Contents. comparison. Multiple comparison problem. Recap & Introduction Inference & multiple. «Take home» message. DISCOS SPM course, CRC, Liège, 2009

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1 DISCOS SPM course, CRC, Liège, 2009 Contents Multiple comparison problem Recap & Introduction Inference & multiple comparison «Take home» message C. Phillips, Centre de Recherches du Cyclotron, ULg, Belgium Based on slides from: T. Nichols image data parameter estimates Statistical Parametric Map corrected p-values Voxel by voxel statistics model specification parameter estimation realignment & motion motion correction General Linear Linear Model Model model model fitting fitting statistic statistic image image correction for for multiple comparisons Time Time hypothesis test statistic normalisation smoothing Random effect effectanalysis anatomical reference kernel design matrix Dynamic causal causal modelling, Functional & effective connectivity, PPI, PPI, single voxel time series Intensity statistic image or SPM

2 General Linear Model (in SPM) Contents Auditory words every 20s (Orthogonalised) Gamma functions ƒ i (u) of peristimulus time u Sampled every TR = 1.7s Design matrix, X [ƒ 1 (u) x(t) ƒ 2 (u) x(t)...] SPM{F} 0 time {secs{ secs} } 30 Recap & Introduction Inference & multiple comparison Single/multiple voxel inference Family wise error rate (FWER) False Discovery rate (FDR) SPM results «Take home» message Inference at a single voxel t-distribution u=2 NULL hypothesis, H: activation is zero α = p(t>u H) p-value: probability of getting a value of t at least as extreme as u. If α is small we reject the null hypothesis H. Hypothesis Testing Null Hypothesis H 0 Test statistic T t observed realization of T α level Acceptable false positive rate Level α = P( T>u α H 0 ) Threshold u α controls false positive rate at level α P-value Assessment of t assuming H 0 P( T > t H 0 ) Null Distribution of T u α α Prob. of of obtaining stat. as as large or or larger in in a new experiment P(Data Null) not P(Null Data) P-val t Null Distribution of T

3 Inference at a single voxel What we d d like t-distribution u=2 NULL hypothesis, H: activation is zero α = p(t>u H) We can choose u to ensure a voxel-wise significance level of α. This is called an uncorrected p-value, for reasons we ll see later. We can then plot a map of above threshold voxels. Don t t threshold, model the signal! Signal location? θˆ Mag. Estimates and CI s s on (x,y,z) ) location Signal magnitude? CI s s on % change Spatial extent? ˆ θ Loc. Estimates and CI s s on activation volume Robust to choice of cluster definition θˆ Ext....but this requires an explicit spatial model space What we need Need an explicit spatial model No routine spatial modeling methods exist High-dimensional mixture modeling problem Activations don t t look like Gaussian blobs Need realistic shapes, sparse representation Some work by Hartvig et al.,, Penny et al. Real-life 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

4 Voxel-level level Inference Retain voxels above α-level threshold u α Gives best spatial specificity The null hypothesis at a single voxel can be rejected Cluster-level level Inference Two step-process process Define clusters by arbitrary threshold u clus Retain clusters larger than α-level threshold k α u α space u clus space Significant Voxels No significant Voxels Cluster not significant k α k α Cluster significant Cluster-level level Inference Typically better sensitivity Worse spatial specificity The null hypothesis of entire cluster is rejected Only means that one or more of voxels in cluster active Set-level Inference Count number of blobs c Minimum blob size k Worst spatial specificity Only can reject global null hypothesis u clus space u clus space Cluster not significant k α k α Cluster significant k k Here c = 1; only 1 cluster larger than k

5 Sensitivity and Specificity fmri Multiple Comparisons Problem TRUTH Don t Reject ACTION Reject H True (o) TN FP H False (x) FN TP At u1 Sens=10/10=100% Spec=7/10=70% At u2 Sens=7/10=70% Spec=9/10=90% Eg. t-scores from regions Sensitivity = TP/(TP+FN) = β that truly do and Specificity = TN/(TN+FP) = 1 - α u1 u2 do not activate FP = Type I error or error o o o o o o o x x x o o x x x o x x x x FN = Type II error α = p-value/fp rate/error rate/significance level β = power 4-Dimensional Data 1,000 multivariate observations, each with 100,000 elements 100,000 time series, each with 1,000 observations Massively Univariate Approach 2 100,000 hypothesis tests 1 Massive MCP! ,000 Inference for Images Multiple comparison problem Noise Use of uncorrected p-value, α=0.1 Signal Signal+Noise 11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5% Percentage of Null Pixels that are False Positives Using an uncorrected p-value of 0.1 will lead us to conclude on average that 10% of voxels are active when they are not. This is clearly undesirable : multiple comparison problem. To correct for this we can define a null hypothesis for images of statistics.

6 Multiple Comparisons Problem Which of 100,000 voxels are sig.? α=0.05 5,000 false positive voxels Which of (random number, say) say) 100 clusters significant? α= false positives clusters Assessing Statistic Images Where s s the signal? High Threshold Med. Threshold Low Threshold t > 5.5 t > 3.5 t > 0.5 t > 0.5 t > 1.5 t > 2.5 t > 3.5 t > 4.5 t > 5.5 t > 6.5 Good Specificity Poor Power (risk of false negatives) Poor Specificity (risk of false positives) Good Power t 59 Gaussian 10mm FWHM (2mm pixels) Multiple comparisons p = 0.05 Threshold at at p? expect (100 p)% by by chance Surprise? extreme voxel values voxel level level inference big big suprathreshold clusters cluster level level inference many suprathreshold clusters set level level inference Power & localisation sensitivity spatial specificity Solutions for Multiple Comparison Problem A MCP Solution must control False Positives How to measure multiple false positives? Familywise Error Rate (FWER) Chance of any false positives Controlled by Bonferroni,, Random Field Methods, non-parametric method (SnPM). False Discovery Rate (FDR) Proportion of false positives among rejected tests

7 Contents Recap & Introduction Inference & multiple comparison Single/multiple voxel inference Family wise error rate (FWER) Bonferroni correction/random Field Theory Non-parametric approach False Discovery rate (FDR) SPM results «Take home» message Family-wise Null Hypothesis FAMILY-WISE NULL HYPOTHESIS: Activation is zero everywhere Family of of hypotheses H k k k Ω= Ω = {1,,K} H Ω = H 1 1 H 2 2 H k k H K If we reject a voxel null hypothesis at any voxel, we reject the family-wise null hypothesis A FP anywhere gives a Family Wise Error (FWE) Family-wise error rate = corrected p-value FWE MCP Solutions: Controlling FWE w/ Max FWE & distribution of maximum FWE = P(FWE) = P( i i {T ii u} H o ) = P( max ii T i i u H o ) 100(1-α)%ile of max dist n controls FWE FWE = P( max ii T i i u α H o ) = α where u -1 α = F -1 max (1-α) max. α u α Multiple comparison problem Example: Experiment with 100,000 «voxels» and 40 d.f. type I error α=0.05 (5% risk) t α = ,000 t values 5000 t values > 1.68 just by chance! Familywise Error I test, P FWE : find threshold t α such that,, in a family of 100,000 t statistics, only 5% probability of one or more t values above that threshold Bonferroni correction: simple method to find the new threshold Random field theory: more accurate for functional imaging

8 The Bonferroni correction Given a family of N independent voxels and a voxel-wise error rate v The probability that all tests are below the threshold, i.e. that H o is true : (1- v) N The Family-Wise Error rate (FWE) or corrected error rate α is α = 1 (1-v) N ~ Nv (for (for small small v) v) Therefore, to ensure a particular FWE we choose v = α/n A Bonferroni correction is appropriate for independent tests. The Bonferroni correction Experiment with N = 100,000 «voxels» and 40 d.f. v = unknown corrected probability threshold, find v such that family-wise error rate α = 0.05 Bonferroni correction: probability that all tests are below the threshold, Use v = α / N Here v=0.05/100000= threshold t = 5.77 Interpretation: Bonferroni procedure gives a corrected p value, i.e. for a t statistics = 5.77, uncorrectd p value = corrected p value = 0.05 Use of uncorrected p-value, α=0.1 Bonferroni correction & independent observations Use of corrected p-value, α=0.1 FWE 100 by 100 voxels, with a z value independent measures Fix the P FWE = 0.05, z threshold? Bonferroni: v = 0.05 / = threshold z = by 100 voxels, with a z value. 100 independent measures Fix the P FWE = 0.05, z threshold? Bonferroni: v = 0.05 / 100 = threshold z = 3.29 v=α/n i where n i is the number of independent observations.

9 Bonferroni correction & smoothed observations Random Field Theory Consider a statistic image as a lattice representation of a continuous random field Use results from continuous random field theory 100 by 100 voxels, with a z value independent measures Fix the P FWE = 0.05, z threshold? Bonferroni: v = 0.05/10000 = threshold z = by 100 voxels, with a z value. How many independent measures? Fix the P FWE = 0.05, z threshold? Bonferroni? Lattice representation Euler Characteristic (EC) Topological measure threshold an image at u excursion set A u χ(α ) = u # blobs - # holes - At high u, χ(α ) = u # blobs Reject H Ω if if Euler characteristic non-zero α Pr(χ(Α ) > u 0 ) Expected Euler chararcteristic p value (at high u) u) α E[χ(Α )] u Euler characteristic Euler characteristic (EC) # blobs in a thresholded image. (True only for high threshold) EC = function of threshold used numberof resels where resels («resolution elements»)~ number of independent obsevations E[EC] P FWE

10 Euler characteristic Euler characteristic Threshold z-map at 2.50 EC = 3 Threshold z-map at 2.75 EC = 1 Euler characteristic (EC) # blobs in a thresholded image. (True only for high threshold) EC = function of threshold used numberof resels where resels («resolution elements»)~ number of independent obsevations E[EC] P FWE For a threshold z t at 2.50, E[EC] = 1.9 at 2.75, E[EC] = 1.1 instead of 3 and 1 as in example Expected Euler characteristic E[χ(Α )] λ(ω) Λ (u u -1) exp(-u 2 /2) / (2π) 2 Ω large search region ΩΩ R R 3 λ(ω) volume Λ smoothness A u excursion set A u = {x R 3 : Z(x) > u} Z(x) Gaussian random field x R 3 A u Ω + Multivariate Normal Finite Dimensional distributions + continuous + strictly stationary + marginal N(0,1) + continuously differentiable + twice differentiable at at 0 + Gaussian ACF (at (at least least near near local local maxima) Unified Theory General form for expected Euler characteristic χ 2,, F,, & ttfields restricted search regions α = Σ R d (Ω) ρ d (u) R d (Ω), RESEL count depends on : the search region how big, how smooth, what shape? Worsley et al. (1996), HBM A u Ω ρ d (υ): EC density depends on : type of field (eg. Gaussian, t) the threshold, u.

11 Unified Theory Estimated component fields General form for expected Euler characteristic χ 2,, F,, & ttfields restricted search regions α = Σ R d (Ω) ρ d (u) R d (Ω), d-dimensional Minkowski functional of Ω R 0 (Ω)=χ(Ω) Euler characteristic of Ω R 1 (Ω)=resel diameter R 2 (Ω)=resel surface area R 3 (Ω)=resel volume Worsley et al. (1996), HBM ρ d (u), d-dimensional EC density : E.g. Gaussian RF: ρ 0 (u) =1-Φ(u) ρ 1 (u) = (4 ln2) 1/2 exp(-u 2 /2) / (2π) ρ 2 (u) = (4 ln2) exp(-u 2 /2) / (2π) 3/2 A u Ω ρ 3 (u) = (4 ln2) 3/2 (u 2-1) exp(-u 2 /2) / (2π) 2 ρ 4 (u) = (4 ln2) 2 (u 3-3u) exp(-u 2 /2) / (2π) 5/2 voxels data matrix scans estimate design matrix = parameters? + errors? ^ β parameter estimates Each row is an estimated component field = residuals estimated variance estimated component fields Random Field Theory Smoothness Parameterization Smoothness, PRF, resels... RESELS Resolution Elements 1 RESEL = FWHM x FWHM y y FWHM zz RESEL Count R R = λ(ω) Λ Λ = (4log2) 3/2 3/2 λ(ω) / /( ( FWHM x x FWHM y y FWHM z z )) Volume of of search region in in units of of smoothness Eg: 10 voxels,, 2.5 FWHM 4 RESELS Beware RESEL misinterpretation RESEL are not number of of independent things in in the image See See Nichols & Hayasaka, 2003, Stat. Meth.. in in Med. Res... Smoothness Λ variance-covariance covariance matrix of of partial derivatives (possibly location Λ dependent) var Λ = cov cov x [ ] [ ] [ ] x cov x, y cov x, z [ ] [ ] [ ] x, y var y cov y, z [ ] [ ] [ ], cov, var Point Response Function PRF Full Width at Half Maximum FWHM. Approximate the peak of of the Covariance function with a Gaussian z y z z Gaussian PRF Σ kernel var/cov matrix ACF 2Σ Λ = (2Σ) -1-1 FWHM f = σ (8ln(2)) f xx 0 0 Σ Σ= = 0 f yy f 1 zz 8ln(2) ignoring covariances Λ = (4ln(2)) 3/2 3/2 / /(f( (f x x f y y f z z ) Resolution Element ((RESEL) Resel dimensions (f (f x x f y f z z ) R R 3 (Ω) ) = λ(ω) / /(f( (f x x f y f z z ) if if strictly stationary E[χ(Α )] u u )]= = R 3 (Ω) 3 (Ω) ) (4ln(2)) 3/2 3/2 (u (u 2 2-1) -1) exp(-u 2 2 /2) /2)/ /(2π) 2 2 R 3 (Ω) 3 (Ω)(1 (1 Φ(u)) for for high high thresholds u

12 RFT Assumptions Random Field Intuition Model fit & assumptions valid distributional results Multivariate normality of of component images Covariance function of component images must be - Can be nonstationary - Twice differentiable Smoothness smoothness» voxel size lattice approximation smoothness estimation practically FWHM 3 VoxDim otherwise conservative Typical applied smoothing: Single Subj fmri: 6mm PET: 12mm Multi Subj fmri: 8-12mm8 PET: 16mm Level of of smoothing should actually depend on what you re looking for Corrected P-value P for voxel value t P c = P(max T > t) E(χ t t ) λ(ω) ) Λ 1/2 t 2 exp(-t 2 /2) Statistic value t increases P c decreases (but only for large t) Search volume increases (bigger Ω) P c increases (more severe MCP) Smoothness increases (roughness Λ Λ 1/2 1/2 decreases) P c decreases (less severe MCP) Small Volume Correction Resel Counts for Brain Structures SVC = correction for multiple comparison in a user s s defined volume of interest. Shape and size of volume become important for small or oddly shaped volume! Example of SVC (900 voxels) compact volume: samples from maximum 16 resels spread volume: sample from up to 36 resels threshold higher for spread volume than compact volume. FWHM=20mm (1) Threshold depends on Search Volume (2) Surface area makes a large contribution

13 Summary Contents We should correct for multiple comparisons We can use Random Field Theory (RFT) or other methods RFT requires a good lattice approximation to underlying multivariate Gaussian fields, that these fields are continuous with a twice differentiable correlation function To a first approximation, RFT is a Bonferroni correction using RESELS. We only need to correct for the volume of interest. Depending on nature of signal we can trade-off anatomical specificity for signal sensitivity with the use of cluster-level inference. Recap & Introduction Inference & multiple comparison Single/multiple voxel inference Family wise error rate (FWER) Bonferroni correction/random Field Theory Non-parametric approach False Discovery rate (FDR) SPM results «Take home» message Nonparametric Permutation Test Permutation Test : Toy Example Parametric methods Assume distribution of statistic under null hypothesis Nonparametric methods Use data to find distribution of statistic under null hypothesis Any statistic! 5% Parametric Null Distribution 5% Nonparametric Null Distribution Data from V1 voxel in visual stim. experiment A: Active, flashing checkerboard B: Baseline, fixation 6 blocks, ABABAB Just consider block averages... A B A Null hypothesis H o No experimental effect, A & B labels arbitrary Statistic Mean difference B A B

14 Permutation Test : Toy Example Under H o Consider all equivalent relabelings Permutation Test : Toy Example Under H o Consider all equivalent relabelings Compute all possible statistic values AAABBB ABABAB BAAABB BABBAA AAABBB 4.82 ABABAB 9.45 BAAABB BABBAA AABABB ABABBA BAABAB BBAAAB AABABB ABABBA 6.97 BAABAB 1.10 BBAAAB 3.15 AABBAB ABBAAB BAABBA BBAABA AABBAB ABBAAB 1.38 BAABBA BBAABA 0.67 AABBBA ABBABA BABAAB BBABAA AABBBA ABBABA BABAAB BBABAA 3.25 ABAABB ABBBAA BABABA BBBAAA ABAABB 6.86 ABBBAA 1.48 BABABA BBBAAA Permutation Test : Toy Example Under H o Consider all equivalent relabelings Compute all possible statistic values Find 95%ile of permutation distribution Permutation Test : Toy Example Under H o Consider all equivalent relabelings Compute all possible statistic values Find 95%ile of permutation distribution AAABBB 4.82 ABABAB 9.45 BAAABB BABBAA AABABB ABABBA 6.97 BAABAB 1.10 BBAAAB 3.15 AABBAB ABBAAB 1.38 BAABBA BBAABA 0.67 AABBBA ABBABA BABAAB BBABAA 3.25 ABAABB 6.86 ABBBAA 1.48 BABABA BBBAAA

15 Permutation Test : Toy Example Under H o Consider all equivalent relabelings Compute all possible statistic values Find 95%ile of permutation distribution AAABBB 4.82 AABABB AABBAB AABBBA ABAABB 6.86 ABABAB 9.45 ABABBA 6.97 ABBAAB 1.38 ABBABA ABBBAA 1.48 BAAABB BAABAB 1.10 BAABBA BABAAB BABABA BABBAA BBAAAB 3.15 BBAABA 0.67 BBABAA 3.25 BBBAAA Controlling FWER: Permutation Test Parametric methods Assume distribution of max statistic under null hypothesis Nonparametric methods Use data to find distribution of max statistic under null hypothesis Again, any max statistic! 5% Parametric Null Max Distribution 5% Nonparametric Null Max Distribution Permutation Test & Exchangeability Permutation Test & Exchangeability Exchangeability is fundamental Def: Distribution of the data unperturbed by permutation Under H 0, exchangeability justifies permuting data Allows us to build permutation distribution Subjects are exchangeable Under Ho, each subject s A/B labels can be flipped Are fmri scans exchangeable under H o? If If no signal, can we permute over time? 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 of correlation E.g. Use wavelets to decorrelate Bullmore et al al 2001, HBM 12:61-78 Intersubject fmri permutation test Create difference image for each subject For each permutation, flip sign of some subjects

16 Permutation Test : Example fmri Study of Working Memory Active 12 subjects, block design Marshuetz et et al 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 UBKDA Baseline XXXXX... N yes no Permutation Test : Example Permute! 2 12 = 4,096 ways to flip 12 A/B labels For each, note maximum of t image. Permutation Distribution Maximum t Maximum Intensity Projection Thresholded t u Perm = sig. vox. t 11 Statistic, Nonparametric Threshold Test Level vs. t 11 Threshold u RF = 9.87 u Bonf = sig. vox. t 11 Statistic, RF & Bonf. Threshold Compare with Bonferroni α = 0.05/110,776 Compare with parametric RFT 110, mm voxels mm FWHM smoothness RESELs Does this Generalize? RFT vs Bonf. vs Perm. t Threshold (0.05 Corrected) df RF Bonf Perm Verbal Fluency Location Switching Task Switching Faces: Main Effect Faces: Interaction Item Recognition Visual Motion Emotional Pictures Pain: Warning Pain: Anticipation

17 RFT vs Bonf. vs Perm. No. Significant Voxels (0.05 Corrected) t df RF Bonf Perm Verbal Fluency Location Switching Task Switching Faces: Main Effect Faces: Interaction Item Recognition Visual Motion Emotional Pictures Pain: Warning Pain: Anticipation Performance Summary Bonferroni Not adaptive to smoothness Not so conservative for low smoothness Random Field Adaptive Conservative for low smoothness & df Permutation Adaptive (Exact) Old Conclusions Contents t random field results conservative for Low df & smoothness 9 df & 12 voxel FWHM; 19 df & < 10 voxel FWHM Bonferroni surprisingly satisfactory for low smoothness Nonparametric methods perform well overall Recap & Introduction Inference & multiple comparison Single/multiple voxel inference Family wise error rate (FWER) False Discovery rate (FDR) SPM results «Take home» message

18 False Discovery Rate False Discovery Rate Illustration: Don t Reject ACTION Reject At u1 FDR=3/17=18% α=3/10=30% Noise TRUTH H True (o) TN FP H False (x) FN TP At u2 FDR=1/12=8% α=1/10=10% Signal FDR = FP/(FP+TP) α = FP/(FP+TN) o o o o o o o x x x o o x x x o x x x x x x x x u1 u2 Eg. t-scores from regions that truly do and do not activate Signal+Noise Control of Per Comparison Rate at 10% Benjamini & Hochberg Procedure 11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5% Percentage of Null Pixels that are False Positives Control of Familywise Error Rate at 10% FWE Occurrence of Familywise Error Control of False Discovery Rate at 10% 6.7% 10.4% 14.9% 9.3% 16.2% 13.8% 14.0% 10.5% 12.2% 8.7% Percentage of Activated Pixels that are False Positives Select desired limit q on E(FDR) Order p-values, p p (1) p (2)... p (V) Let r be largest i such that p (i) i/v*q Reject all hypotheses corresponding to p (1),..., p (r). p (1) JRSS-B (1995) 57: p-value 0 1 i/v p (i) i/v q/c(v) 0 1

19 Benjamini & Hochberg: Varying Signal Extent Benjamini & Hochberg: Varying Signal Extent p = z = p = z = Signal Intensity 3.0 Signal Extent 1.0 Noise Smoothness Signal Intensity 3.0 Signal Extent 2.0 Noise Smoothness Benjamini & Hochberg: Varying Signal Extent Benjamini & Hochberg: Varying Signal Extent p = z = p = z = 3.48 Signal Intensity 3.0 Signal Extent 3.0 Noise Smoothness Signal Intensity 3.0 Signal Extent 5.0 Noise Smoothness 3.0 4

20 Benjamini & Hochberg: Varying Signal Extent p = z = 2.94 Benjamini & Hochberg: Varying Signal Extent p = z = 2.45 Signal Intensity 3.0 Signal Extent 9.5 Noise Smoothness Signal Intensity 3.0 Signal Extent16.5 Noise Smoothness Benjamini & Hochberg: Varying Signal Extent p = z = 2.07 Benjamini & Hochberg: Properties Adaptive Larger the signal, the lower the threshold Larger the signal, the more false positives False positives constant as fraction of rejected tests Not a problem with imaging s s sparse signals Smoothness OK Smoothing introduces positive correlations Signal Intensity 3.0 Signal Extent25.0 Noise Smoothness 3.0 7

21 Contents Summary: Levels of inference & power SPM intensity Recap & Introduction Inference & multiple comparison Single/multiple voxel inference Family wise error rate (FWER) False Discovery rate (FDR) SPM results «Take home» message h u L 1 L 2 SPM position : significant at the set level : significant at the cluster level L 1 > spatial extent threshold : significant at the voxel level L 2 < spatial extent threshold Sensitivity Test based on The intensity of a voxel The spatial extent above u or the spatial extent and the maximum peak height The number of clusters above u with size greater than n The sum of square of the SPM or a MANOVA Parameters set by the user Low pass filter Low pass filter intensity threshold u Low pass filter intensity thres. u spatial threshold n Low pass filter Regional specificity Levels of inference voxel-level level P(c 1 n 0, t 4.37) = (corrected) P(t 4.37) = 1 - Φ{4.37} < (uncorrected) t=4.37 n=82 n=12 omnibus P(c 7 7 n 0, t 3.09) = set-level P(c 3 n 12, t 3.09) = SPM results... n=32 cluster-level level P(c 1 n 82, t 3.09) = (corrected) P(n 82 t 3.09) = (uncorrected) Parameters u k - 12 voxels S voxels FWHM voxels D - 3

22 SPM results... SPM results... fmri : Activations significant at voxel and cluster level Contents Conclusions: FWER vs FDR Must account for multiplicity Otherwise have a fishing expedition Recap & Introduction Inference & multiple comparison «Take home» message FWER Very specific, less sensitive FDR Less specific, more sensitive Trouble with cluster inference

23 More Power to Ya! Statistical Power the probability of rejecting the null hypothesis when it is actually false if there s an effect, how likely are you to find it? Effect size bigger effects, more power e.g., MT localizer (moving rings - stationary runs) -- 1 run is usually enough looking for activation during imagined motion might require many more runs Sample size larger n, more power more subjects - longer runs - more runs Signal:Noise Ratio better SNR, more power stronger, cleaner magnet - more focal coil - fewer artifacts - more filtering