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FMRI – Week 9 – Analysis I Scott Huettel, Duke University FMRI Data Analysis: I. Basic Analyses and the General Linear Model FMRI Graduate Course (NBIO 381, PSY 362) Dr. Scott Huettel, Course Director

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University When do we not need statistical analysis? Inter-ocular Trauma Test (Lockhead, personal communication)

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Why use statistical analyses? Replaces simple subtractive methods –Signal highly corrupted by noise Typical SNRs: 0.2 – 0.5 –Sources of noise Thermal variation (unstructured) Physiological, task variability (structured) Assesses quality of data –How reliable is an effect? –Allows distinction of weak, true effects from strong, noisy effects

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University What do our analyses generate? Statistical Parametric Maps Brain maps of statistical quality of measurement –Examples: correlation, regression approaches –Displays likelihood that the effect observed is due to chance factors –Typically expressed in probability (e.g., p < 0.001), or via t or z statistics

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University What are our statistics for?

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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Key Concepts Within-subjects analyses –Simple non-GLM approaches (older) –General Linear Model (GLM) Across-subjects analyses –Fixed vs. Random effects Correction for Multiple Comparisons Displaying Data

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Simple Hypothesis-Driven Analyses t-test across conditions Time point analysis (i.e., t-test) Correlation Fourier analysis

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Correlation Approaches (old-school) How well does our data match an expected hemodynamic response? Special case of General Linear Model Limited by choice of HDR –Assumes particular correlation template –Does not model task-unrelated variability –Does not model interactions between events

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Fourier Analysis Fourier transform: converts information in time domain to frequency domain –Used to change a raw time course to a power spectrum –Hypothesis: any repetitive/blocked task should have power at the task frequency BIAC function: FFTMR –Calculates frequency and phase plots for time series data. Equivalent to correlation in frequency domain Subset of general linear model –Same as if used sine and cosine as regressors

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University 12s on, 12s offFrequency (Hz) Power

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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The General Linear Model (GLM)

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Basic Concepts of the GLM GLM treats the data as a linear combination of model functions plus noise –Model functions have known shapes –Amplitude of functions are unknown –Assumes linearity of HDR; nonlinearities can be modeled explicitly GLM analysis determines set of amplitude values that best account for data –Usual cost function: least-squares deviance of residual after modeling (noise)

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Signal, noise, and the General Linear Model Measured Data Amplitude (solve for) Design Model Noise Cf. Boynton et al., 1996

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Form of the GLM Data = N Time Points Model N Time Points Model Functions * Amplitudes Model Functions + Noise

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Design Matrices Model Parameters Images

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Regressors (How much of the variance in the data does each explain?) Contrasts (Does one regressor explain more variance than another?)

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Task and Nuisance Regressors Task Regressors Nuisance (Motion) Regressors

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Hemodynamic and Basis Functions Double Gamma Gaussian Gamma

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University The optimal relation between regressors depends on our research question

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Suppose that we have two correlated regressors. R1: Motor? R2: Visual? Value of R1 (at each point in time) Value of R2 (at each point in time) Because of their correlation, the design is inefficient at distinguishing the contributions of R1 and R2 to the activation of a voxel. X = Y

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Let’s now make the regressors anti-correlated. Value of R1 (at each point in time) Value of R2 (at each point in time) Now, the design allows us to separate the contributions of each regressor, but cannot look at their common effect. X = -Y

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Value of R1 (at each point in time) Value of R2 (at each point in time) This makes the activation uncorrelated, but doesn’t efficiently use the space. We can shift our block design in time, so that the regressors are off-set. X = -Y X = Y

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University And, we can make the regressors uncorrelated with each other through randomization. Value of R1 (at each point in time) Value of R2 (at each point in time) Now, we get more of a “cloud” arrangement of the time points. (Squareness and lack of evenness is caused by my simulation approach)

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Orthogonalization of Regressors Cue Regressor Target Regressor (Orthogonalized) Cue Regressor Non- Orthogonal Orthogonal

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Constant Effect Parametric Effect Setting up Parametric Effects

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Fixed and Random Effects Comparisons

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Fixed Effects Fixed-effects Model –Assumes that effect is constant (“fixed”) in the population –Uses data from all subjects to construct statistical test –Examples Averaging across subjects before a t-test Taking all subjects’ data and then doing an ANOVA –Allows inference to subject sample

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Random Effects Random-effects Model –Assumes that effect varies across the population –Accounts for inter-subject variance in analyses –Allows inferences to population from which subjects are drawn –Especially important for group comparisons –Required by many reviewers/journals

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Key Concepts of Random Effects Assumes that activation parameters may vary across subjects –Since subjects are randomly chosen, activation parameters may vary within group –(Fixed-effects models assume that parameters are constant across individuals) Calculates descriptive statistic for each subject –i.e., parameter estimate from regression model Uses all subjects’ statistics in a higher-level analysis –i.e., group significance based on the distribution of subjects’ values.

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University The Problem of Multiple Comparisons P < (32 voxels)P < 0.01 (364 voxels)P < 0.05 (1682 voxels)

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University A BC t = 2.10, p < 0.05 (uncorrected)t = 3.60, p < (uncorrected)t = 7.15, p < 0.05, Bonferroni Corrected

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Options for Multiple Comparisons Statistical Correction (e.g., Bonferroni) –Family-wise Error Rate –False Discovery Rate (FDR) Cluster Analyses ROI Approaches

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Statistical Corrections If more than one test is made, then the collective alpha value is greater than the single-test alpha –That is, overall Type I error increases One option is to adjust the alpha value of the individual tests to maintain an overall alpha value at an acceptable level –This procedure controls for overall Type I error –Known as Bonferroni Correction

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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Bonferroni Correction Very severe correction –Results in very strict significance values –Typical brain may have up to ~30,000 functional voxels P(Type I error) ~ 1.0 ; Corrected alpha ~ Greatly increases Type II error rate Is not appropriate for correlated data –If data set contains correlated data points, then the effective number of statistical tests may be greatly reduced –Most fMRI data has significant correlation

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Gaussian Field Theory Approach developed by Worsley and colleagues to account for multiple comparisons Provides false positive rate for fMRI data based upon the smoothness of the data –If data are very smooth, then the chance of noise points passing threshold is reduced Recommendation: Use a combination of voxel and cluster correction methods

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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Cluster Analyses Assumptions –Assumption I: Areas of true fMRI activity will typically extend over multiple voxels –Assumption II: The probability of observing an activation of a given voxel extent can be calculated Cluster size thresholds can be used to reject false positive activity –Forman et al., Mag. Res. Med. (1995) –Xiong et al., Hum. Brain Map. (1995)

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University How many foci of activation? Data from motor/visual event-related task (used in laboratory)

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University How large should clusters be? At typical alpha values, even small cluster sizes provide good correction –Spatially Uncorrelated Voxels At alpha = 0.001, cluster size 3 Type 1 rate to << per voxel –Highly correlated Voxels Smoothing (FW = 0.5 voxels) Increases needed cluster size to 7 or more voxels Efficacy of cluster analysis depends upon shape and size of fMRI activity –Not as effective for non-convex regions –Power drops off rapidly if cluster size > activation size Data from Forman et al., 1995

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University False Discovery Rate Controls the expected proportion of false positive values among suprathreshold values –Genovese, Lazar, and Nichols (2002, NeuroImage) –Does not control for chance of any face positives FDR threshold determined based upon observed distribution of activity –So, sensitivity increases because metric becomes more lenient as voxels become significant

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Genovese, et al., 2002 (sum)

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University ROI Comparisons Changes basis of statistical tests –Voxels: ~16,000 –ROIs : ~ 1 – 100 Each ROI can be thought of as a very large volume element (e.g., voxel) –Anatomically-based ROIs do not introduce bias Potential problems with using functional ROIs –Functional ROIs result from statistical tests –Therefore, they cannot be used (in themselves) to reduce the number of comparisons

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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Voxel and ROI analyses are similar, in concept

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Summary of Multiple Comparison Correction Basic statistical corrections are often too severe for fMRI data What are the relative consequences of different error types? –Correction decreases Type I rate: fewer false positives –Correction increases Type II rate: more misses Alternate approaches may be more appropriate for fMRI –Cluster analyses –Region of interest approaches –Smoothing and Gaussian Field Theory –False Discovery Rate

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Displaying Data

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University Never Mask!

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FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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Summary of Basic Analysis Methods Simple experimental designs –Blocked: t-test, Fourier analysis –Event-related: correlation, t-test at time points Complex experimental designs –Regression approaches (GLM) Critical problem: Minimization of Type I Error –Strict Bonferroni correction is too severe –Cluster analyses improve –Accounting for smoothness of data also helps Use random-effects analyses to allow generalization to the population

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