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Data Modeling General Linear Model & Statistical Inference Thomas Nichols, Ph.D. Assistant Professor Department of Biostatistics http://www.sph.umich.edu/~nichols Brain Function and fMRI ISMRM Educational Course July 11, 2002 1 Motivations • Data Modeling – Characterize Signal – Characterize Noise • Statistical Inference – Detect signal – Localization (Where’s the blob?) 2 Outline • Data Modeling – – – – General Linear Model Linear Model Predictors Temporal Autocorrelation Random Effects Models • Statistical Inference – Statistic Images & Hypothesis Testing – Multiple Testing Problem 3 Basic fMRI Example • Data at one voxel – Rest vs. passive word listening • Is there an effect? 4 Time • “Linear” in parameters b1 & b2 = b1 Intensity + b2 x1 error A Linear Model + e x2 5 Linear model, in image form… = b1 Y b1 x1 + b2 + b2 x2 e 6 Linear model, in image form… Estimated = bˆ1 Y bˆ1 x1 + bˆ2 + ˆ b 2 x2 eˆ 7 … in image matrix form… bˆ1 ˆ b 2 = Y X bˆ + eˆ 8 … in matrix form. p 1 1 1 Y Xb e b Y N = N X p + e N N: Number of scans, p: Number of regressors 9 Linear Model Predictors • Signal Predictors – Block designs – Event-related responses • Nuisance Predictors – Drift – Regression parameters 10 Signal Predictors • Linear Time-Invariant system • LTI specified solely by – Stimulus function of experiment Blocks Events – Hemodynamic Response Function (HRF) • Response to instantaneous impulse 11 Convolution Examples Block Design Event-Related Experimental Stimulus Function Hemodynamic Response Function Predicted Response 12 HRF Models • Canonical HRF – Most sensitive if it is correct – If wrong, leads to bias and/or poor fit • E.g. True response may be faster/slower • E.g. True response may have smaller/ bigger undershoot SPM’s HRF 13 HRF Models • Smooth Basis HRFs – More flexible – Less interpretable • No one parameter explains the response Gamma Basis – Less sensitive relative to canonical (only if canonical is correct) Fourier Basis 14 HRF Models • Deconvolution – Most flexible • Allows any shape • Even bizarre, non-sensical ones – Least sensitive relative to canonical (again, if canonical is correct) Deconvolution Basis 15 Drift Models • Drift – Slowly varying – Nuisance variability • Models – Linear, quadratic – Discrete Cosine Transform Discrete Cosine 16 Transform Basis General Linear Model Recap • Fits data Y as linear combination of predictor columns of X Y Xb e • Very “General” – Correlation, ANOVA, ANCOVA, … • Only as good as your X matrix 17 Temporal Autocorrelation • Standard statistical methods assume independent errors – Error ei tells you nothing about ej i j • fMRI errors not independent – Autocorrelation due to – Physiological effects – Scanner instability 18 Temporal Autocorrelation In Brief • Independence • Precoloring • Prewhitening 19 Autocorrelation: Independence Model • Ignore autocorrelation • Leads to – Under-estimation of variance – Over-estimation of significance – Too many false positives 20 Autocorrelation: Precoloring • Temporally blur, smooth your data – This induces more dependence! – But we exactly know the form of the dependence induced – Assume that intrinsic autocorrelation is negligible relative to smoothing • Then we know autocorrelation exactly • Correct GLM inferences based on “known” autocorrelation 21 [Friston, et al., “To smooth or not to smooth…” NI 12:196-208 2000] Autocorrelation: Prewhitening • Statistically optimal solution • If know true autocorrelation exactly, can undo the dependence – De-correlate your data, your model – Then proceed as with independent data • Problem is obtaining accurate estimates of autocorrelation – Some sort of regularization is required • Spatial smoothing of some sort 22 Autocorrelation Redux Advantage Disadvantage Software Indep. Simple Inflated significance All Precoloring Avoids Statistically autocorr. est. inefficient Whitening Statistically optimal SPM99 Requires precise FSL, autocorr. est. SPM2 23 Autocorrelation: Models • Autoregressive – Error is fraction of previous error plus “new” error – AR(1): ei = ei-1 + I • Software: fmristat, SPM99 • AR + White Noise or ARMA(1,1) – AR plus an independent WN series • Software: SPM2 • Arbitrary autocorrelation function – k = corr( ei, ei-k ) • Software: FSL’s FEAT 24 Statistic Images & Hypothesis Testing • For each voxel Y Xb e – Fit GLM, estimate betas • Write b for estimate of b – But usually not interested in all betas • Recall b is a length-p vector 25 Building Statistic Images Predictor of interest b1 b2 b3 b4 = b5 + b6 b7 b8 b9 Y = X b + e 26 Building Statistic Images c’ = 1 0 0 0 0 0 0 0 • Contrast – A linear combination of parameters – c’b contrast of estimated parameters T= c’b T= variance estimate b1 b2 b3 b4 b5 .... s2c’(X’X)+c 27 Hypothesis Test • So now have a value T for our statistic • How big is big – Is T=2 big? T=20? 28 Hypothesis Testing • Assume Null Hypothesis of no signal • Given that there is no signal, how likely is our measured T? • P-value measures this T P-val – Probability of obtaining T as large or larger • level – Acceptable false positive rate 29 Random Effects Models • GLM has only one source of randomness Y Xb e – Residual error • But people are another source of error – Everyone activates somewhat differently… 30 Fixed vs. Random Effects • Fixed Effects – Intra-subject variation suggests all these subjects different from zero • Random Effects – Intersubject variation suggests population not very different from zero Distribution of each subject’s effect Subj. 1 Subj. 2 Subj. 3 Subj. 4 Subj. 5 Subj. 6 0 31 Random Effects for fMRI • Summary Statistic Approach – Easy • Create contrast images for each subject • Analyze contrast images with one-sample t – Limited • Only allows one scan per subject • Assumes balanced designs and homogeneous meas. error. • Full Mixed Effects Analysis – Hard • Requires iterative fitting • REML to estimate inter- and intra subject variance – SPM2 & FSL implement this, very differently – Very flexible 32 Random Effects for fMRI Random vs. Fixed • Fixed isn’t “wrong”, just usually isn’t of interest • If it is sufficient to say “I can see this effect in this cohort” then fixed effects are OK • If need to say “If I were to sample a new cohort from the population I would get the same result” then random effects are needed 33 Multiple Testing Problem • Inference on statistic images – Fit GLM at each voxel – Create statistic images of effect • Which of 100,000 voxels are significant? – =0.05 5,000 false positives! t > 0.5 t > 1.5 t > 2.5 t > 3.5 t > 4.5 t > 5.5 t > 6.5 34 MCP 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) 35 FWER MCP Solutions • Bonferroni • Maximum Distribution Methods – Random Field Theory – Permutation 36 FWER MCP Solutions • Bonferroni • Maximum Distribution Methods – Random Field Theory – Permutation 37 FWER MCP 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 FWER FWER = P(Max voxel u | Ho) u 38 FWER MCP Solutions: Random Field Theory • Euler Characteristic u – Topological Measure • #blobs - #holes Threshold – At high thresholds, Random Field just counts blobs – FWER = P(Max voxel u | Ho) = P(One or more blobs | Ho) P(u 1 | Ho) E(u | Ho) 39 Sets Suprathreshold Controlling FWER: Permutation Test • Parametric methods – Assume distribution of max statistic under null hypothesis • Nonparametric methods 5% Parametric Null Max Distribution – Use data to find distribution of max statistic 5% under null hypothesis Nonparametric Null Max Distribution – Any max statistic! 40 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) 41 Measuring False Positives FWER vs FDR Noise Signal Signal+Noise 42 Control of Per Comparison Rate at 10% 11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% Percentage of Null Pixels that are False Positives 9.5% Control of Familywise Error Rate at 10% Occurrence of Familywise Error FWE 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% Percentage of Activated Pixels that are False Positives 8.7% 43 Controlling FDR: Benjamini & Hochberg 1 • Select desired limit q on E(FDR) • Order p-values, p(1) p(2) ... p(V) • Let r be largest i such that p(i) i/V q p-value i/V q 0 • Reject all hypotheses corresponding to p(1), ... , p(r). p(i) 0 1 i/V 44 Conclusions • Analyzing fMRI Data – Need linear regression basics – Lots of disk space, and time – Watch for MTP (no fishing!) 45 Thanks • Slide help – Stefan Keibel, Rik Henson, JB Poline, Andrew Holmes 46