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## Statistical Analysis

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**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 • 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**Types of Errors**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)**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**5%**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 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**Power**12s on, 12s off Frequency (Hz)**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**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 A B C**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**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**Amplitude (solve for) Measured Data Noise Design Model Cf. Boynton et al., 1996**Form of the GLM**Model Functions Model Functions Model * Amplitudes = + Data Noise N Time Points N Time Points**Implementation of GLM in SPM**Model Parameters Images**The Problem of Multiple Comparisons**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**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**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**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**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**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**A**B**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, 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