
Doper expelled
(hit)
Allow to compete
Innocent competes (correct rejection)
Doper sneaks through (miss) Type II error
Decision

29. Errors With noisy data, we will make mistakes. Statistics allows us to
Estimate our confidence
Bias the type of mistake we make (e.g. we can decide whether we will tend to make false alarms or misses)
We can be liberal: avoiding misses
We can be conservative: avoiding false alarms.
We want liberal tests for airport weapons detection (X-ray often leads to innocent cases being opened).
Our society wants conservative tests for criminal conviction: avoid sending innocent people to jail.

30. Liberal vs Conservative Thresholds
A low threshold, we will accuse innocent people (high rate of false alarms, Type I).
CONSERVATIVE
A high threshold, we will fail to detect dopers (high rate of misses, Type II).

31. Statistical Power is our probability of making a Hit.
It reflects our ability to detect real effects.
To make new discoveries, we need to optimize power.
There are 4 ways to increase power…
Type II error
Correct rejection
Accept Ho
Hit
Type I error
Reject Ho
Ho false
Ho true
Decision
Reality

32. 1.) Alpha and Power
By making alpha less strict, we can increase power. (e.g. p < 0.05 instead of 0.01)
However, we increase the chance of a Type I error!

33. 2.) Effect Size and Power
Power will increase if the effect size increases. (e.g. higher dose of drug, 7T MRI instead of 1.5T).
Unfortunately, effect sizes are often small and fixed.

34. 3.) Variability and Power
Reducing variability increases the relative effect size.
Most measures of brain activity noisy.

35. 4.) Sample Size
A final way to increase our power is to collect more data.
We can sample a person’s brain activity on many similar trials.
We can test more people.
The disadvantage is time and money.
Increasing the sample size is often our only option for increasing statistical power.

36. In graphs below, same  is used.
Reflection
Statistically, relative ‘effect size’ and ‘variability’ are equivalent.
Our confidence is the ratio of effect size versus variability (signal versus noise).
In graphs below, same  is used. =

37. Alpha level  Statistics allow us to estimate our confidence.
 is our statistical threshold: it measures our chance of Type I error.
An alpha level of 5% means only 1/20 chance of false alarm (we will only accept p < 0.05).
An alpha level of 1% means only 1/100 chance of false alarm (p< 0.01).
Therefore, a 1% alpha is more conservative than a 5% alpha.

38. Multiple Comparison Problem
Assume a 1% alpha for drug testing.
An innocent athlete only has 1% chance of being accused.
Problem: 10,500 athletes in the Olympics.
If all innocent, and  = 1%, we will wrongly accuse 105 athletes (0.01*10500)!
This is the multiple comparison problem.

39. Multiple Comparisons
The gray matter volume ~900cc (900,000mm3)
Typical fMRI voxel is 3x3x3mm (27mm3)
Therefore, we will conduct >30,000 tests
With 5% alpha, we will make >1500 false alarms!

40. Multiple Comparison Problem
If we conduct 20 tests, with an  = 5%, we will on average make one false alarm (20x0.05).
If we make twenty comparisons, it is possible that we may be making 0, 1, 2 or in rare cases even more errors.
The chance we will make at least one error is given by the formula: 1- (1- )C: if we make twenty comparisons at p < .05, we have a 1-(.95) 20 = 64% chance that we are reporting at least one erroneous finding. This is our familywise error (FWE) rate.

41. Bonferroni Correction
Bonferroni Correction: controls FWE.
For example: if we conduct 10 tests, and want a 5% chance of any errors, we will adjust our threshold to be p < (0.05/10).
Benefits: Controls for FWE.
Problem: Very conservative = very little chance of detecting real effects = low power.

42. Random Field Theory
5mm
We spatially smooth our data – peaks due to noise should be attenuated by neighbors.
Worsley et al, HBM 4:58-73, 1995.
RFT uses resolution elements (resels) instead of voxels.
If we smooth our data with 8mm FWHM, then resel size is 8mm.
SPM uses RFT for FWE correction: only requires statistical map, smoothness and cluster size threshold.
Euler characteristic: unsmoothed noise will have high peaks but few clusters, smoothed data will be have lower peaks but show clustering.
RFT has many unchecked assumptions (Nichols)
Works best for heavily smoothed data (x3 voxel size)
10mm
15mm
Image from Nichols

43. Permutation Thresholding
Group 1
Group 2
Prediction: Label ‘Group 1’ and ‘Group 2’ mean something.
Null Hypothesis (Ho): Labels are meaningless.
If Ho true, we should get similar t-scores if we randomly scramble order.

44. Permutation Thresholding
Group 1
Group 2
Observed, max T = 4.1
Permutation 1, max T = 3.2
Permutation 2, max T = 2.9
Permutation 3, max T = 3.3
Permutation 4, max T = 2.8
Permutation 5, max T = 3.5 …
Permutation 1000, max T = 3.1 … …

45. Permutation Thresholding
Compute maximum T-score for 1000 permutations.
Find 5th Percentile max T.
Any voxel in our observed dataset that exceeds this threshold has only 5% probability of being noise. 5
T= 3.9
Max T
Percentile 5%

46. Permutation Thresholding
Permutation Thresholding offers the same protection against false alarms as Bonferroni.
Typically, much more powerful than Bonferroni.
Implementations include SnPM, FSL’s randomise, and my own NPM.
Disadvantage: computing 1000 permutations means it takes x1000 times longer than typical analysis!
Simulation data from Nichols et al.: Permutation always optimal. Bonferroni typically conservative. Random Fields only accurate with high DF and heavily smoothed.

47. False Discovery Rate
Traditional statistics attempts to control the False Alarm rate.
‘False Discovery Rate’ controls the ratio of false alarms to hits.
It often provides much more power than Bonferroni correction.

48. Assume Olympics where no athletes took EPO:
FDR
Assume Olympics where no athletes took EPO:
Assume Olympics where some cheat:
When we conduct many tests, we can estimate the amount of real signal

49. Bonferroni FWE applies same threshold to each data set
FDR vs FWE
Bonferroni FWE applies same threshold to each data set
FDR is dynamic: threshold based on signal detected.
5% Bonferroni: only a 5% chance an innocent athlete will be accused.
5% FDR: only 5% of expelled athletes are innocent.

50. Controlling for multiple comparisons
Bonferroni correction
We will often fail to find real results.
RFT correction
Typically less conservative than Bonferroni.
Requires large DF and broad smoothing.
Permutation Thresholding
Offers same inference as Bonferroni correction.
Typically much less conservative than Bonferroni.
Computationally very slow
FDR correction
At FDR of .05, about 5% of ‘activated’ voxels will be false alarms.
If signal is only tiny proportion of data, FDR will be similar to Bonferroni.

51. Alternatives to voxelwise analysis
Conventional fMRI statistics compute one statistical comparison per voxel.
Advantage: can discover effects anywhere in brain.
Disadvantage: low statistical power due to multiple comparisons.
Small Volume Comparison: Only test a small proportion of voxels. (Still have to adjust for RFT).
Region of Interest: Pool data across anatomical region for single statistical test.
Example: how many comparisons on this slice?
SPM: 1600
SVC: 57
ROI: 1
SPM
SVC
ROI

52. ROI analysis
In voxelwise analysis, we conduct an indepent test for every voxel
Each voxel is noisy
Huge number of tests, so severe penalty for multiple comparisons
Alternative: pool data from region of interest.
Averaging across meaningful region should reduce noise.
One test per region, so FWE adjustment less severe.
Region must be selected independently of statistical contrast!
Anatomically predefined
Defined based on previous localizer session
Selected based on combination of conditions you will contrast.
M1: movement
S1: sensation

53. Inference from fMRI statistics
fMRI studies have very low power.
Correction for multiple comparisons
Poor signal to noise
Variability in functional anatomy between people.
Null results impossible to interpret. (Hard to say an area is not involved with task).

54. Between and Within Subject Variance
Consider experiment to see if music influences typing speed.
Possible effect will be small.
Large variability between people: some people much better typist than others.
Solution: repeated measure design to separate between and within subject variability. 70 60
Alice
Bob 50
Donna 40
Nick
Typing speed: words per minute
Sam 30 20 10
Bach
Rock
Silent

55. Multiple Subject Analysis: Mixed Model
Model all of the data at once
Between and within subject variation is accounted for
Can’t apply mixed model directly to fMRI data because there is so much data!
Z stats
Group
Sub 1
Sub 2
Sub 3
Sub 4

56. Multiple Subject Analysis: SPM2
First estimate each subject’s contrast effect sizes (copes)
Run a t-test on the copes
Holmes and Friston assume within subject variation is same for all subjects, this allows them to ignore it at the group level
Not equivalent to a mixed model
Results
Group: T-test
copes
copes
copes
copes
Sub 1
Sub 2
Sub 3
Sub 4

57. Multiple Subject Analysis: FSL
First estimate each subject’s copes and cope variability (varcopes)
Then enter the copes and varcopes into group model
varcopes supply within subject variation
Between subject variation and group level means are then estimated
Equivalent to mixed model
Much slower than SPM
Z stats
Group
copes varcopes
copes varcopes
copes varcopes
copes varcopes
Sub 1
Sub 2
Sub 3
Sub 4