Date post: | 21-Jan-2016 |
Category: |
Documents |
Upload: | marilynn-ellis |
View: | 216 times |
Download: | 0 times |
Advanced Statistics
Inference Methods & Issues:
Multiple Testing, Nonparametrics, Conjunctions & Bayes
Thomas Nichols, Ph.D.
Department of BiostatisticsUniversity of Michigan
http://www.sph.umich.edu/~nichols
OHBM fMRI Course June 12, 2005
NIH Neuroinformatics / Human Brain Project
© 2005 Thomas Nichols
2
Overview
• Multiple Testing Problem– Which of my 100,000 voxels are “active”?
• Nonparametric Inference– Can I trust my P-value at this voxel?
• Conjunction Inference– Is this voxel “active” in all of my tasks?
• Bayesian vs. Classical Inference– What in the world is a posterior probability?
© 2005 Thomas Nichols
3
Overview
• Multiple Testing Problem– Which of my 100,000 voxels are “active”?
• Nonparametric Inference– Can I trust my P-value at this voxel?
• Conjunction Inference– Is this voxel “active” in all of my tasks?
• Bayesian vs. Classical Inference– What in the world is a posterior probability?
© 2005 Thomas Nichols
4
• Null Hypothesis H0
• Test statistic T– t observed realization of T
level– Acceptable false positive rate
– Level = P( T > u | H0 )
– Threshold u controls false positive rate at level
• P-value– Assessment of t assuming H0
– P( T > t | H0 )• Prob. of obtaining stat. as large
or larger in a new experiment
– P(Data|Null) not P(Null|Data)
Hypothesis Testing
u
Null Distribution of T
t
P-val
Null Distribution of T
© 2005 Thomas Nichols
5
Hypothesis Testing in fMRI
• Massively Univariate Modeling– Fit model at each voxel– Create statistic images of effect
• Then find the signal in the image...
© 2005 Thomas Nichols
6
Assessing Statistic Images
Where’s the signal?
t > 0.5t > 3.5t > 5.5
High Threshold Med. Threshold Low Threshold
Good Specificity
Poor Power(risk of false negatives)
Poor Specificity(risk of false positives)
Good Power
...but why threshold?!
© 2005 Thomas Nichols
7
• Don’t threshold, model the signal!– Signal location?
• Estimates and CI’s on(x,y,z) location
– Signal magnitude?• CI’s on % change
– Spatial extent?• Estimates and CI’s on activation volume
• Robust to choice of cluster definition
• ...but this requires an explicit spatial model
Blue-sky inference:What we’d like
space
Loc. Ext.
Mag.
© 2005 Thomas Nichols
8
Blue-sky inference:What we need
• Need an explicit spatial model
• No routine spatial modeling methods exist– High-dimensional mixture modeling problem– Activations don’t look like Gaussian blobs– Need realistic shapes, sparse representation
• Some work by Hartvig et al., Penny et al.
© 2005 Thomas Nichols
9
Real-life inference:What we get
• Signal location– Local maximum – no inference– Center-of-mass – no inference
• Sensitive to blob-defining-threshold
• Signal magnitude– Local maximum intensity – P-values (& CI’s)
• Spatial extent– Cluster volume – P-value, no CI’s
• Sensitive to blob-defining-threshold
© 2005 Thomas Nichols
10
Voxel-level Inference
• Retain voxels above -level threshold u• Gives best spatial specificity
– The null hyp. at a single voxel can be rejected
Significant Voxels
space
u
No significant Voxels
© 2005 Thomas Nichols
11
Cluster-level Inference
• Two step-process– Define clusters by arbitrary threshold uclus
– Retain clusters larger than -level threshold k
Cluster not significant
uclus
space
Cluster significantk k
© 2005 Thomas Nichols
12
Cluster-level Inference
• Typically better sensitivity
• Worse spatial specificity– The null hyp. of entire cluster is rejected– Only means that one or more of voxels in
cluster active
Cluster not significant
uclus
space
Cluster significantk k
© 2005 Thomas Nichols
14
Voxel-wise Inference & Multiple Testing Problem (MTP)
• Standard Hypothesis Test– Controls Type I error of each test,
at say 5%
– But what if I have 100,000 voxels?• 5,000 false positives on average!
• Must control false positive rate– What false positive rate?
– Chance of 1 or more Type I errors?
– Proportion of Type I errors?
5%0
© 2005 Thomas Nichols
15
MTP Solutions:Measuring False Positives
• Familywise Error Rate (FWER)– Familywise Error
• Existence of one or more false positives
– FWER is probability of familywise error
• False Discovery Rate (FDR)– R voxels declared active, V falsely so
• Observed false discovery rate: V/R
– FDR = E(V/R)
© 2005 Thomas Nichols
16
FWER MTP Solutions
• Bonferroni
• Maximum Distribution Methods– Random Field Theory– Permutation
© 2005 Thomas Nichols
17
FWER MTP Solutions:Bonferroni
• V voxels to test
• Corrected Threshold– Threshold corresponding to = 0.05/V
• Corrected P-value– min{ P-value V, 1 }
© 2005 Thomas Nichols
18
FWER MTP Solutions
• Bonferroni
• Maximum Distribution Methods– Random Field Theory– Permutation
© 2005 Thomas Nichols
19
FWER MTP Solutions: Controlling FWER w/ Max
• FWER & distribution of maximum
FWER= P(FWE)= P(One or more voxels u |
Ho)= P(Max voxel u | Ho)
• 100(1-)%ile of max distn controls FWERFWER = P(Max voxel u | Ho)
u
© 2005 Thomas Nichols
20
FWER MTP Solutions
• Bonferroni
• Maximum Distribution Methods– Random Field Theory– Permutation
© 2005 Thomas Nichols
21
FWER MTP Solutions:Random Field Theory
• Euler Characteristic u
– Topological Measure• #blobs - #holes
– At high thresholds,just counts blobs
– FWER = P(Max voxel u | Ho)= P(One or more blobs | Ho) P(u 1 | Ho) E(u | Ho)
Random Field
Suprathreshold Sets
Threshold
No holes
Never more than 1 blob
© 2005 Thomas Nichols
22
RFT Details:Expected Euler Characteristic
E(u) () || (u 2 -1) exp(-u 2/2) / (2)2
– Search region R3
– ( volume– || roughness
• Assumptions– Multivariate Normal– Stationary*– ACF twice differentiable at 0
* Stationary– Results valid w/out stationary– More accurate when stat. holds
Only very upper tail approximates1-Fmax(u)
© 2005 Thomas Nichols
25
Random Field Intuition
• Corrected P-value for voxel value t Pc = P(max T > t) E(t) () || t2 exp(-t2/2)
• Statistic value t increases– Pc decreases (of course!)
• Search volume () increases– Pc increases (more severe MCP)
• Smoothness increases (|| smaller)– Pc decreases (less severe MCP)
© 2005 Thomas Nichols
26
Random Field TheoryStrengths & Weaknesses
• Closed form results for E(u)– Z, t, F, Chi-Squared Continuous RFs
• Results depend only on volume & smoothness
• Smoothness assumed known• Sufficient smoothness required
– Results are for continuous random fields– Smoothness estimate becomes biased
• Multivariate normality• Several layers of approximations
Lattice ImageData
Continuous Random Field
© 2005 Thomas Nichols
27
Real Data
• fMRI Study of Working Memory – 12 subjects, block design Marshuetz et al (2000)
– Item Recognition• Active:View five letters, 2s pause,
view probe letter, respond
• Baseline: View XXXXX, 2s pause,view Y or N, respond
• Second Level RFX– Difference image, A-B constructed
for each subject– One sample t test
...
D
yes
...
UBKDA
Active
...
N
no
...
XXXXX
Baseline
© 2005 Thomas Nichols
28
Real Data:RFT Result
• Threshold– S = 110,776– 2 2 2 voxels
5.1 5.8 6.9 mmFWHM
– u = 9.870• Result
– 5 voxels above the threshold
– 0.0063 minimumFWE-correctedp-value-lo
g 10 p
-va
lue
© 2005 Thomas Nichols
29
MTP Solutions:Measuring False Positives
• Familywise Error Rate (FWER)– Familywise Error
• Existence of one or more false positives
– FWER is probability of familywise error
• False Discovery Rate (FDR)– FDR = E(V/R)– R voxels declared active, V falsely so
• Realized false discovery rate: V/R
© 2005 Thomas Nichols
30
False Discovery Rate• For any threshold, all voxels can be cross-classified:
• Realized FDR
rFDR = V0R/(V1R+V0R) = V0R/NR
– If NR = 0, rFDR = 0
• But only can observe NR, don’t know V1R & V0R – We control the expected rFDR
FDR = E(rFDR)
Accept Null Reject Null
Null True (no effect) V0A V0R m0
Null False (true effect) V1A V1R m1
NA NR V
© 2005 Thomas Nichols
31
False Discovery RateIllustration:
Signal
Signal+Noise
Noise
© 2005 Thomas Nichols
32
FWE
6.7% 10.4% 14.9% 9.3% 16.2% 13.8% 14.0% 10.5% 12.2% 8.7%
Control of Familywise Error Rate at 10%
11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5%
Control of Per Comparison Rate at 10%
Percentage of Null Pixels that are False Positives
Control of False Discovery Rate at 10%
Occurrence of Familywise Error
Percentage of Activated Pixels that are False Positives
© 2005 Thomas Nichols
33
Benjamini & HochbergProcedure
• Select desired limit q on FDR• Order p-values, p(1) p(2) ... p(V)
• Let r be largest i such that
• Reject all hypotheses corresponding to p(1), ... , p(r).
p(i) i/V qp(i)
i/V
i/V qp-
valu
e
0 1
01
JRSS-B (1995)57:289-300
© 2005 Thomas Nichols
34
Benjamini & Hochberg Procedure Details
• Method is valid under smoothness– Positive Regression Dependency on Subsets
P(X1c1, X2c2, ..., Xkck | Xi=xi) is non-decreasing in xi
• Only required of test statistics for which null true
• Special cases include– Independence
– Multivariate Normal with all positive correlations
– Same, but studentized with common std. err.
• For arbitrary covariance structure– Replace q with q c(V)
c(V) = i=1,...,V 1/i log(V)+0.5772
Benjamini &Yekutieli (2001).Ann. Stat.29:1165-1188
© 2005 Thomas Nichols
35
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
When no signal:P-value
threshold /V
When allsignal:P-value
threshold
Ord
ered
p-v
alue
s p
(i)
Fractional index i/V
Adaptiveness of Benjamini & Hochberg FDR
...FDR adapts to the amount of signal in the data
© 2005 Thomas Nichols
37
FDR Example
FDR Threshold = 3.833,073 voxels
FWER Perm. Thresh. = 7.6758 voxels
• Threshold– Indep/PosDep
u = 3.83
• Result– 3,073 voxels above
Indep/PosDep u– <0.0001 minimum
FDR-correctedp-value
© 2005 Thomas Nichols
38
Overview
• Multiple Testing Problem– Which of my 100,000 voxels are “active”?
• Nonparametric Inference– Can I trust my P-value at this voxel?
• Conjunction Inference– Is this voxel “active” in all of my tasks?
• Bayesian vs. Classical Inference– What in the world is a posterior probability?
© 2005 Thomas Nichols
39
Nonparametric Permutation Test
• Parametric methods– Assume distribution of
statistic under nullhypothesis
• Nonparametric methods– Use data to find
distribution of statisticunder null hypothesis
– Any statistic!
5%
Parametric Null Distribution
5%
Nonparametric Null Distribution
© 2005 Thomas Nichols
40
Permutation TestToy Example
• Data from V1 voxel in visual stim. experimentA: Active, flashing checkerboard B: Baseline, fixation6 blocks, ABABAB Just consider block averages...
• Null hypothesis Ho – No experimental effect, A & B labels arbitrary
• Statistic– Mean difference
A B A B A B
103.00 90.48 99.93 87.83 99.76 96.06
© 2005 Thomas Nichols
44
Permutation TestToy Example
• Under Ho
– Consider all equivalent relabelings– Compute all possible statistic values– Find 95%ile of permutation distribution
AAABBB 4.82 ABABAB 9.45 BAAABB -1.48 BABBAA -6.86
AABABB -3.25 ABABBA 6.97 BAABAB 1.10 BBAAAB 3.15
AABBAB -0.67 ABBAAB 1.38 BAABBA -1.38 BBAABA 0.67
AABBBA -3.15 ABBABA -1.10 BABAAB -6.97 BBABAA 3.25
ABAABB 6.86 ABBBAA 1.48 BABABA -9.45 BBBAAA -4.82
© 2005 Thomas Nichols
45
Permutation TestToy Example
• Under Ho
– Consider all equivalent relabelings– Compute all possible statistic values– Find 95%ile of permutation distribution
0 4 8-4-8
© 2005 Thomas Nichols
46
Controlling FWER: Permutation Test
• Parametric methods– Assume distribution of
max statistic under nullhypothesis
• Nonparametric methods– Use data to find
distribution of max statisticunder null hypothesis
– Again, any max statistic!
5%
Parametric Null Max Distribution
5%
Nonparametric Null Max Distribution
© 2005 Thomas Nichols
47
Permutation Test& Exchangeability
• Exchangeability is fundamental– Def: Distribution of the data unperturbed by permutation
– Under H0, exchangeability justifies permuting data
– Allows us to build permutation distribution
• Subjects are exchangeable– Under Ho, each subject’s A/B labels can be flipped
• fMRI scans are not exchangeable under Ho– If no signal, can we permute over time?
– No, permuting disrupts order, temporal autocorrelation
© 2005 Thomas Nichols
48
Permutation Test& Exchangeability
• fMRI scans are not exchangeable– Permuting disrupts order, temporal autocorrelation
• Intrasubject fMRI permutation test– Must decorrelate data, model before permuting– What is correlation structure?
• Usually must use parametric model of correlation
– E.g. Use wavelets to decorrelate• Bullmore et al 2001, HBM 12:61-78
• Intersubject fMRI permutation test– Create difference image for each subject– For each permutation, flip sign of some subjects
© 2005 Thomas Nichols
52
Permutation TestExample
• fMRI Study of Working Memory – 12 subjects, block design Marshuetz et al (2000)
– Item Recognition• Active:View five letters, 2s pause,
view probe letter, respond
• Baseline: View XXXXX, 2s pause,view Y or N, respond
• Second Level RFX– Difference image, A-B constructed
for each subject
– One sample, smoothed variance t test
...
D
yes
...
UBKDA
Active
...
N
no
...
XXXXX
Baseline
© 2005 Thomas Nichols
53
Permutation TestExample
• Permute!– 212 = 4,096 ways to flip 12 A/B labels– For each, note maximum of t image.
Permutation DistributionMaximum t
Maximum Intensity Projection Thresholded t
© 2005 Thomas Nichols
54
Permutation TestExample
• Compare with Bonferroni = 0.05/110,776
• Compare with parametric RFT– 110,776 222mm voxels– 5.15.86.9mm FWHM smoothness– 462.9 RESELs
© 2005 Thomas Nichols
55
t11 Statistic, RF & Bonf. Thresholdt11 Statistic, Nonparametric Threshold
uRF = 9.87uBonf = 9.805 sig. vox.
uPerm = 7.67
58 sig. vox.
Smoothed Variance t Statistic,Nonparametric Threshold
378 sig. vox.
Test Level vs. t11 Threshold
© 2005 Thomas Nichols
Does this Generalize?RFT vs Bonf. vs Perm. No. Significant Voxels
(0.05 Corrected) t SmVar t df RF Bonf Perm Perm
Verbal Fluency 4 0 0 0 0 Location Switching 9 0 0 158 354 Task Switching 9 4 6 2241 3447 Faces: Main Effect 11 127 371 917 4088 Faces: Interaction 11 0 0 0 0 Item Recognition 11 5 5 58 378 Visual Motion 11 626 1260 1480 4064 Emotional Pictures 12 0 0 0 7 Pain: Warning 22 127 116 221 347 Pain: Anticipation 22 74 55 182 402
© 2005 Thomas Nichols
58
Overview
• Multiple Testing Problem– Which of my 100,000 voxels are “active”?
• Nonparametric Inference– Can I trust my P-value at this voxel?
• Conjunction Inference– Is this voxel “active” in all of my tasks?
• Bayesian vs. Classical Inference– What in the world is a posterior probability?
© 2005 Thomas Nichols
59
Conjunction Inference
• Consider several working memory tasks– N-Back tasks with different stimuli– Letter memory: D J P F D R A T F M R I B K– Number memory: 4 2 8 4 4 2 3 9 2 3 5 8 9 3 1 4– Shape memory:
• Interested in stimuli-generic response– What areas of the brain respond to all 3 tasks?– Don’t want areas that only respond in 1 or 2 tasks
© 2005 Thomas Nichols
61
Conjunction Inference• For working memory example, K=3...
– Letters H1 T1
– Numbers H2 T2
– Shapes H3 T3
– Test
• At least one of the three effects not present
– versus
• All three effects present
}0{}0{}0{: 3210 HHHH
}1{}1{}1{: 321 HHHH A
© 2005 Thomas Nichols
64
Conjunction Inference Methods: Friston et al
• Use the minimum of the K statistics– Idea: Only declare a conjunction if all of the
statistics are sufficiently large– only when for all kuTk
kmin uTk
© 2005 Thomas Nichols
71
Valid Conjunction Inference With the Minimum Statistic
• For valid inference, compare min stat to u– Assess mink Tk image as if it were just T1
– E.g. u0.05=1.64 (or some corrected threshold)
• Equivalently, take intersection mask– Thresh. each statistic image at, say, 0.05 FWE corr.– Make mask: 0 = below thresh., 1 = above thresh.– Intersection of masks: conjunction-significant voxels
© 2005 Thomas Nichols
72
Overview
• Multiple Testing Problem– Which of my 100,000 voxels are “active”?
• Nonparametric Inference– Can I trust my P-value at this voxel?
• Conjunction Inference– Is this voxel “active” in all of my tasks?
• Bayesian vs. Classical Inference– What in the world is a posterior probability?
© 2005 Thomas Nichols
73
Classical Statistics: Model
Y
Likelihood of Y
p(Y|)
• Estimation– n = 12 subj. fMRI study
• Data at one voxel– Y, sample average
% BOLD change
• Model– Y ~ N(, /√n) is true population
mean BOLD % change
– Likelihood p(Y|) • Relative frequency of
observing Y for one given value of
© 2005 Thomas Nichols
74
Classical Statistics: MLE
• Estimating – Don’t know in
practice
• Maximum Likelihood Estimation– Find that makes
data most likely– The MLE ( ) is
our estimate of
Y
Likelihood of Y
p(Y|)
Actual y observed in the experiment
y
– Here, the MLE of the population mean is simply the BOLD sample mean, y
© 2005 Thomas Nichols
75
Classical Statistical Inference
• Level 95% Confidence Interval– Y ± 1.96/√n– With many
replications of the experiment, CI will contain 95% of the time
Y
Likelihood of Y
p(Y|)
CI observed
© 2005 Thomas Nichols
76
Classical Statistics Redux
• Grounded in long-run frequency of observable phenomena– Data, over theoretical replications– Inference: Confidence intervals, P-values
• Estimation based on likelihood
• Parameters are fixed– Can’t talk about probability of parameters– P( Pop mean > 0 ) ???
• True population mean % BOLD is either > 0 or not
• Only way to know is to scan everyone in population
© 2005 Thomas Nichols
77
Bayesian Statistics
• Grounded in degrees of belief– “Belief” expressed with the grammar of
probability– No problem making statements about
unobservable parameters• Parameters are regarded random, not fixed
• Data is regarded as fixed, since you only have one dataset
© 2005 Thomas Nichols
78
Bayesian Essentials• Prior Distribution
– Expresses belief on parameters before seeing the data
• Likelihood– Same as with Classical
• Posterior Distribution– Expresses belief on parameters after the seeing the data
• Bayes Theorem– Tells how to combine prior with likelihood (data) to
create posterior
)()|(')'()'|(
)()|()|(
pyp
dpyp
pypyp
Posterior Likelihood Prior
© 2005 Thomas Nichols
79
Bayesian Statistics:From Prior to Posterior
• Prior p( ) ~ N(0 , ) 0 = 0 %: a priori belief
that activation & de-activation are equally likely
= 1 % : a priori belief that activation is small
• Data: y = 5 %• Posterior
– Somewhere between prior and likelihood
Prior
LikelihoodPosterior
0-50
5 10
Population Mean % BOLD Change
© 2005 Thomas Nichols
80
Bayesian Statistics:Posterior Inference
• All Inference based on posterior
• E.g. Posterior Mean(instead of MLE)
Prior
LikelihoodPosterior
0-50
5 10
Population Mean % BOLD Change
yn
n
n /11
/1
0
/11
1
22
2
22
2
PriorMean
Data (Sample mean)Posterior
Mean
0 y
– Weighted sum of prior & data mean
– Weights based on prior & data precision
© 2005 Thomas Nichols
81
Bayesian Inference:Posterior Inference
• But posterior is just another distribution– Can ask any probability question
• E.g. “What’s the probability, after seeing the data, that > 0”, or P( > 0 | y )– Here P( > 0 | y ) ≈ 1
• “Credible Intervals”– Here 4 ± 0.9 has
95% posterior prob.– No reference to
repetitions of theexperiment
Posterior
0-50
5 10
Population Mean % BOLD Change
© 2005 Thomas Nichols
82
Bayesian vs. Classical• Foundations
– ClassicalHow observable statistics behave in long-run
– BayesianMeasuring belief about unobservable parameters
• Inference– Classical
References other possible datasets not observed• Requires awkward explanations for CI’s & P-values
– BayesianBased on posterior, combination of prior and data
• Allows intuitive probabilistic statements (posterior probabilities)
© 2005 Thomas Nichols
83
Bayesian vs. Classical• Bayesian Challenge: Priors
– I can set my prior to always find a result– “Objective” priors can be found; results then often
similar to Classical inference
• When are the two similar?– When n large, the prior can be overwhelmed by
likelihood– One-sided P-value ≈ Posterior probability of > 0 – Doesn’t work with 2-sided P-value!
[ P( 0 | y ) = 1 ]
© 2005 Thomas Nichols
84
Bayesian vs. ClassicalSPM T vs SPM PPM
• Auditory experiment
SPM:Voxels with T > 5.5
PPM:Voxels with Posterior Probability > 0.95
Slide: Alexis Roche, CEA, SHFJ
SPM
mip
[0, 0
, 0]
<
< <
SPM{T39.0
}
SPMresults:Height threshold T = 5.50
Extent threshold k = 0 voxelsDesign matrix
1 4 7 10 13 16 19 22
147
1013161922252831343740434649525560
contrast(s)
3
SP
Mm
ip[0
, 0, 0
]
<
< <
PPM 2.06
SPMresults:Height threshold P = 0.95
Extent threshold k = 0 voxelsDesign matrix
1 4 7 10 13 16 19 22
147
1013161922252831343740434649525560
contrast(s)
4
• Qualitatively similar, but hard to equate thresholds
© 2005 Thomas Nichols
85
Conclusions
• Multiple Testing Problem– Choose a MTP metric (FDR, FWE)– Use a powerful method that controls the metric
• Nonparametric Inference– More power for small group FWE inferences
• Conjunction Inference– Use intersection mask, or treat mink Tk as single T
• Bayesian Inference– Conceptually different, but simpler than Classical– Priors controversial, but objective ones can be used
© 2005 Thomas Nichols
86
References• Multiple Testing Problem
– Worsley, Marrett, Neelin, Vandal, Friston and Evans, A Unified Statistical Approach for Determining Significant Signals in Images of Cerebral Activation. Human Brain Mapping, 4:58-73, 1996.
– Nichols & Hayasaka, Controlling the Familywise Error Rate in Functional Neuroimaging: A Comparative Review. Statistical Methods in Medical Research, 12:419-446, 2003.
– CR Genovese, N Lazar and TE Nichols. Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate. NeuroImage, 15:870-878, 2002.
• Nonparametric Inference– TE Nichols and AP Holmes. Nonparametric Permutation Tests for Functional Neuroimaging: A Primer with
Examples. Human Brain Mapping, 15:1-25, 2002.– Bullmore, Long and Suckling. Colored noise and computational inference in neurophysiological (fMRI) time
series analysis: resampling methods in time and wavelet domains. Human Brain Mapping, 12:61-78, 2001.
• Conjunction Inference– TE Nichols, M Brett, J Andersson, TD Wager, J-B Poline. Valid Conjunction Inference with the Minimum
Statistic. NeuroImage, 2005.– KJ Friston, WD Penny and DE Glaser. Conjunction Revisited. NeuroImage, NeuroImage 25:661– 667, 2005.
• Bayesian Inference– L.R. Frank, R.B. Buxton, E.C. Wong. Probabilistic analysis of functional magnetic resonance imaging data.
Magnetic Resonance in Medicine, 39:132–148, 1998.– Friston,, Penny, Phillips, Kiebel, Hinton and Ashbuarner, Classical and Bayesian inference in neuroimagining:
theory. NeuroImage, 16: 465-483, 2002. (See also, 484-512)– Woolrich, M., Behrens, T., Beckmann, C., Jenkinson, M., and Smith, S. (2004). Multi-Level Linear Modelling
for FMRI Group Analysis Using Bayesian Inference. NeuroImage, 21(4):1732-1747