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FMRI Analysis Experiment Design Scanning Pre-Processing Individual Subject Analysis Group Analysis Post-Processing -1 - FMRI Analysis Experiment Design Scanning Pre-Processing Individual Subject Analysis Group Analysis Post-Processing -1 -

Group Analysis Basic analysis Design Program Contrasts 3 dttest, 3 d. ANOVA/2/3, 3 d. Group Analysis Basic analysis Design Program Contrasts 3 dttest, 3 d. ANOVA/2/3, 3 d. Reg. Ana, Group. Ana, 3 d. LME Simple Correlation 3 d. Deconvolve Connectivity Analysis Context-Dependent Correlation Path Analysis 3 d. Decovolve 1 d. SEM -2 -

 • Group Analysis: Basic concepts H Group analysis å Make general conclusions about • Group Analysis: Basic concepts H Group analysis å Make general conclusions about some population å Partition/untangle H Why two tiers of analysis? å High computation cost å Within-subject H data variability into various sources variation relatively small compared to cross-subject Mess in terminology å Fixed: factor, analysis/model/effects Ø Fixed-effects å Random: analysis (sometimes): averaging a few subjects factor, analysis/model/effects Ø Random-effects analysis (sometimes): subject as a random factor But really a mixed-effects analysis å Mixed: design, model/effects Ø Mixed design: crossed [e. g. , AXBXC] and nested [e. g. , BXC(A)] Psychologists: Within-subject (repeated measures) / between-subjects factor Ø Mixed-effects: model with both types of factors; model with both inter/intra-subject variances -3 -

 • Group Analysis: Basic concepts H Fixed factor å Treated as a fixed • Group Analysis: Basic concepts H Fixed factor å Treated as a fixed variable in the model Ø Categorization Ø Group å All of experiment conditions (mode: Face/House) of subjects (male/female, normal/patient) levels of the factor are of interest and included for all replications å Fixed in the sense inferences Ø apply Ø don’t H only to the specific levels of the factor extend to other potential levels that might have been included Random factor å Exclusively å Treated as a random variable in the model Ø average å Each subject in FMRI + random effects uniquely attributable to each subject: N(0, σ2) subject is of NO interest å Random in the sense Ø subjects serve as a random sample of a population Ø inferences can be generalized to a population -4 -

 • Group Analysis: Types H Averaging across subjects (fixed-effects analysis) å Number å • Group Analysis: Types H Averaging across subjects (fixed-effects analysis) å Number å Case study: can’t generalize to whole population å Simple ØT of subjects n < 6 approach (3 dcalc) = ∑tii/√n å Sophisticated approach ØB = ∑(bi/√vi)/∑(1/√vi), T = B∑(1/√vi)/√n, vi = variance for i-th regressor ØB = ∑(bi/vi)/∑(1/vi), T = B√[∑(1/vi)] Ø Combine H individual data and then run regression Mixed-effects analysis å Number å Random of subjects n > 10 effects of subjects å Individual and group analyses: separate å Within-subject å Main variation ignored focus of this talk -5 -

 • Group Analysis: Programs in AFNI H Non-parametric analysis å 4 < number • Group Analysis: Programs in AFNI H Non-parametric analysis å 4 < number of subjects < 10 å No assumption of normality; statistics based on ranking å Programs Ø 3 d. Wilcoxon (~ paired t-test) Ø 3 d. Mann. Whitney (~ two-sample t-test) Ø 3 d. Kruskal. Wallis (~ between-subjects with 3 d. ANOVA) Ø 3 d. Friedman (~one-way within-subject with 3 d. ANOVA 2) Ø Permutation test å Multiple testing correction with FDR (3 d. FDR) å Less sensitive to outliers (more robust) å Less flexible than parametric tests å Can’t handle complicated designs with more than one fixed factor -6 -

 • Group Analysis: Programs in AFNI H Parametric tests (mixed-effects analysis) å Number • Group Analysis: Programs in AFNI H Parametric tests (mixed-effects analysis) å Number of subjects > 10 å Assumption: Gaussian random effects å Programs Ø 3 dttest (one-sample, two-sample and paired t) Ø 3 d. ANOVA (one-way between-subject) Ø 3 d. ANOVA 2 (one-way within-subject, 2 -way between-subjects) Ø 3 d. ANOVA 3 (2 -way within-subject and mixed, 3 -way between-subjects) Ø 3 d. Reg. Ana (regression/correlation, simple unbalanced ANOVA, simple ANCOVA) Ø Group. Ana (Matlab package for up to 5 -way ANOVA) Ø 3 d. LME (R package for all sorts of group analysis) -7 -

 • Group Analysis: Planning H How many subjects? å Power/efficiency: å Balance: H • Group Analysis: Planning H How many subjects? å Power/efficiency: å Balance: H proportional to √n; n > 10 Equal number of subjects across groups if possible Input files å Common å% brain in tlrc space (resolution doesn’t have to be 1 x 1 x 1 mm 3 ) signal change (not statistics) or normalized variables Ø HRF magnitude: Regression coefficients Ø Contrasts H Design å Number of factors å Number of levels for each factor å Factor types Ø Fixed (factors of interest) vs. random (subject) Ø Cross/nesting: Balanced? Within-subject/repeated-measures vs. between- subjects å Which program? Ø 3 dttest, 3 d. ANOVA/2/3, Group. Ana, 3 d. Reg. Ana, 3 d. LME -8 -

 • Group Analysis: Planning H Output å Main Ø F: effect F general • Group Analysis: Planning H Output å Main Ø F: effect F general information about all levels of a factor Ø Any difference response between two sexes å Interaction F Ø Mutual/reciprocal Ø Effect å General influence among 2 or more factors for each factor depends on levels of other factors linear test Ø Contrast Ø General linear test (e. g. , trend analysis) å Example Ø Dependent Ø Factor Ø Main Ø Is variable: income A: sex (men vs. women); factor B: race (whites vs. blacks) effects: men > women; whites > blacks it fair to only focus on main effects? Interaction! Black men < black women; Black women almost the same as white women; Black men << white men -9 -

 • Group Analysis: Main effect and interaction -10 - • Group Analysis: Main effect and interaction -10 -

 • Group Analysis: Planning H Thresholding å Two-tail å If H by default • Group Analysis: Planning H Thresholding å Two-tail å If H by default in AFNI one-tail p is desirable, look for 2 p on AFNI Scripting – 3 d. ANOVA 3 å Three-way Ø 3 between-subjects (type 1) categorizations of groups: sex, disease, age å Two-way within-subject (type 4): Crossed design AXBXC Ø One group of subjects: 16 subjects Ø Two categorizations of conditions: A – category; B - affect å Two-way Ø Nesting Ø One mixed (type 5): BXC(A) (between-subjects) factor (A): subject classification, e. g. , sex category of condition (within-subject factor B): condition (visual vs. auditory) Ø Nesting: balanced -11 -

 • Group Analysis: Example 3 d. ANOVA 3 -type 4 -alevels 3 -blevels • Group Analysis: Example 3 d. ANOVA 3 -type 4 -alevels 3 -blevels 3 -clevels 16 Model type, Factor levels -dset 1 1 1 stats. sb 04. beta+tlrc’[0]’ -dset 1 2 1 stats. sb 04. beta+tlrc’[1]’ -dset 1 3 1 stats. sb 04. beta+tlrc’[2]’ Input for each cell in ANOVA table: totally 3 X 3 X 16 = 154 -dset 2 1 1 stats. sb 04. beta+tlrc’[4]’ … -fa Category -fb Affect F tests: Main effects & interaction -fab Cat. XAff 1 -acontr 1 0 -1 Tvs. F (coding with coefficients) -bcontr 0. 5 -1 non-neu (coefficients) -a. Bcontr 1 -1 0 : 1 Tvs. E-pos (coefficients) -Abcontr 2 : 1 -1 0 HMvs. HP (coefficients) -bucket anova 33 T (coding with indices) -amean t tests: 1 st order Contrasts t tests: 2 nd order Contrasts Output: bundled -12 -

 • Group Analysis: Group. Ana H Multi-way ANOVA å Matlab script package for • Group Analysis: Group. Ana H Multi-way ANOVA å Matlab script package for up to 5 -way ANOVA å Can handle both volume and surface data å Can handle up to 4 -way unbalanced designs Ø No missing data allowed å Downsides Ø Requires Ø Slow: Matlab plus Statistics Toolbox GLM approach - regression through dummy variables Ø Complicated å Heavy duty computation Ø Minutes Ø Input to hours with lower resolution recommended Ø Resample å See H design, and compromised power with adwarp -dxyz # or 3 dresample http: //afni. nimh. nih. gov/sscc/gangc for more info Alternative: 3 d. LME -13 -

 • Group Analysis: ANCOVA (ANalysis of COVAriances) H Why ANCOVA? å Subjects or • Group Analysis: ANCOVA (ANalysis of COVAriances) H Why ANCOVA? å Subjects or cross-regressors effects might not be an ideally randomized å If not controlled, such variability will lead to loss of power and accuracy å Different from amplitude modulation: cross-regressor vs. within-regressor variation å Direct control via design: balanced selection of subjects (e. g. , age group) å Indirect (statistical) control: add covariates in the model å Covariate (variable of no interest): uncontrollable/confounding, usually continuous Ø Age, IQ, cortex thickness Ø Behavioral data, e. g. , response time, correct/incorrect rate, symptomatology score, … H ANCOVA = Regression + ANOVA å Assumption: å GLM H linear relation between HDR and the covariate approach: accommodate both categorical and quantitative variables Programs å 3 d. Reg. Ana: Ø If for simple ANCOVA the analysis can be handled with 3 dttest without covariates å 3 d. LME: R package (versatile) -14 -

 • Group Analysis: ANCOVA Example H Example: Running ANCOVA å Two groups: 15 • Group Analysis: ANCOVA Example H Example: Running ANCOVA å Two groups: 15 normal vs. 13 patients å Analysis Ø Compare two group: without covariates, two-sample t with 3 dttest Ø Controlling å GLM Ø Yi age effect model = β 0 + β 1 X 1 i + β 2 X 2 i + β 3 X 3 i + εi, i = 1, 2, . . . , n (n = 28) Ø Code the factor (group) with dummy coding 0, when the subject is a patient – control/reference group; X 2 i = { 1, when the subject is normal. Ø Centralize covariate (age) X 1 so that β 0 = patient effect; β 1 = age effect (correlation coef); β 2 = normal vs patient Ø X 3 i = X 1 i X 2 i models interaction (optional) between covariate and factor (group) β 3 = interaction -15 -

 • Group Analysis: ANCOVA Example 3 d. Reg. Ana -rows 28 -cols 3 • Group Analysis: ANCOVA Example 3 d. Reg. Ana -rows 28 -cols 3 -xydata 0. 1 0 0 patient/Pat 1+tlrc. BRIK -xydata 7. 1 0 0 patient/Pat 2+tlrc. BRIK … -xydata … -xydata 7. 1 0 0 patient/Pat 13+tlrc. BRIK 2. 1 1 2. 1 normal/Norm 1+tlrc. BRIK 2. 1 1 2. 1 normal/Norm 2+tlrc. BRIK Input: Covariates, factor levels, interaction, and input files 0. 1 1 0. 1 normal/Norm 15+tlrc. BRIK -model 1 2 3 : 0 -bucket 0 Pat_vs_Norm -brick -brick Model parameters: 28 subjects, 3 independent variables 0 1 2 3 4 5 6 7 coef 0 ‘Pat’ tstat 0 ‘Pat t' coef 1 'Age Effect' tstat 1 'Age Effect t' coef 2 'Norm-Pat' tstat 2 'Norm-Pat t' coef 3 'Interaction' tstat 3 'Interaction t' Specify model for F and R 2 Output: #subbriks=2*#coef+F + R 2 Label output subbricks for β 0, β 1, β 2, β 3 See http: //afni. nimh. nih. gov/sscc/gangc/ANCOVA. html for more information -16 -

 • Group Analysis: 3 d. LME H An R package å Open å • Group Analysis: 3 d. LME H An R package å Open å Linear source platform mixed-effects (LME) modeling å Versatile: handles almost all situations in one package Ø Unbalanced Ø ANOVA Ø Able and ANCOVA, but unlimited factors and covariates to handle HRF modeling with basis functions Ø Violation Ø Model å No designs (unequal number of subjects, missing data, etc. ) of sphericity: heteroscedasticity, variance-covariance structure fine-tuning scripting å Disadvantages Ø High computation cost (lots of repetitive calculation) Ø Sometimes difficult to compare with traditional ANOVA å Still under development å See http: //afni. nimh. nih. gov/sscc/gangc/lme. html for more information -17 -

 • Group Analysis: 3 d. LME H Linear model å yi = β • Group Analysis: 3 d. LME H Linear model å yi = β 0+β 1 x 1 i + … + βpxpi + εi , εi ~ NID(0, σ2) åY = Xβ + ε, ε ~ Nn(0, σ2 In) å Only H one random effect, residual ε Linear mixed-effects (LME) model å yij = β 0+β 1 x 1 ij+ … +βpxpij+bi 1 z 1 ij+…+biqzqij+εij, å bik~N(0, ψk 2), å Yi = Xiβ +Zibi+εi, bi~ Nq(0, ψ), εi ~ Nni(0, σ2Λi) å Two å In cov(bk, bk’)=ψkk’, εij ~ N(0, σ2λijj), cov(εij, εij’)= σ2λijj’ random effect components: Zibi nd εi f. MRI, usually q=1, Zi= Ini – subject: one parameter ψ -18 -

 • Group Analysis: 3 d. LME H Linear mixed-effects (LME) model å For • Group Analysis: 3 d. LME H Linear mixed-effects (LME) model å For each subject Yi = Xiβ+Zibi+εi, bi~ Nq(0, ψ), εi ~ Nni(0, σ2Λi) å AN(C)OVA Ø ni can be incorporated as a special case is constant (>1, repeated-measures), Λi = Inxn (iid) å LME is much more flexible Ø No differentiation between categorical and continuous variables (ANOVA vs. ANCOVA) Ø ni can vary (unequal number of subjects, missing beta values) Ø Don’t have to include an intercept: basis functions! Ø Residual variance-covariance σ2Λi can be any structure -19 -

 • Group Analysis: 3 d. LME H LME: correlation structure in σ2Λi - • Group Analysis: 3 d. LME H LME: correlation structure in σ2Λi - off-diagonals Ø iid Λi = Inxn: traditional AN(C)OVA; one parameter σ2 Ø Compound symmetry: 2 parameters σ2 and σ1 Assume equal correlation across factor levels: fixed variance/covariance First-order autoregressive structure AR(1): 2 parameters σ2 and r Equally-spaced longitudinal observations across factor levels ARMA(p, q): p+q parameters -20 -

 • Group Analysis: 3 d. LME H LME: variance structure in σ2Λi - • Group Analysis: 3 d. LME H LME: variance structure in σ2Λi - diagonals Ø Ø H iid Λi = Inxn: traditional AN(C)OVA; one parameter σ2 Heteroscedasticity: different σ2 across factor levels; ni+1 parameters HRF modeled with basis functions Ø Traditional approach: AUC Ø Can’t detect shape difference Ø Difficult to handle betas with mixed signs Ø LME approach Ø Usually H 0: b 1=b 2=…=bk Ø But now we don’t care about the differences among bs Ø H 0: b 1=b 2=…=bk=0 Ø Solution: take all bs and model with no intercept Ø But we have to deal with temporal correlations among bs! -21 -

 • Group Analysis: 3 d. LME H Running LME å http: //afni. nimh. • Group Analysis: 3 d. LME H Running LME å http: //afni. nimh. nih. gov/sscc/gangc/lme. html å Install 3 d. LME. R and a few packages å Create a text file model. txt (3 fixed factors plus 1 covariate) Data. Format <-- either Volume or Surface Output. File. Name <-- any string (no suffix needed) MASK: Mask+tlrc. BRIK <-- mask dataset Gender*Object*Modality+Age <-- model formula for fixed effects COV: Age <-- covariate list Saved. For. Random. Effects <-- space reserved for future MFace-FFace <-- contrast label Male*Face*0*0 -Female*Face*0*0 <-- contrast specification MVisual-Maudial Male*0*Visual*0 -Male*0*Audial*0. . . Subj Gender Object Modality Age Input. File Jim Male Face Visual 25 file 1+tlrc. BRIK Carol Female House Audial 23 file 2+tlrc. BRIK Karl Male House Visual 26 file 3+tlrc. BRIK Casey Female Face Audial 24 file 4+tlrc. BRIK . . . å Run R CMD BATCH $LME/3 d. LME. R My. Out & -22 -

 • Group Analysis: 3 d. LME H Running LME: A more complicated example • Group Analysis: 3 d. LME H Running LME: A more complicated example (still testing) å HRF modeled with 6 tents å Null hypothesis: no HRF difference between two conditions Data: Volume <-- either Volume or Surface Output: test <-- any string (no suffix needed) MASK: Mask+tlrc. BRIK <-- mask dataset Fix. Eff: Time-1 <-- model formula for fixed effects COV: <-- covariate list Ran. Eff: TRUE <-- random effect specification Var. Str: weights=var. Ident(form=~1|Time) <-- heteroscedasticity? Cor. Str: correlation=cor. AR 1(form=~Order|Subj) <-- correlation structure SS: sequential <-- sequential or marginal Subj Time. Order Input. File Jim t 1 1 contrast. T 1+tlrc. BRIK Jim t 2 2 contrast. T 2+tlrc. BRIK Jim t 3 3 contrast 3+tlrc. BRIK Jim t 4 4 contrast 4+tlrc. BRIK . . . -23 -

 • Group Analysis: 3 d. LME H Running LME: å model fine-tuning (planning) • Group Analysis: 3 d. LME H Running LME: å model fine-tuning (planning) How to specify 4 structures: Fix. Eff: Time-1 <-- model formula for fixed effects Ran. Eff: TRUE <-- random effect specification Var. Str: weights=var. Ident(form=~1|Time) <-- heteroscedasticity? Cor. Str: correlation=cor. AR 1(form=~Order|Subj) <-- correlation å Pick up a most interesting voxel å Start with a reasonably simple model, and compare alternatives Ø Ø å Add or reduce fixed and random effects Vary variance and correlation structures Problems Ø The best model at one voxel might not be true for other voxels Ø More sophisticated model means more parameters and longer running time Ø Solution: ROI analysis – analyze each ROI separately! -24 -

Group Analysis Basic analysis Design Program Contrasts 3 dttest, 3 d. ANOVA/2/3, 3 d. Group Analysis Basic analysis Design Program Contrasts 3 dttest, 3 d. ANOVA/2/3, 3 d. Reg. Ana, Group. Ana, 3 d. LME Simple Correlation Connectivity Analysis 3 d. Deconvolve Context-Dependent Correlation Path Analysis 3 d. Decovolve 1 d. SEM -25 -

 • Connectivity: Correlation Analysis H Correlation analysis å Similarity å Says between a • Connectivity: Correlation Analysis H Correlation analysis å Similarity å Says between a seed region and the rest of the brain not much about causality/directionality å Voxel-wise å Both å Two H analysis individual subject and group levels types: simple and context-dependent correlation (a. k. a. PPI) Steps at individual subject level å Create ROI å Isolate signal for a condition/task å Extract seed time series å Correlation å More H analysis through regression analysis accurately, partial (multiple) correlation Steps at group level å Convert correlation coefficients to Z (Fisher transformation): 3 dcalc å One-sample H t test on Z scores: 3 dttest More details: http: //afni. nimh. nih. gov/sscc/gangc -26 -

 • Connectivity: Path Analysis or SEM H Causal modeling (a. k. a. structural • Connectivity: Path Analysis or SEM H Causal modeling (a. k. a. structural connectivity) å Start with a network of ROI’s å Path analysis Ø Assess the network based on correlations (covariances) of ROI’s Ø Minimize discrepancies between correlations based on data and estimated from model Ø Input: Model specification, correlation matrix, residual error variances, DF Ø Output: Path coefficients, various fit indices å Caveats Ø H 0: It is a good model Ø Valid Ø No proof: modeled through correlation analysis Ø Even Ø If only with the data and model specified with the same data, an alternative model might be equally good or better one critical ROI is left out, things may go awry Ø Interpretation of path coefficient -27 -

 • Connectivity: Path Analysis or SEM H Path analysis with 1 d. SEM • Connectivity: Path Analysis or SEM H Path analysis with 1 d. SEM å Model validation: ‘confirm’ a theoretical model Ø Accept, å Model reject, or modify the model? search: look for ‘best’ model Ø Start with a minimum model (1): can be empty Ø Some paths can be excluded (0), and some optional (2) Ø Model grows by adding one extra path a time Ø ‘Best’ in terms of various fit criteria å More H information http: //afni. nimh. nih. gov/sscc/gangc/Path. Ana. html Difference between causal and correlation analysis å Predefined å Modeling: å ROI network (model-based) vs. network search (data-based) causation (and directionality) vs. correlation vs. voxel-wise å Input: correlation (condensed) vs. original time series å Group analysis vs. individual + group -28 -

Statistical Analysis It is easy to lie with statistics. It is hard to tell Statistical Analysis It is easy to lie with statistics. It is hard to tell the truth without it. ----Andrejs Dunkels -29 -