Statistical Analysis I. Basic Hypothesis Driven Analyses

fMRI Graduate Course November 13, 2002

When do we not need statistical analysis?

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1 51 101 151 201 251 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

– 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

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

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

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

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means to population variability – Uses

t

– Defined as the likely distribution drawn from a single population conditions in blocked designs Comparisons 0.8

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Drift Artifact and T-Test

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

12s on, 12s off Frequency (Hz)

3% 2% 1% 0% -1% -2% -3% Left-Right Right-Left 1 12 23 34 45 56 67 78 89 100 111 122 Image 1600 1200 800 400 0 0.005 0.047 0.089 0.130 0.172 0.214 0.255 0.297

Frequency 1400 1200 1000 800 600 400 200 0 0 30 60 90 120 150 180 210 240 270 300 330 Phase Angle (Degrees) 3% 2% 1% 0% -1% -2% -3% Left-Right Right-Left 1 12 23 34 45 56 67 78 89 100 111 122 Image 1400 1200 1000 800 600 400 200 0 1600 1200 800 400 0 0.005 0.047 0.089 0.130 0.172 0.214 0.255 0.297

Frequency Phase Angle (Degrees)

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

35 30 25 20 15 10 5 0 -5 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 -10 -15 -20

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

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

Kolmogorov – Smirnov (KS) Test

• Statistical evaluation of differences in cumulative density function – Cf. t-test evaluates differences in mean p < 10 -30 p < 10 -30 ns

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

Use of Continuous Tasks to Evaluate Functional Connectivity

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

The General Linear Model

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)

Signal, noise, and the General Linear Model

Measured Data

Y

 

M

  Amplitude (solve for) Design Model Noise Cf. Boynton et al., 1996

=

Form of the GLM

Model Functions Model Functions

Model

* Amplitudes +

Implementation of GLM in SPM

The Problem of Multiple Comparisons

The Problem of Multiple Comparisons

P < 0.05 (1682 voxels) P < 0.01 (364 voxels) P < 0.001 (32 voxels)

Options for Multiple Comparisons

• Statistical Correction (e.g., Bonferroni) – Gaussian Field Theory • Cluster Analyses • ROI Approaches

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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Type I Probability Adjusted Alpha 0.01

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0.00001

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0.0000001

0 1 2 4 8 16 32 64 128 256 512 10 24 20 48 40 96

Number of Comparisons

81 92 16 384 32 768 65 536 13 107 2 26 214 4 52 428 8 10 485 76 0.00000001

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

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

Figures from http://www.irsl.org/~fet/Presentations/wavestatfield/wavestatfield.html

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)

How many foci of activation?

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

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

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

Are there differences between voxel-wise and ROI analyses?

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

Fixed and Random Effects Comparisons

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

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

Summary of Hypothesis Tests

• Simple experimental designs – Blocked:

t

-test, Fourier analysis – Event-related: correlation, – Cluster analyses improve

t

-test at time points • Complex experimental designs – Regression approaches (GLM) • Critical problem: Minimization of Type I Error – Strict Bonferroni correction is too severe – Accounting for smoothness of data also helps • Use random-effects analyses to allow generalization to the population