# FMRI Data Analysis: I. Basic Analyses and the General Linear Model

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

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

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

fmri-fig jpg FMRI – Week 9 – Analysis I Scott Huettel, Duke University

Key Concepts Within-subjects analyses Across-subjects analyses
Simple non-GLM approaches (older) General Linear Model (GLM) Across-subjects analyses Fixed vs. Random effects Correction for Multiple Comparisons Displaying Data 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 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 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 FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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

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

The General Linear Model (GLM)
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) FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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

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

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

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

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

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

We can shift our block design in time, so that the regressors are off-set.
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. X = -Y X = Y 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) FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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

Setting up Parametric Effects
1 2 3 4 Constant Effect Parametric Effect FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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

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

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

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

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

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

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

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

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

fmri-fig jpg FMRI – Week 9 – Analysis I Scott Huettel, Duke University

Voxel and ROI analyses are similar, in concept
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 FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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

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

fmri-fig jpg FMRI – Week 9 – Analysis I Scott Huettel, Duke University

fmri-fig jpg FMRI – Week 9 – Analysis I Scott Huettel, Duke University

fmri-fig jpg FMRI – Week 9 – Analysis I Scott Huettel, Duke University

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

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