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# Statistical Analysis - PowerPoint PPT Presentation

Statistical Analysis. fMRI Graduate Course October 29, 2003. When do we not need statistical analysis?. Inter-ocular Trauma Test (Lockhead, personal communication). Why use statistical analyses?. Replaces simple subtractive methods Signal highly corrupted by noise Typical SNRs: 0.2 – 0.5

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## PowerPoint Slideshow about 'Statistical Analysis' - melodie-carpenter

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### Statistical Analysis

October 29, 2003

Inter-ocular Trauma Test (Lockhead, personal communication)

• 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

• 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

• Signal change divided by noise (SNR)

• Typically expressed as t or z statistics

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

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

• 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

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

• 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

• 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

• 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

• Can see connections between functional areas (e.g., between Broca’s and Wernicke’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)

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

• 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 – 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 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

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 subjects’ 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