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## Group Analysis

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**FMRI Analysis**Experiment Design Scanning Pre-Processing Individual Subject Analysis Group Analysis Post-Processing**Group Analysis**Background Basics ANOVA Program Contrasts Design 3dttest, 3dANOVA/2/3, 3dRegAna Conjunction Cluster Analysis MTC Clusters 3dmerge, 3dclust 3dcalc AlphaSim, 3dFDR Simple Correlation Connectivity Analysis Context-Dependent Correlation Path Analysis**Group Analysis: types**• Fixed effect • Only a few subjects • Case study: can’t generalize to whole population • Simple approach: T = ∑ti/sqrt(n) • Sophisticated approach • B = ∑[bi/sqrt(vi)]/∑[1/sqrt(vi)], T = B∑[1/sqrt(vi)]/sqrt(n) • B = ∑(bi/vi)/∑(1/vi), T = Bsqrt[∑(1/vi)] • Concatenate individual subject data • Random effect • Individual subject and group analysis: separate • Assumption: within-subject variation is negligible compared to between-subjects • Focus of this talk • Mixed effect • Bring within-subject variances to group analysis • Currently not easy to do in FMRI analysis**Group Analysis: Overview**• Parametric Tests • 3dttest (one-sample, unpaired and paired t) • 3dANOVA (one-way between-subject) • 3dANOVA2 (one-way within-subject, 2-way between-subjects) • 3dANOVA3 (2-way between-subjects, within-subject and mixed, 3-way between-subjects) • 3dRegAna (regression/correlation, unbalanced ANOVA, ANCOVA) • GroupAna (Matlab script for up to 5-way ANOVA) • Non-Parametric Analysis • No assumption of normality; Statistics based on ranking • Appropriate when number of subjects too few • Programs • 3dWilcoxon (~ paired t-test) • 3dMannWhitney (~ two-sample t-test) • 3dKruskalWallis (~3dANOVA) • 3dFriedman (~3dANOVA2) • Permutation test plugin on AFNI under Define Datamode / Plugins / C program by Tom Holroyd • Can’t handle complicated designs • Less sensitive to outliers (more robust) and less flexible than parametric tests**Group Analysis: Overview**• How many subjects? • Power: proportional to √n; n > 10 • Efficiency increases by the square root of # subjects • Balance: Equal number of subjects across groups if possible • Input • % signal change (not statistics) or normalized variables • HRF magnitude: Regression coefficients • Contrasts • Normalized values: connectivity analysis • Common brain in tlrc space (resolution doesn’t have to be 1x1x1 mm3 ) • Design • Number of factors • Number of levels for each factor • Within-subject / repeated-measures vs. between-subjects • Fixed (factors of interest) vs. random (subject) • Nesting: Balanced? • Which program? • Contrasts and trend analysis: One-tail or two-tail?**Group Analysis : 3dttest**• Basic usage • One-samplet • One group: simple effect • Example: 15 subjects under condition A with H0:μA= 0 • Two-samplet • Two groups: Compare one group with another • ~ 1-way between-subject (3dANOVA) • Unequal sample sizes allowed • Assumption of equal variance • Example: 15 subjects under A and 13 other subjects under B - H0:μA = μB • Pairedt • Two conditions of one group: Compare one condition with another • ~ one-way within-subject (3dANOVA2 -type 3) • ~ one-sample t on individual contrasts • Example: Difference between conditions A and B for 15 subjects with H0:μA = μB • Output: 2 values (% and t) at each voxel • Versatile program: Most tests can be done with 3dttest -piecemeal vs. bundled**Group Analysis: 3dANOVA**• Generalization of two-sample t-test • One-way between-subject • H0: no difference across all levels (groups) • Examples of groups: gender, age, genotype, disease, etc. • Unequal sample sizes allowed • Assumptions • Normally distributed with equal variances across groups • Results: 2 values (% and t) • 3dANOVA vs. 3dttest • Equivalent with 2 levels (groups) • More than 2 levels (groups): Can run multiple two-sample t-test**Group Analysis: 3dANOVA2**• Designs • One-way within-subject (type 3) • Major usage • Compare conditions in one group • Extension and equivalence of paired t • Two-way between-subjects (type 1) • 1 condition, 2 classifications of subjects • Extension and equivalence two-sample t • Unbalanced designs disallowed: Equal number of subjects across groups • Output • Main effect (-fa): F • Interaction for two-way between-subjects (-fab): F • Contrast testing • Simple effect (-amean) • 1st level (-acontr, -adiff): one-sample or paired t among factor levels • 2nd level (interaction) for two-way between-subjects • 2 values per contrast: % and t**Group Analysis:3dANOVA3**• Designs • Three-way between-subjects (type 1) • 3 categorizations of groups • Two-way within-subject (type 4): Crossed design AXBXC • Generalization of paired t-test • One group of subjects • Two categorizations of conditions: A and B • Two-way mixed (type 5): Nested design BXC(A) • Nesting factor: ≥ 2 groups of subjects (Factor A): subject classification, e.g., gender • One category of condition (Factor B) • Nesting: balanced • Output • Main effect (-fa and -fb) and interaction (-fab): F • Contrast testing • 1st level: -amean, -adiff, -acontr, -bmean, -bdiff, -bcontr • 2nd level: -abmean, -aBdiff, -aBcontr, -Abdiff, -Abcontr • 2 values per contrast : % and t**Group Analysis: GroupAna**• Multi-way ANOVA • Matlab script package for up to 5-way ANOVA • Requires Matlab plus Statistics Toolbox • GLM approach (slow) • Powerful: Test for interactions • Downside • Difficult to test and interpret simple effects/contrasts • Complicated design, and compromised power • Heavy duty computation: minutes to hours • Input with lower resolution recommended • Resample with adwarp -dxyz # or 3dresample • Can handle both volume and surface data • Can handle following unbalanced designs (two-sample t type): • 3-way ANOVA type 3: BXC(A) • 4-way ANOVA type 3: BXCXD(A) • 4-way ANOVA type 4: CXD(AXB) • See http://afni.nimh.nih.gov/sscc/gangc for more info • Alternative: 3dRegAna**Group Analysis: Example**• Design • 4 conditions (TM, TP, HM, HP) and 8 subjects • 2-way within-subject: 2x2x8 • A (Object), 2 levels: Tool vs Human • B (Animation), 2 levels: Motion vs Point • C (subject), 8 levels • AxBxC: Program?3dANOVA3 -type 4 • Main effects (A and B): 2 F values • Interaction AXB: 1 F • Contrasts • 1st order: TvsH, MvsP • 2nd order: TMvsTP, HMvsHP, TMvsHM, TPvsHP • 6 contrasts x 2 values/contrast = 12 values • Logistic • Input: 2x2x8 = 32 files (4 from each subject) • Output: 18 subbricks**Model type, number of**levels for each factor Input for each cell in ANOVA table: totally 2X2X8 = 32 • Group Analysis: Example • Script 3dANOVA3 -type 4 -alevels 2 -blevels 2 -clevels 8 \ -dset 1 1 1 ED_TM_irf_mean+tlrc \ -dset 1 2 1 ED_TP_irf_mean+tlrc \ -dset 2 1 1 ED_HM_irf_mean+tlrc \ -dset 2 2 1 ED_HP_irf_mean+tlrc \ … -adiff 1 2 TvsH1 \ (indices for difference) -acontr 1 -1 TvsH2 \ (coefficients for contrast) -bdiff 1 2 MvsP1 \ -aBdiff 1 2 : 1 TMvsHM \ (indices for difference) -aBcontr 1 -1 : 1 TMvsHM \ (coefficients for contrast) -aBcontr -1 1 : 2 HPvsTP \ -Abdiff 1 : 1 2 TMvsTP \ -Abcontr 2 : 1 -1 HMvsHP \ -fa ObjEffect \ -fb AnimEffect \ -fab ObjXAnim \ -bucket Group 1st order Contrasts, paired t test 2nd order Contrasts, paired t test Main effects & interaction F test; Equivalent to contrasts Output: bundled**Group Analysis: Example**• Alternative approaches • GroupAna • 3dRegAna • Paired t: 6 tests • Program: 3dttest -paired • For TM vs HM: 16 (2x8) input files (β coefficients: %) from each subject 3dttest -paired -prefix TMvsHM \ -set1 ED_TM_irf_mean+tlrc ... ZS_TM_irf_mean+tlrc \ -set2 ED_HM_irf_mean+tlrc ... ZS_HM_irf_mean+tlrc • One-sample t : 6 tests • Program: 3dttest • For TM vs HM: 8 input files (contrasts: %) from each subject 3dttest -prefix TMvsHM \ -base1 0 \ -set2 ED_TMvsHM_irf_mean+tlrc ... ZS_TMvsHM_irf_mean+tlrc**Group Analysis: ANCOVA (ANalysis of COVAriances)**• Why ANCOVA? • Subjects might not be an ideally randomized representation of a population • If no controlled, cross-subject variability will lead to loss of power and accuracy • Direct control: balanced selection of subjects • Indirect (statistical) control: untangling covariate effect • Factor of no interest - covariate: uncontrollable/confounding variable, usually continuous • Age • Behavioral data, e.g., response time • Cortex thickness • Gender • ANCOVA = Regression + ANOVA • Assumption: linear relation between % signal change and the covariate • GLM approach • Centralize covariate so that it would not confound with other effects • 3dRegAna • Flexible program that can run all sorts of group analysis • Miserable to write script, but hopeful: python scripting in future**Group Analysis: ANCOVA Example**• Example: Running ANCOVA • Two groups: 15 normal vs. 13 patients • Analysis: comparing the two groups • Running what test? • Two-sample t with 3dttest • Controlling age effect? • GLM model • Yi = β0 + β1X1i + β2X2i +β3X3i +εi, i = 1, 2, ..., n (n = 28) • Demean covariate (age) X1 • Code the factor (group) with a dummy variable 0, when the subject is a patient; X2i = { 1, when the subject is normal. • With covariate X1 centralized: β0 = effect of patient; β1 = age effect (correlation coef); β2 = effect of normal • X3i= X1i X2i models interaction (optional) between covariate and factor (group) β3 = interaction**Model parameters: 28 subjects,**3 independent variables Memory Input: Covariates, factor levels, interaction, and input files • Group Analysis: ANCOVA Example 3dRegAna -rows 28 -cols 3 \ -workmem 1000 \ -xydata 0.1 0 0 patient/Pat1+tlrc.BRIK \-xydata 7.1 0 0 patient/Pat2+tlrc.BRIK \… -xydata 7.1 0 0 patient/Pat13+tlrc.BRIK \-xydata 2.1 1 2.1 normal/Norm1+tlrc.BRIK \-xydata 2.1 1 2.1 normal/Norm2+tlrc.BRIK \… -xydata -8.9 1 -8.9 normal/Norm14+tlrc.BRIK \-xydata 0.1 1 0.1 normal/Norm15+tlrc.BRIK \ -model 1 2 3 : 0 \ -bucket 0 Pat_vs_Norm \ -brick 0 coef 0 ‘Pat’ \-brick 1 tstat 0 ‘Pat t' \-brick 2 coef 1 'Age Effect' \-brick 3 tstat 1 'Age Effect t' \-brick 4 coef 2 'Norm-Pat' \-brick 5 tstat 2 'Norm-Pat t' \-brick 6 coef 3 'Interaction' \-brick 7 tstat 3 'Interaction t' See http://afni.nimh.nih.gov/sscc/gangc/ANCOVA.html for more information Specify model for F and R2 Output: #subbriks = 2*#coef + F + R2 Label output subbricks for β0,β1,β2,β3**Group Analysis: A Complicated ANCOVA Example**3 groups, 13 subjects in each group; 2 conditions; 1 covariate 3dANOVA3 –type 5 if no covariate 3dRegAna -workmem 2000 -rows 78 -cols 42 \ -xydata 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 121 gs.acw.cf+tlrc \ -xydata 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 96 ew.acw.cf+tlrc \ … \ -xydata 1 0 1 1 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 115 kb.acw.cf+tlrc \ -xydata 1 0 -1 -1 -0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 121 gs.aril.cf+tlrc \ -xydata 1 0 -1 -1 -0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 96 ew.aril.cf+tlrc \ … \ -xydata 1 0 -1 -1 -0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 115 kb.aril.cf+tlrc \ -xydata 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 100 bd.acw.cf+tlrc \ -xydata 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 106 05.acw.cf+tlrc \ … \ -xydata 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 78 dw.acw.cf+tlrc \ -xydata 0 1 -1 -0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 100 bd.aril.cf+tlrc \ -xydata 0 1 -1 -0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 106 05.aril.cf+tlrc \ … \ -xydata 0 1 -1 -0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 78 dw.aril.cf+tlrc \ -xydata -1 -1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 86 pw.acw.cf+tlrc \ -xydata -1 -1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 109 an.acw.cf+tlrc \ …\ -xydata -1 -1 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 91 jf.acw.cf+tlrc \ -xydata -1 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 86 pw.aril.cf+tlrc \ -xydata -1 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 109 an.aril.cf+tlrc \ … \ -xydata -1 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 91 jf.aril.cf+tlrc \ -model 1 2 : 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 \ -bucket 0 GrpEff**Group Analysis**Background Basics Conjunction MTC Cluster Analysis Clusters 3dmerge, 3dclust 3dcalc AlphaSim, 3dFDR ANOVA Program Contrasts Design 3dttest, 3dANOVA/2/3, 3dRegAna Simple Correlation Connectivity Analysis Context-Dependent Correlation Path Analysis**Cluster Analysis: Multiple testing correction**• 2 types of errors in statistical tests • What is H0 in FMRI studies? • Type I = P (reject H0|when H0is true) = false positive = p value Type II = P (accept H0|when H1is true) = false negative = β • Usual strategy: controlling type I error (power = 1- β = probability of detecting true activation) • Significance level = α: p < α • Family-Wise Error (FWE) • Birth rate H0: sex ratio at birth = 1:1 • What is the chance there are 5 boys (or girls) in a family? (2)5 ~ 0.03 • In a community with 100 families with 5 kids, expected #families with 5 boys =? 100X(2)5 ~ 3 • In fMRI H0: no activation at a voxel • What is the chance a voxel is mistakenly labeled as activated (false +)? • Multiple testing problem: With n voxels, what is the chance to mistakenly label at least one voxel? Family-Wise Error: αFW = 1-(1- p)n →1 as n increases • Bonferroni correction: αFW = 1-(1- p)n ~ np, if p << 1/n Use p=α/n as individual voxel significance level to achieve αFW = α**Cluster Analysis: Multiple testing correction**• Multiple testing problem in fMRI: voxel-wise statistical analysis • Increase of chance at least one detection is wrong in cluster analysis • 3 occurrences of multiple testings: individual, group, and conjunction • Group analysis is the most concerned • Two approaches • Control FWE: αFW = P (≥ one false positive voxel in the whole brain) • Making αFW small but without losing too much power • Bonferroni correction doesn’t work: p=10-8~10-6 *Too stringent and overly conservative: Lose statistical power • Something to rescue? Correlation and structure! *Voxels in the brain are not independent *Structures in the brain • Control false discovery rate (FDR) • FDR = expected proportion of false + voxels among all detected voxels • Concrete example: individual voxel p = 0.001 for a brain of 25,000 EPI voxels • Uncorrected → 25 false + voxels in the brain • FWE: corrected p = 0.05 → 5% false + hypothetical brains for a fixed voxel location • FDR: corrected p = 0.05 → 5% voxels in those positively labeled ones are false +**Cluster Analysis: AlphaSim**• FWE: Monte Carlo simulations • Named for Monte Carlo, Monaco, where the primary attractions are casinos • Program: AlphaSim • Randomly generate some number (e.g., 1000) of brains with white noise • Count the proportion of voxels are false + in all brains • Parameters: * ROI - mask * Spatial correlation - FWHM * Connectivity - radium * Individual voxel significant level - uncorrected p • Output * Simulated (estimated) overall significance level (corrected p-value) * Corresponding minimum cluster size • Decision: Counterbalance among * Uncorrected p • * Minimum cluster size • * Corrected p**Program**Restrict correcting region: ROI Spatial correlation • Cluster Analysis: AlphaSim • See detailed steps at http://afni.nimh.nih.gov/sscc/gangc/mcc.html • Example AlphaSim \ -mask MyMask+orig \ -fwhmx 4.5 -fwhmy 4.5 -fwhmz 6.5 \ -rmm 6.3 \ -pthr 0.0001 \ -iter 1000 • Output: 5 columns • Focus on the 1st and last columns, and ignore others • 1st column: minimum cluster size in voxels • Last column: alpha (α), overall significance level (corrected p value) Cl Size Frequency Cum Prop p/Voxel Max Freq Alpha2 1226 0.999152 0.00509459 831 0.859 5 25 0.998382 0.00015946 25 0.137 10 3 1.0 0.00002432 3 0.03 • May have to run several times with different uncorrected p: uncorrected p↑↔ cluster size↑ Connectivity: how clusters are defined Uncorrected p Number of simulations**Cluster Analysis: 3dFDR**• Definition: FDR = proportion of false + voxels among all detected voxels • Doesn’t consider • spatial correlation • cluster size • connectivity • Again, only controls the expected % false positives among declared active voxels • Algorithm: statistic (t) p value FDR (q value) z score • Example: 3dFDR -input ‘Group+tlrc[6]' \ -mask_file mask+tlrc \ -cdep -list \ -output test One statistic ROI Arbitrary distribution of p Output**Cluster Analysis: FWE or FDR?**Correct type I error in different sense FWE: αFW = P (≥ one false positive voxel in the whole brain) Frequentist’s perspective: Probability among many hypothetical activation brains Used usually for parametric testing FDR = expected % false + voxels among all detected voxels Focus: controlling false + among detected voxels in one brain More frequently used in non-parametric testing Fail to survive correction? At the mercy of reviewers Analysis on surface Tricks One-tail? ROI – cheating? Many factors along the pipeline Experiment design: power? Sensitivity (power) vs specificity (small regions) Poor spatial alignment among subjects**Cluster Analysis: Conjunction analysis**• Conjunction analysis: HM vs TM • Common activation area • Exclusive activations • Double/dual thresholding with AFNI GUI • Tricky • Only works for two contrasts • Common but not exclusive areas • Conjunction analysis with 3dcalc • Flexible and versatile • Heaviside unit (step function) defines a On/Off event**Cluster Analysis: Conjunction analysis**• Example with 3 contrasts: A vs D, B vs D, and C vs D • Map 3 contrasts based on binary system: A > D: 000(1); B > D: 010(2); C > D: 100(4) • Create a mask with 3 subbricks of t (threshold = 4.2) 3dcalc -a func+tlrc'[5]' -b func+tlrc'[10]' -c func+tlrc'[15]‘ \ -expr 'step(a-4.2)+2*step(b-4.2)+4*step(c-4.2)' \ -prefix ConjAna • Interpret output - 8 (=23) scenarios: 000(0): none; 001(1): A > D but no others; 010(2): B > D but no others; 011(3): A > D and B > D but not C > D; 100(4): C > D but no others; 101(5): A > D and C > D but not B > D; 110(6): B > D and C > D but not A > D; 111(7): A > D, B > D and C > D • Downside: no p associated conjunctions with and no MTC**Group Analysis**Background Basics Conjunction MTC Cluster Analysis Clusters 3dmerge, 3dclust 3dcalc AlphaSim, 3dFDR ANOVA Program Contrasts Design 3dttest, 3dANOVA/2/3, 3dRegAna Simple Correlation Connectivity Analysis Context-Dependent Correlation Path Analysis**Connectivity: Path Analysis**• Causal model approach on a network of ROI’s • Minimizing discrepancies • btw correlation based on data and one estimated from model • Input: Model specification, correlation matrix, residual error variances, DF • Output: Path coefficients, various fit indices**Connectivity: Path Analysis – 1dSEM**• AFNI program 1dSEM • Written in C • Not dependent on FMRI analysis platform • Two modes • Validate a theoretical model • Accept, reject, or modify the model? • Search for ‘best’ model • Start with a minimum model (can be empty) • Some paths can be excluded • ‘Best’ in terms of various fit criteria • Script: 1dSEM -theta testthetasfull.1D -C testcorr.1D -psi testpsi.1D -DF 30 • Pitfalls • Causal relationship modeled through correlation (covariance) analysis • Valid only with the data and model specified • If one critical ROI is left out, things may go awry • More details • http://afni.nimh.nih.gov/sscc/gangc/PathAna.html • 1dSEM -help**Need Help?**Command with “-help” 3dANOVA3 -help Manuals http://afni.nimh.nih.gov/afni/doc/manual/ Web http://afni.nimh.nih.gov/sscc/gangc Examples: HowTo#5 http://afni.nimh.nih.gov/afni/doc/howto/ Message board http://afni.nimh.nih.gov/afni/community/board/ Appointment Contact us @1-800-NIH-AFNI