Statistical Analysis I' Basic HypothesisDriven Analyses

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Statistical Analysis I. Basic Hypothesis-Driven
Analyses

• fMRI Graduate Course
• November 13, 2002

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When do we not need statistical analysis?
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 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 lt
  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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Why use effect size measures?

• Dissociate size of signal change from reliability
  of signal change
• Understanding reliability of change allows
  quantification of error probabilities
• Types of Errors
• Type I Rejecting null hypothesis when it is true
• Calling active voxels that really have no
  activity
• To minimize false positives, adopt a high
  threshold for significance
• Type II Accepting null hypothesis when it is
  false
• Calling inactive voxels that are really
  associated with the task
• To minimize incorrect rejections, adopt a low
  threshold for significance

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Simple Statistical 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
• Potential problem Multiple Comparisons

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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
• At short durations, like a sine wave (single
  frequency)
• At long durations, like a trapezoid (multiple
  frequencies)
• Subset of multiple regression
• Same as if used sine and cosine as regressors

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Power
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 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

p lt 10-30
ns
p lt 10-30
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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 Brocas and Wernickes 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
Amplitude (solve for)
Measured Data
Noise
Design Model
Cf. Boynton et al., 1996
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Form of the GLM
Model Functions
Model Functions
Model

Amplitudes


Data
Noise
N Time Points
N Time Points
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Implementation of GLM in SPM
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The Problem of Multiple Comparisons
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The Problem of Multiple Comparisons
P lt 0.001 (32 voxels)
P lt 0.01 (364 voxels)
P lt 0.05 (1682 voxels)
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Options for Multiple Comparisons

• Statistical Correction (e.g., Bonferroni)
• Gaussian Field Theory
• 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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Figures from http//www.irsl.org/fet/Presentation
s/wavestatfield/wavestatfield.html
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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 ltlt 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 gt
  activation size

Data from Forman et al., 1995
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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

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Fixed and Random Effects Comparisons
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How do we compare across subjects?

• Fixed-effects Model
• 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
• Accounts for inter-subject variance in analyses
• Allows inferences to population from which
  subjects are drawn
• Especially important for group comparisons
• Beginning to be required by 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