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Statistical Analysis

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Statistical Analysis

fMRI Graduate Course

October 29, 2003

Inter-ocular Trauma Test (Lockhead, personal communication)

- Replaces simple subtractive methods
- Signal highly corrupted by noise
- Typical SNRs: 0.2 â€“ 0.5

- Sources of noise
- Thermal variation (unstructured)
- Physiological variability (structured)

- Signal highly corrupted by noise
- Assesses quality of data
- How reliable is an effect?
- Allows distinction of weak, true effects from strong, noisy effects

- 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

C

A

B

Hypothesis Truth?

H1 (Active)

H0 (Inactive)

Type I Error

HIT

Reject H0 (Active)

Output of Statistical Test

Type II Error

Correct Rejection

Accept H0 (Inactive)

- Common
- t-test across conditions
- Fourier
- t-test at time points
- Correlation

- General Linear Model
- Other tests
- Kolmogorov-Smirnov
- Iterative Connectivity Mapping

5%

- 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

- 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

Power

12s on, 12s off

Frequency (Hz)

- 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

- Creates:

- 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

- Limited by choice of HDR
- Poorly chosen HDR can significantly impair power
- Examples from previous weeks

- May require different correlations across subjects

- Poorly chosen HDR can significantly impair power
- 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

- Do not model variability contributing to noise (e.g., scanner drift)

- Statistical evaluation of differences in cumulative density function
- Cf. t-test evaluates differences in mean

A

B

C

- 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 Brocaâ€™s and Wernickeâ€™s Areas)

Hampson et al., Hum. Brain. Map., 2002

Hampson et al., Hum. Brain. Map., 2002

- 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)

Amplitude (solve for)

Measured Data

Noise

Design Model

Cf. Boynton et al., 1996

Model Functions

Model Functions

Model

*

Amplitudes

=

+

Data

Noise

N Time Points

N Time Points

Model Parameters

Images

P < 0.05 (1682 voxels)

P < 0.01 (364 voxels)

P < 0.001 (32 voxels)

B

C

A

t = 2.10, p < 0.05 (uncorrected)

t = 3.60, p < 0.001 (uncorrected)

t = 7.15, p < 0.05, Bonferroni Corrected

- Statistical Correction (e.g., Bonferroni)
- Gaussian Field Theory

- Cluster Analyses
- ROI Approaches

- 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

- 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

- 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

- 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)

Data from motor/visual event-related task (used in laboratory)

- 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

- Spatially Uncorrelated 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

- Changes basis of statistical tests
- Voxels: ~16,000
- ROIs : ~ 1 â€“ 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

- 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

- Fixed-effects Model
- Uses data from all subjects to construct statistical test
- Examples
- Averaging across subjects before a t-test
- Taking all subjectsâ€™ 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

A

B

- 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 subjectsâ€™ statistics in a one-sample t-test
- i.e., another t-test based only on significance maps

- 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