Inference - FSL

Inference how surprising is your statistic? (thresholding)

But ... can I trust it?

Outline

• Null-hypothesis and Null-distribution

• Multiple comparisons and Family-wise error

• Different ways of being surprised

• Voxel-wise inference (Maximum z)

• Cluster-wise inference (Maximum size)

• Parametric vs non-parametric tests

• Enhanced clusters

• FDR - False Discovery Rate

Parametric vs non-parametric• As we described earlier, one of the

great things about for example the t-test is that we know the null-distribution

• But most distributions are not that simple

• And errors are not always normal-distributed

Provided that e ~ N(0,σ2)

Example: VBM-style analysis• Our data is segmented grey matter maps

• A voxel is either grey matter, or not.

Group #1(Oxford students)

Group #2(Train spotters)

Ok!

hist(e) ~ N ?L

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Parametric vs non-parametric

• There are approximations to the Max-z and Max-size statistics

• These are valid under certain sets of assumptions

• But can be a problem when applied outside of that set of assumptions

Cluster failure: Why fMRI inferences for spatial extenthave inflated false-positive ratesAnders Eklunda,b,c,1, Thomas E. Nicholsd,e, and Hans Knutssona,c

aDivision of Medical Informatics, Department of Biomedical Engineering, Linköping University, S-581 85 Linköping, Sweden; bDivision of Statistics andMachine Learning, Department of Computer and Information Science, Linköping University, S-581 83 Linköping, Sweden; cCenter for Medical ImageScience and Visualization, Linköping University, S-581 83 Linköping, Sweden; dDepartment of Statistics, University of Warwick, Coventry CV4 7AL, UnitedKingdom; and eWMG, University of Warwick, Coventry CV4 7AL, United Kingdom

Edited by Emery N. Brown, Massachusetts General Hospital, Boston, MA, and approved May 17, 2016 (received for review February 12, 2016)

The most widely used task functional magnetic resonance imaging(fMRI) analyses use parametric statistical methods that depend on avariety of assumptions. In this work, we use real resting-state dataand a total of 3 million random task group analyses to computeempirical familywise error rates for the fMRI software packages SPM,FSL, and AFNI, as well as a nonparametric permutation method. For anominal familywise error rate of 5%, the parametric statisticalmethods are shown to be conservative for voxelwise inferenceand invalid for clusterwise inference. Our results suggest that theprincipal cause of the invalid cluster inferences is spatial autocorre-lation functions that do not follow the assumed Gaussian shape. Bycomparison, the nonparametric permutation test is found to producenominal results for voxelwise as well as clusterwise inference. Thesefindings speak to the need of validating the statistical methods beingused in the field of neuroimaging.

fMRI | statistics | false positives | cluster inference | permutation test

Since its beginning more than 20 years ago, functional magneticresonance imaging (fMRI) (1, 2) has become a popular tool

for understanding the human brain, with some 40,000 publishedpapers according to PubMed. Despite the popularity of fMRI as atool for studying brain function, the statistical methods used haverarely been validated using real data. Validations have insteadmainly been performed using simulated data (3), but it is obviouslyvery hard to simulate the complex spatiotemporal noise that arisesfrom a living human subject in an MR scanner.Through the introduction of international data-sharing initia-

tives in the neuroimaging field (4–10), it has become possible toevaluate the statistical methods using real data. Scarpazza et al.(11), for example, used freely available anatomical images from396 healthy controls (4) to investigate the validity of parametricstatistical methods for voxel-based morphometry (VBM) (12).Silver et al. (13) instead used image and genotype data from 181subjects in the Alzheimer’s Disease Neuroimaging Initiative(8, 9), to evaluate statistical methods common in imaging ge-netics. Another example of the use of open data is our previouswork (14), where a total of 1,484 resting-state fMRI datasets fromthe 1,000 Functional Connectomes Project (4) were used as nulldata for task-based, single-subject fMRI analyses with the SPMsoftware. That work found a high degree of false positives, up to70% compared with the expected 5%, likely due to a simplistictemporal autocorrelation model in SPM. It was, however, notclear whether these problems would propagate to group studies.Another unanswered question was the statistical validity of otherfMRI software packages. We address these limitations in thecurrent work with an evaluation of group inference with the threemost common fMRI software packages [SPM (15, 16), FSL (17),and AFNI (18)]. Specifically, we evaluate the packages in theirentirety, submitting the null data to the recommended suite ofpreprocessing steps integrated into each package.The main idea of this study is the same as in our previous one

(14). We analyze resting-state fMRI data with a putative taskdesign, generating results that should control the familywise error

(FWE), the chance of one or more false positives, and empiricallymeasure the FWE as the proportion of analyses that give rise toany significant results. Here, we consider both two-sample andone-sample designs. Because two groups of subjects are randomlydrawn from a large group of healthy controls, the null hypothesisof no group difference in brain activation should be true. More-over, because the resting-state fMRI data should contain noconsistent shifts in blood oxygen level-dependent (BOLD) activity,for a single group of subjects the null hypothesis of mean zeroactivation should also be true. We evaluate FWE control for bothvoxelwise inference, where significance is individually assessed ateach voxel, and clusterwise inference (19–21), where significanceis assessed on clusters formed with an arbitrary threshold.In brief, we find that all three packages have conservative

voxelwise inference and invalid clusterwise inference, for bothone- and two-sample t tests. Alarmingly, the parametric methodscan give a very high degree of false positives (up to 70%, com-pared with the nominal 5%) for clusterwise inference. By com-parison, the nonparametric permutation test (22–25) is found toproduce nominal results for both voxelwise and clusterwise in-ference for two-sample t tests, and nearly nominal results for one-sample t tests. We explore why the methods fail to appropriatelycontrol the false-positive risk.

ResultsA total of 2,880,000 random group analyses were performed tocompute the empirical false-positive rates of SPM, FSL, andAFNI; these comprise 1,000 one-sided random analyses repeatedfor 192 parameter combinations, three thresholding approaches,

Significance

Functional MRI (fMRI) is 25 years old, yet surprisingly its mostcommon statistical methods have not been validated using realdata. Here, we used resting-state fMRI data from 499 healthycontrols to conduct 3 million task group analyses. Using this nulldata with different experimental designs, we estimate the in-cidence of significant results. In theory, we should find 5% falsepositives (for a significance threshold of 5%), but insteadwe foundthat the most common software packages for fMRI analysis (SPM,FSL, AFNI) can result in false-positive rates of up to 70%. Theseresults question the validity of a number of fMRI studies and mayhave a large impact on the interpretation of weakly significantneuroimaging results.

Author contributions: A.E. and T.E.N. designed research; A.E. and T.E.N. performed re-search; A.E., T.E.N., and H.K. analyzed data; and A.E., T.E.N., and H.K. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.

See Commentary on page 7699.1To whom correspondence should be addressed. Email: [email protected].

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1602413113/-/DCSupplemental.

7900–7905 | PNAS | July 12, 2016 | vol. 113 | no. 28 www.pnas.org/cgi/doi/10.1073/pnas.1602413113


• Those approximations were based on Gaussian Random Field Theory, and was an impressive body of work

• They served us fantastically well at a time when we had little choice

• But the future is non-parametric

• We can permute the data itself to create a distribution that we can use to test our statistic.

+ Makes very few assumptions about the data

+ Works for any test statistic

A simple permutation test

We have performed an experiment

And calculated a statistic, e.g. a t-value

t = 2.27If the null-hypothesis is true, there is no

difference between the groups. That means we should be able to “re-label” the individual points without changing

anything.

One re-labelling t-value after re-labelling

t = 0.67

Let’s start collecting them

Original labelling

A simple permutation test• We can permute the data itself to create a distribution

that we can use to test our statistic.



Second re-labelling t-value after re-labelling

t = 1.97Original labelling

And another one





Of the 5000 re-labellings, only 90 had a t-value > 2.27 (the original labelling).

I.e. there is only a ~1.8% (90/5000) chance of obtaining a value > 2.27 if there is no

difference between the groups

C.f. p(x 2.27) = 1.79% for t185000 re-labellings. Phew!

�

Original labelling





And we can use this for any statistic

We compared activation by painful stimuli in two groups of 5 subjects each.

This is what we gotVery intriguing activation. t8 = 4.65

But, can we trust it?

Group 1

Group 2

2nd level model

Our group difference map

max(t)=4.65





Permuted model

Permuted group difference map

max(t)=8.23



Group 1

Group 2



2nd Permutation

2nd permuted map

max(t)=5.43



Group 1

Group 2



3rd Permutation

3rd permuted map

max(t)=5.84



Group 1

Group 2



5000 permutations

3925 permutations yielded higher

max(t)-value than original labelling.We cannot reject

the null-hypothesis.

Original labelling



Group 1

Group 2



But beware the “exchangeability”

• When we swap the labels of two data-points we need to make sure that they are “exchangeable”

• I will start to explain “exchangeability” through a case that is not

• But first we need to learn about covariance matrices

Height and weight of a random

sample of Swedish men




Covariance matrices

Mean height ≈181 cm

Mean weight ≈79.4 kgCharacterised by two means




And a covariance -

matrix⌃ =

130 5252 165

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Covariance matrices




⌃ =

2

4130 52 4.852 165 694.8 69 156

3

5

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Covariance matrices




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Covariance matrices

• You may, or may not, have seen this slide in the 1st level GLM talk.

1st level fMRI data is not exchangeable

= X �

��1

�2

�

+Gaussian noise

(temporal autocorrelation)Design MatrixData from a voxel

Regressor, Explanatory Variable (EV)

Regression parameters, Effect sizes

y

=

e

+

x1 x2This time we will look more closely

at this part

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Our old friend “the covariance matrix”

• One important component of noise in fMRI consists of physiological/neuronal events convolved by the HRF




…

…

If we sample this every 20 seconds it no longer looks “smooth”



Variance at point 1

Variance at point 2

Covariance between points

1 and 2

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…

But that is not a realistic TR. What about every 3 seconds?



Variance at point 1

Variance at point 2

Covariance between points

1 and 2

• Let us now return to our model again


= X �

��1

�2

�

+Gaussian noise

(temporal autocorrelation)Design MatrixData from

a voxel

Regressor, Explanatory Variable (EV)

Regression parameters, Effect sizes

y

=

e

+

x1 x2

• The model consists of our regressors X and the noise model

• All permutations must result in “equivalent models”

• Let us now see what happens if we swap two data-points (points 5 and 10)

e ⇠ N(0,⌃)<latexit sha1_base64="WmKqLrvjHkdWK5/0n5uVhTSlh/k=">AAACDXicbVDLSsNAFJ3UV62vqEs3g1WoICURQZcFN11JRfuAppTJdNIOnZmEmYlQQn7Ajb/ixoUibt2782+ctBG09cCFwzn3cu89fsSo0o7zZRWWlldW14rrpY3Nre0de3evpcJYYtLEIQtlx0eKMCpIU1PNSCeSBHGfkbY/vsr89j2RiobiTk8i0uNoKGhAMdJG6ttHHkd65AcJSaGnKIfXlR/FSU+9Wzrk6KRvl52qMwVcJG5OyiBHo29/eoMQx5wIjRlSqus6ke4lSGqKGUlLXqxIhPAYDUnXUIE4Ub1k+k0Kj40ygEEoTQkNp+rviQRxpSbcN53ZpWrey8T/vG6sg8teQkUUayLwbFEQM6hDmEUDB1QSrNnEEIQlNbdCPEISYW0CLJkQ3PmXF0nrrOo6VffmvFyr53EUwQE4BBXgggtQA3XQAE2AwQN4Ai/g1Xq0nq03633WWrDymX3wB9bHN4uFmzc=</latexit><latexit sha1_base64="WmKqLrvjHkdWK5/0n5uVhTSlh/k=">AAACDXicbVDLSsNAFJ3UV62vqEs3g1WoICURQZcFN11JRfuAppTJdNIOnZmEmYlQQn7Ajb/ixoUibt2782+ctBG09cCFwzn3cu89fsSo0o7zZRWWlldW14rrpY3Nre0de3evpcJYYtLEIQtlx0eKMCpIU1PNSCeSBHGfkbY/vsr89j2RiobiTk8i0uNoKGhAMdJG6ttHHkd65AcJSaGnKIfXlR/FSU+9Wzrk6KRvl52qMwVcJG5OyiBHo29/eoMQx5wIjRlSqus6ke4lSGqKGUlLXqxIhPAYDUnXUIE4Ub1k+k0Kj40ygEEoTQkNp+rviQRxpSbcN53ZpWrey8T/vG6sg8teQkUUayLwbFEQM6hDmEUDB1QSrNnEEIQlNbdCPEISYW0CLJkQ3PmXF0nrrOo6VffmvFyr53EUwQE4BBXgggtQA3XQAE2AwQN4Ai/g1Xq0nq03633WWrDymX3wB9bHN4uFmzc=</latexit><latexit sha1_base64="WmKqLrvjHkdWK5/0n5uVhTSlh/k=">AAACDXicbVDLSsNAFJ3UV62vqEs3g1WoICURQZcFN11JRfuAppTJdNIOnZmEmYlQQn7Ajb/ixoUibt2782+ctBG09cCFwzn3cu89fsSo0o7zZRWWlldW14rrpY3Nre0de3evpcJYYtLEIQtlx0eKMCpIU1PNSCeSBHGfkbY/vsr89j2RiobiTk8i0uNoKGhAMdJG6ttHHkd65AcJSaGnKIfXlR/FSU+9Wzrk6KRvl52qMwVcJG5OyiBHo29/eoMQx5wIjRlSqus6ke4lSGqKGUlLXqxIhPAYDUnXUIE4Ub1k+k0Kj40ygEEoTQkNp+rviQRxpSbcN53ZpWrey8T/vG6sg8teQkUUayLwbFEQM6hDmEUDB1QSrNnEEIQlNbdCPEISYW0CLJkQ3PmXF0nrrOo6VffmvFyr53EUwQE4BBXgggtQA3XQAE2AwQN4Ai/g1Xq0nq03633WWrDymX3wB9bHN4uFmzc=</latexit><latexit sha1_base64="WmKqLrvjHkdWK5/0n5uVhTSlh/k=">AAACDXicbVDLSsNAFJ3UV62vqEs3g1WoICURQZcFN11JRfuAppTJdNIOnZmEmYlQQn7Ajb/ixoUibt2782+ctBG09cCFwzn3cu89fsSo0o7zZRWWlldW14rrpY3Nre0de3evpcJYYtLEIQtlx0eKMCpIU1PNSCeSBHGfkbY/vsr89j2RiobiTk8i0uNoKGhAMdJG6ttHHkd65AcJSaGnKIfXlR/FSU+9Wzrk6KRvl52qMwVcJG5OyiBHo29/eoMQx5wIjRlSqus6ke4lSGqKGUlLXqxIhPAYDUnXUIE4Ub1k+k0Kj40ygEEoTQkNp+rviQRxpSbcN53ZpWrey8T/vG6sg8teQkUUayLwbFEQM6hDmEUDB1QSrNnEEIQlNbdCPEISYW0CLJkQ3PmXF0nrrOo6VffmvFyr53EUwQE4BBXgggtQA3XQAE2AwQN4Ai/g1Xq0nq03633WWrDymX3wB9bHN4uFmzc=</latexit>



“Point” 10 now covaries with points 4 and 6

“Point 5” now covaries with

points 9 and 11

And the models are no longer

equivalent



And the models are no longer

equivalent

And for a random permutation …

• Data-points are not “exchangeable” if swapping them means that the noise covariance-matrix ends up looking different.

• Formally we say that “The joint distribution of the data must be unchanged by the permutations under the null-hypothesis”.

• If the noise covariance-matrix has non-zero off-diagonal elements (covariances) you need to beware.

• You typically never estimate or see the covariance-matrix. You need to “imagine it” and determine from that if there is a problem.

Back to exchangeability

Examples of exchangeability: Two groups unpaired

This is the “exchangeability group”. Here all scans are in the same group, which means any scan can be

exchanged for any other.

N.B. The “group” labelling is used for completely

different purposes when using FLAME/GRFT

The implicit assumption here is that data from all subjects have the same uncertainty and are all

independent

Assumed covariance matrix


123456789

10

Original

63785124910

Perm 1

61749583102

Perm 2 …


123456789

10

Original

63785124910

Perm 1

61749583102

Perm 2

17694285310

Perm 3

N.B. Equivalent{ <latexit sha1_base64="RG03HEwJX1WBjc+2nWlsTV1XmgQ=">AAAB6XicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxvO+ndLa+sbmVnm7srO7t3/gHh61dJIphk2WiER1QqpRcIlNw43ATqqQxqHAdji+nfntJ1SaJ/LRTFIMYjqUPOKMGis99PK+W/Vq3hxklfgFqUKBRt/96g0SlsUoDRNU667vpSbIqTKcCZxWepnGlLIxHWLXUklj1EE+v3RKzqwyIFGibElD5urviZzGWk/i0HbG1Iz0sjcT//O6mYmug5zLNDMo2WJRlAliEjJ7mwy4QmbExBLKFLe3EjaiijJjw6nYEPzll1dJ66LmezX//rJavyniKMMJnMI5+HAFdbiDBjSBQQTP8Apvzth5cd6dj0VrySlmjuEPnM8fm4SNZQ==</latexit><latexit sha1_base64="RG03HEwJX1WBjc+2nWlsTV1XmgQ=">AAAB6XicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxvO+ndLa+sbmVnm7srO7t3/gHh61dJIphk2WiER1QqpRcIlNw43ATqqQxqHAdji+nfntJ1SaJ/LRTFIMYjqUPOKMGis99PK+W/Vq3hxklfgFqUKBRt/96g0SlsUoDRNU667vpSbIqTKcCZxWepnGlLIxHWLXUklj1EE+v3RKzqwyIFGibElD5urviZzGWk/i0HbG1Iz0sjcT//O6mYmug5zLNDMo2WJRlAliEjJ7mwy4QmbExBLKFLe3EjaiijJjw6nYEPzll1dJ66LmezX//rJavyniKMMJnMI5+HAFdbiDBjSBQQTP8Apvzth5cd6dj0VrySlmjuEPnM8fm4SNZQ==</latexit><latexit sha1_base64="RG03HEwJX1WBjc+2nWlsTV1XmgQ=">AAAB6XicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxvO+ndLa+sbmVnm7srO7t3/gHh61dJIphk2WiER1QqpRcIlNw43ATqqQxqHAdji+nfntJ1SaJ/LRTFIMYjqUPOKMGis99PK+W/Vq3hxklfgFqUKBRt/96g0SlsUoDRNU667vpSbIqTKcCZxWepnGlLIxHWLXUklj1EE+v3RKzqwyIFGibElD5urviZzGWk/i0HbG1Iz0sjcT//O6mYmug5zLNDMo2WJRlAliEjJ7mwy4QmbExBLKFLe3EjaiijJjw6nYEPzll1dJ66LmezX//rJavyniKMMJnMI5+HAFdbiDBjSBQQTP8Apvzth5cd6dj0VrySlmjuEPnM8fm4SNZQ==</latexit><latexit sha1_base64="RG03HEwJX1WBjc+2nWlsTV1XmgQ=">AAAB6XicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvTAXXxvO+ndLa+sbmVnm7srO7t3/gHh61dJIphk2WiER1QqpRcIlNw43ATqqQxqHAdji+nfntJ1SaJ/LRTFIMYjqUPOKMGis99PK+W/Vq3hxklfgFqUKBRt/96g0SlsUoDRNU667vpSbIqTKcCZxWepnGlLIxHWLXUklj1EE+v3RKzqwyIFGibElD5urviZzGWk/i0HbG1Iz0sjcT//O6mYmug5zLNDMo2WJRlAliEjJ7mwy4QmbExBLKFLe3EjaiijJjw6nYEPzll1dJ66LmezX//rJavyniKMMJnMI5+HAFdbiDBjSBQQTP8Apvzth5cd6dj0VrySlmjuEPnM8fm4SNZQ==</latexit>


Examples of exchangeability:Single group average

Here we model a single mean and want to know if that is different from zero

But there isn’t really anything to permute, or

is there?


++++++++++

Original

t = �0.17<latexit sha1_base64="z0uMqgB7mpi8OruR79tohv3kRHo=">AAAB8HicbVBNS8NAEJ3Ur1q/qh69LBbBiyURoV6EgpceK9haaUPZbDft0t0k7E6EEvorvHhQxKs/x5v/xm2bg7Y+GHi8N8PMvCCRwqDrfjuFtfWNza3idmlnd2//oHx41DZxqhlvsVjGuhNQw6WIeAsFSt5JNKcqkPwhGN/O/Icnro2Io3ucJNxXdBiJUDCKVnpEckMu3KpX65crbtWdg6wSLycVyNHsl796g5ilikfIJDWm67kJ+hnVKJjk01IvNTyhbEyHvGtpRBU3fjY/eErOrDIgYaxtRUjm6u+JjCpjJiqwnYriyCx7M/E/r5tieO1nIkpS5BFbLApTSTAms+/JQGjOUE4soUwLeythI6opQ5tRyYbgLb+8StqXVc9GdndVqTfyOIpwAqdwDh7UoA4NaEILGCh4hld4c7Tz4rw7H4vWgpPPHMMfOJ8/P6qOvg==</latexit><latexit sha1_base64="z0uMqgB7mpi8OruR79tohv3kRHo=">AAAB8HicbVBNS8NAEJ3Ur1q/qh69LBbBiyURoV6EgpceK9haaUPZbDft0t0k7E6EEvorvHhQxKs/x5v/xm2bg7Y+GHi8N8PMvCCRwqDrfjuFtfWNza3idmlnd2//oHx41DZxqhlvsVjGuhNQw6WIeAsFSt5JNKcqkPwhGN/O/Icnro2Io3ucJNxXdBiJUDCKVnpEckMu3KpX65crbtWdg6wSLycVyNHsl796g5ilikfIJDWm67kJ+hnVKJjk01IvNTyhbEyHvGtpRBU3fjY/eErOrDIgYaxtRUjm6u+JjCpjJiqwnYriyCx7M/E/r5tieO1nIkpS5BFbLApTSTAms+/JQGjOUE4soUwLeythI6opQ5tRyYbgLb+8StqXVc9GdndVqTfyOIpwAqdwDh7UoA4NaEILGCh4hld4c7Tz4rw7H4vWgpPPHMMfOJ8/P6qOvg==</latexit><latexit sha1_base64="z0uMqgB7mpi8OruR79tohv3kRHo=">AAAB8HicbVBNS8NAEJ3Ur1q/qh69LBbBiyURoV6EgpceK9haaUPZbDft0t0k7E6EEvorvHhQxKs/x5v/xm2bg7Y+GHi8N8PMvCCRwqDrfjuFtfWNza3idmlnd2//oHx41DZxqhlvsVjGuhNQw6WIeAsFSt5JNKcqkPwhGN/O/Icnro2Io3ucJNxXdBiJUDCKVnpEckMu3KpX65crbtWdg6wSLycVyNHsl796g5ilikfIJDWm67kJ+hnVKJjk01IvNTyhbEyHvGtpRBU3fjY/eErOrDIgYaxtRUjm6u+JjCpjJiqwnYriyCx7M/E/r5tieO1nIkpS5BFbLApTSTAms+/JQGjOUE4soUwLeythI6opQ5tRyYbgLb+8StqXVc9GdndVqTfyOIpwAqdwDh7UoA4NaEILGCh4hld4c7Tz4rw7H4vWgpPPHMMfOJ8/P6qOvg==</latexit><latexit sha1_base64="z0uMqgB7mpi8OruR79tohv3kRHo=">AAAB8HicbVBNS8NAEJ3Ur1q/qh69LBbBiyURoV6EgpceK9haaUPZbDft0t0k7E6EEvorvHhQxKs/x5v/xm2bg7Y+GHi8N8PMvCCRwqDrfjuFtfWNza3idmlnd2//oHx41DZxqhlvsVjGuhNQw6WIeAsFSt5JNKcqkPwhGN/O/Icnro2Io3ucJNxXdBiJUDCKVnpEckMu3KpX65crbtWdg6wSLycVyNHsl796g5ilikfIJDWm67kJ+hnVKJjk01IvNTyhbEyHvGtpRBU3fjY/eErOrDIgYaxtRUjm6u+JjCpjJiqwnYriyCx7M/E/r5tieO1nIkpS5BFbLApTSTAms+/JQGjOUE4soUwLeythI6opQ5tRyYbgLb+8StqXVc9GdndVqTfyOIpwAqdwDh7UoA4NaEILGCh4hld4c7Tz4rw7H4vWgpPPHMMfOJ8/P6qOvg==</latexit>


+-+--++-++

First flip

t = �0.09<latexit sha1_base64="fL5QFtG/p4x7ilO8c6QMjSKxNtk=">AAAB8HicbVBNSwMxEJ31s9avqkcvwSJ4seyKoB6EgpceK9gPaZeSTbNtaDa7JLNCWforvHhQxKs/x5v/xrTdg7Y+CHm8N8PMvCCRwqDrfjsrq2vrG5uFreL2zu7efungsGniVDPeYLGMdTughkuheAMFSt5ONKdRIHkrGN1N/dYT10bE6gHHCfcjOlAiFIyilR6R3JJzt+Le9Epl+81AlomXkzLkqPdKX91+zNKIK2SSGtPx3AT9jGoUTPJJsZsanlA2ogPesVTRiBs/my08IadW6ZMw1vYpJDP1d0dGI2PGUWArI4pDs+hNxf+8TorhtZ8JlaTIFZsPClNJMCbT60lfaM5Qji2hTAu7K2FDqilDm1HRhuAtnrxMmhcVz61495flai2PowDHcAJn4MEVVKEGdWgAgwie4RXeHO28OO/Ox7x0xcl7juAPnM8fQS2Ovw==</latexit><latexit sha1_base64="fL5QFtG/p4x7ilO8c6QMjSKxNtk=">AAAB8HicbVBNSwMxEJ31s9avqkcvwSJ4seyKoB6EgpceK9gPaZeSTbNtaDa7JLNCWforvHhQxKs/x5v/xrTdg7Y+CHm8N8PMvCCRwqDrfjsrq2vrG5uFreL2zu7efungsGniVDPeYLGMdTughkuheAMFSt5ONKdRIHkrGN1N/dYT10bE6gHHCfcjOlAiFIyilR6R3JJzt+Le9Epl+81AlomXkzLkqPdKX91+zNKIK2SSGtPx3AT9jGoUTPJJsZsanlA2ogPesVTRiBs/my08IadW6ZMw1vYpJDP1d0dGI2PGUWArI4pDs+hNxf+8TorhtZ8JlaTIFZsPClNJMCbT60lfaM5Qji2hTAu7K2FDqilDm1HRhuAtnrxMmhcVz61495flai2PowDHcAJn4MEVVKEGdWgAgwie4RXeHO28OO/Ox7x0xcl7juAPnM8fQS2Ovw==</latexit><latexit sha1_base64="fL5QFtG/p4x7ilO8c6QMjSKxNtk=">AAAB8HicbVBNSwMxEJ31s9avqkcvwSJ4seyKoB6EgpceK9gPaZeSTbNtaDa7JLNCWforvHhQxKs/x5v/xrTdg7Y+CHm8N8PMvCCRwqDrfjsrq2vrG5uFreL2zu7efungsGniVDPeYLGMdTughkuheAMFSt5ONKdRIHkrGN1N/dYT10bE6gHHCfcjOlAiFIyilR6R3JJzt+Le9Epl+81AlomXkzLkqPdKX91+zNKIK2SSGtPx3AT9jGoUTPJJsZsanlA2ogPesVTRiBs/my08IadW6ZMw1vYpJDP1d0dGI2PGUWArI4pDs+hNxf+8TorhtZ8JlaTIFZsPClNJMCbT60lfaM5Qji2hTAu7K2FDqilDm1HRhuAtnrxMmhcVz61495flai2PowDHcAJn4MEVVKEGdWgAgwie4RXeHO28OO/Ox7x0xcl7juAPnM8fQS2Ovw==</latexit><latexit sha1_base64="fL5QFtG/p4x7ilO8c6QMjSKxNtk=">AAAB8HicbVBNSwMxEJ31s9avqkcvwSJ4seyKoB6EgpceK9gPaZeSTbNtaDa7JLNCWforvHhQxKs/x5v/xrTdg7Y+CHm8N8PMvCCRwqDrfjsrq2vrG5uFreL2zu7efungsGniVDPeYLGMdTughkuheAMFSt5ONKdRIHkrGN1N/dYT10bE6gHHCfcjOlAiFIyilR6R3JJzt+Le9Epl+81AlomXkzLkqPdKX91+zNKIK2SSGtPx3AT9jGoUTPJJsZsanlA2ogPesVTRiBs/my08IadW6ZMw1vYpJDP1d0dGI2PGUWArI4pDs+hNxf+8TorhtZ8JlaTIFZsPClNJMCbT60lfaM5Qji2hTAu7K2FDqilDm1HRhuAtnrxMmhcVz61495flai2PowDHcAJn4MEVVKEGdWgAgwie4RXeHO28OO/Ox7x0xcl7juAPnM8fQS2Ovw==</latexit>

t = 1.54<latexit sha1_base64="6q819vkUgiYHlbL7dyf4a4CC5DQ=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBU0hE0YtQ8NJjBfsBbSib7aZdutnE3YlQSv+EFw+KePXvePPfuG1z0NYHA4/3ZpiZF6ZSGPS8b6ewtr6xuVXcLu3s7u0flA+PmibJNOMNlshEt0NquBSKN1Cg5O1UcxqHkrfC0d3Mbz1xbUSiHnCc8iCmAyUiwShaqY3klvju1WWvXPFcbw6ySvycVCBHvVf+6vYTlsVcIZPUmI7vpRhMqEbBJJ+WupnhKWUjOuAdSxWNuQkm83un5MwqfRIl2pZCMld/T0xobMw4Dm1nTHFolr2Z+J/XyTC6CSZCpRlyxRaLokwSTMjsedIXmjOUY0so08LeStiQasrQRlSyIfjLL6+S5oXre65/f1mp1vI4inACp3AOPlxDFWpQhwYwkPAMr/DmPDovzrvzsWgtOPnMMfyB8/kD19+OiQ==</latexit><latexit 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sha1_base64="1LiVteQRsPztfHLyLh3pSCPRNaY=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0YtQ8NJjBfsBbSib7aZdutnE3YlQQv+EFw+KePXvePPfuG1z0NYHA4/3ZpiZFyRSGHTdb6ewtr6xuVXcLu3s7u0flA+PWiZONeNNFstYdwJquBSKN1Gg5J1EcxoFkreD8d3Mbz9xbUSsHnCScD+iQyVCwShaqYPklnjVq8t+ueJW3TnIKvFyUoEcjX75qzeIWRpxhUxSY7qem6CfUY2CST4t9VLDE8rGdMi7lioaceNn83un5MwqAxLG2pZCMld/T2Q0MmYSBbYzojgyy95M/M/rphje+JlQSYpcscWiMJUEYzJ7ngyE5gzlxBLKtLC3EjaimjK0EZVsCN7yy6ukdVH13Kp371Zq9TyOIpzAKZyDB9dQgzo0oAkMJDzDK7w5j86L8+58LFoLTj5zDH/gfP4A1p+OhQ==</latexit><latexit 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sha1_base64="6q819vkUgiYHlbL7dyf4a4CC5DQ=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBU0hE0YtQ8NJjBfsBbSib7aZdutnE3YlQSv+EFw+KePXvePPfuG1z0NYHA4/3ZpiZF6ZSGPS8b6ewtr6xuVXcLu3s7u0flA+PmibJNOMNlshEt0NquBSKN1Cg5O1UcxqHkrfC0d3Mbz1xbUSiHnCc8iCmAyUiwShaqY3klvju1WWvXPFcbw6ySvycVCBHvVf+6vYTlsVcIZPUmI7vpRhMqEbBJJ+WupnhKWUjOuAdSxWNuQkm83un5MwqfRIl2pZCMld/T0xobMw4Dm1nTHFolr2Z+J/XyTC6CSZCpRlyxRaLokwSTMjsedIXmjOUY0so08LeStiQasrQRlSyIfjLL6+S5oXre65/f1mp1vI4inACp3AOPlxDFWpQhwYwkPAMr/DmPDovzrvzsWgtOPnMMfyB8/kD19+OiQ==</latexit><latexit 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----+++++-

Second flip


++++++++++

----+++++-

+-+--++-++

Etc …

…


• Symmetric errors

• Errors independent

• Subjects drawn from a single population

And the assumptions are:

Examples of exchangeability:Two groups paired

Here we can only exchange scans within

each subject. I.e. Input 1 for Input 2, Input 3 for

Input 4 etc


The implicit assumption here is that data from all subjects have the same uncertainty

and that there is no dependence between subjects

Allowed swap



The implicit assumption here is that data from all subjects have the same uncertainty

and that there is no dependence between subjects

Disallowed swap


123456789

10

Original

21345687910

Perm 1

12346587910

Perm 2 …


Examples of exchangeability:blocked ANOVA

Same as previous: We can only swap labels within

each subject


Assumptions: All subjects from the same “population”,

no dependence between subjects and “compound

symmetry” within subjects

Allowed swap



Assumptions: All subjects from the same “population”,

no dependence between subjects and “compound

symmetry” within subjects

No longer allowed swap


My advice: Keep it simple!Each subject

scanned like this

Taking 4 contrasts to 2nd level

We want to find areas that respond “linearly” to pain.

My advice: Keep it simple!

Each subject scanned like this

Taking 4 contrasts to 2nd level

Repeating this for four subjects


You have to assume this covariance matrix

And figure out this contrast

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Why put yourself through all that pain?


Much nicer, no?When you can take a single contrast from

the first level

And get this at the second level

Assuming only symmetric errors

Warning pertaining to FSL 6.0.1

Do not use the Model setup wizard together with

Randomise in FSL 6.0.1

Inference - FSL

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