statistical analysis fmri graduate course november 2, 2005
TRANSCRIPT
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Statistical Analysis
fMRI Graduate Course
November 2, 2005
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When do we not need statistical analysis?
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Inter-ocular Trauma Test (Lockhead, personal communication)
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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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Statistical Parametric Maps
• 1. 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)
• 2. Effect size– Determined by comparing task-related variability and
non-task-related variability– Signal change divided by noise (SNR)– Typically expressed as t or z statistics
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What are our statistics for?
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Which is more important to avoid: Type I or Type II errors?
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Simple Hypothesis-Driven Analyses
• Common– t-test across conditions– Fourier– t-test at time points – Correlation
• General Linear Model • Other tests
– Kolmogorov-Smirnov – Iterative Connectivity Mapping
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t – Tests across Conditions
• Compares difference between means to population variability– Uses t distribution– Defined as the likely distribution
due to chance between samples drawn from a single population
• Commonly used across conditions in blocked designs
• Subset of general linear model 5%
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Drift Artifact and t-Test
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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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12s on, 12s off Frequency (Hz)
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t / z – Tests across Time Points
• Determines whether a single data point in an epoch is significantly different from baseline
• BIAC Tool: tstatprofile– Creates:
• Avg_V*.img• StdDev_V*.img• ZScore_V*.img
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Correlation
• Special case of General Linear Model– Blocked t-test is equivalent to correlation with square
wave function– Allows use of any reference waveform
• Correlation coefficient describes match between observation and expectation– Ranges from -1 to 1– Amplitude of response does not affect correlation
directly
• BIAC tool: tstatprofile
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Problems with Correlation Approaches
• Limited by choice of HDR– Poorly chosen HDR can significantly impair power
• Examples from previous weeks
– May require different correlations across subjects
• Assume that correlation template is Gaussian • Assume random variation around HDR
– Do not model variability contributing to noise (e.g., scanner drift)
• Such variability is usually removed in preprocessing steps
– Do not model interactions between successive events
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Kolmogorov – Smirnov (KS) Test
• Statistical evaluation of differences in cumulative density function– Cf. t-test evaluates differences in mean
• Useful if distributions have same mean but different shape
A B C
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Iterative Connectivity Mapping
• Acquire two data sets– 1: Defines regions of interest
and hypothetical connections– 2: Evaluates connectivity
based on low frequency correlations
• Use of Continuous Data Sets– Null Data– Task Data– Can see connections
between functional areas (e.g., between Broca’s and Wernicke’s Areas)
Hampson et al., Hum. Brain. Map., 2002
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Use of Continuous Tasks to Evaluate Functional Connectivity
Hampson et al., Hum. Brain. Map., 2002
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The General Linear Model
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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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Signal, noise, and the General Linear Model
MYMeasured Data
Amplitude (solve for)
Design Model
Noise
Cf. Boynton et al., 1996
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Form of the GLMD
ata=
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* Amplitudes
Model Functions
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Implementation of GLM in SPM
Model Parameters
Ima
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The Problem of Multiple Comparisons
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The Problem of Multiple Comparisons
P < 0.001 (32 voxels)P < 0.01 (364 voxels)P < 0.05 (1682 voxels)
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A B C
t = 2.10, p < 0.05 (uncorrected) t = 3.60, p < 0.001 (uncorrected) t = 7.15, p < 0.05, Bonferroni Corrected
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Options for Multiple Comparisons
• Statistical Correction (e.g., Bonferroni)– Gaussian Field Theory– False discovery rate
• Cluster Analyses
• ROI Approaches
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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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Bonferroni Correction
• Very severe correction– Results in very strict significance values for even
medium data sets– Typical brain may have about 15,000-20,000
functional voxels• PType1 ~ 1.0 ; Corrected alpha ~ 0.000003
• 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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Gaussian Field Theory
• Approach developed by Worsley and colleagues to account for multiple comparisons– Forms basis for much of SPM
• 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
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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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How many foci of activation?
Data from motor/visual event-related task (used in laboratory)
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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 reduces Type 1 rate to << 0.00001 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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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– Weak familywise Type I error rate
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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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Are there differences between voxel-wise and ROI analyses?
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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: false positives– Correction increases Type II rate: 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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Fixed and Random Effects Comparisons
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How do we compare across subjects?
• 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
• 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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How are random-effects models run?
• 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., t-test for each subject based on correlation
• Uses all subjects’ statistics in a one-sample t-test – i.e., another t-test based only on significance maps
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Summary of Hypothesis Tests
• 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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Data Driven Analyses
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Independent Components
Analysis
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Why conduct data-driven analyses?
• Powerful tools for exploring data– PCA, ICA: Intrinsic, spatially stationary patterns of
activity in dataset– Clustering: Collections of voxels with similar time
courses of activity– PLS: How those patterns of activity maximally
differentiate experimental conditions
• Allows segmentation of nuisance factors• Provides check on hypothesis-driven analyses