1. fMRI Statistics Part I John VanMeter, Ph.D. Center for Functional and Molecular Imaging Georgetown University Medical Center

2. What Constitutes Activation Statistics help us to answer the following: –How do we determine whether an area of the brain is activated by our task? –How confident are we that the areas we find are activated by our task? –Are the results in my group of subjects applicable to the wider population? –Are the differences between groups of subjects significant? –and more

3. Formal Statement of a Hypothesis Research question is framed as a hypothesis Null hypothesis assumes that the hypothesis is not true Statistics aim to disprove the null hypothesis and thus accept the research hypothesis In fMRI we are testing difference in BOLD signal between two conditions H 1 : Condition 1 ≠ Condition 2 H 0 : Condition 1 = Condition 2

4. False Positives and Negatives H 0 is FalseH 0 is True Reject H 0 Correct Decision False Positive (Type I Error) Accept H 0 False Negative (Type II Error) Correct Decision Hypothesis Truth Statistical Result

5. False Positives and Negatives H 0 is FalseH 0 is True Reject H 0 1 - β = power  Accept H 0 β 1 -  Hypothesis Truth Statistical Result

6. Types of Error Type I error - falsely accept a voxel as being active Alpha value is probability that we have committed a Type I error Type II error - falsely reject a voxel as active Almost always trying to minimize Type I errors at the cost of Type II errors When might you want to minimize Type II errors?

7. fMRI Example Hypothesis: “Moving the thumb will cause an increase in neuronal activity which we detect with BOLD signal changes” Null Hypothesis: “Moving the thumb will NOT cause an increase in neuronal activity which we detect with BOLD signal changes” –Experimental condition – moving thumb –Control condition – thumb not moving –Outcome measure – MRI signal changes

8. Hypothesis Testing MRI Intensity Number of Scans Rest Move

9. Hypothesis Testing MRI Intensity Number of Scans Rest Move

10. Hypothesis Testing MRI Intensity Number of Scans Rest Move

11. Hypothesis Testing MRI Intensity Number of Scans Rest Move

12. Hypothesis Testing Two factors describe how much effect the experimental condition had: –Difference between the mean intensities of each condition –Degree of overlap in intensities

13. Hypothesis Testing Experimental condition has an effect

14. Hypothesis Testing Experimental condition has no effect

15. The t-test Formally incorporates our intuitive sense of when there is an effect Based on a measure of the distance between the two means and the spread of each condition t = (m 1 – m 2 ) √( 1 2 +  2 2 )

16. t-statistics and p-values The p-value for a t-statistic gives the probability that the difference between the experimental and control conditions arose by chance Typically p < 0.05 is considered minimum cut- off for significance (i.e. alpha is set at p < 0.05) Statistics tables list the p-values for each t- statistic based on the df, degrees of freedom, (single subject analysis df=total number of scans minus 1)

17. One-Tailed Test Yellow area under the curve is about 0.025 (for t=2).

18. Two-Tailed Test Yellow area under the curve is about 0.05 (for t=2).

19. t-statistics and p-values Ex. Suppose we find a voxel in which the t- statistic is 3.3 and there were 20 scans at rest and 20 scans while moving thumb. The probability that the difference in the MRI signal in this voxel is not due to the movement of the thumb is 0.002 two-tail or 0.001 one-tail.

20. Examples of the Affect of Variance

21. t-statistics, p-values, & Z- scores Unlike t-statistics, the p-value for Z- scores, which are based on the normal distribution, does not change depending on the number of scans In functional imaging it is common to convert the t-statistic to a Z-score since it is easier to compare across studies (not dependent on degrees of freedom)

22. Problems with the t-test Systematic differences such as artifacts can create apparent significant differences where none exists Disregards any temporal characteristics of the data since only means are compared Assumption of t-test is that the data for both conditions is normally distributed - usually though not always true –Smoothing helps make data normally distributed

23. Kolmogorov-Smirnov KS-test (Non-parametric) Compares cumulative distribution function (CDF) of both conditions CDF is proportion of data at each value of the dependent variable (i.e. fMRI signal) KS statistic is D k the maximal difference between the two conditions for any given value of the dependent variable –Advantage: non-parametric test (makes no assumptions about distribution of intensities) –Disadvantage: less sensitive than t-test

24. t-map in a Phantom Due to warming of the gradients, the center frequency of the scanner changes slightly from time-point to time-point Results in apparent motion as center frequency is assumed to be constant (123.3 for 3T) Image reconstruction inaccurately localizes the data because of this t-map of a phantom shows activation!

25. Correlation/Regression Use a representative waveform representing the on-off periodicity of the design convolved with HRF and correlate that with MRI signal change across each of the scans

26. Correlation with External Variables In addition to the stimulation paradigm could correlate fMRI signal with other variables such as reaction time In group analyses variables like age, IQ, accuracy on a test can be interesting variables to examine Common to include motion correction transformation estimates (translations and rotations) as covariate of no-interest (aka confounds) Technically both the fMRI signal and the variable being correlated against should be normally distributed

27. General Linear Model (GLM) Convenient method for examining multiple aspects of the same data Model fMRI data as a linear combination of effects –y(t) is the fMRI data –x(t) is a regressor or factor –  weights are the contribution of each effect or regressor –  (t) are residuals y(t) =  0 +  1 x 1 (t) +  2 x 2 (t) + … +  (t) In Matrix form: Y = X + 

28. Simple fMRI GLM x(t) is the block design convolved with a model of the HRF H o : 1* = 0

29. GLM - Adding Regressors x 2 (t) adds Temporal Derivates - allows for shifts in transition points of block design H o : 1*    = 0

30. SPM Incorporates Temporal Filtering in GLM s j (t) are each of the cosine basis functions of increasing frequency H o :  1*  = 0

31. Design Matrix x(t) (and s(t)) components of the GLM is your design matrix and are typically referred to as regressors Minimally, the design matrix will include a regressor that models the expected hemodynamic response to the task –On-off periodicity of the block design convolved with HRF –A regressor corresponding to a vector of 1’s at the onset time of a given trial type convolved with HRF for ER designs Could also include additional regressors that identify trials within the block design with a correct response, reaction time, etc… Defining your design matrix correctly is essential to getting interpretable results

32. Block Design Matrix (Single Subject Analysis) Depicted pictorially with each column corresponding to a separate regressor each row corresponds to different scan Brightness indicates timepoints that are expected to have a higher response Example shows –On-off of block design in 1 st column convolved with HRF –Trials with correct responses in 2 nd column –Trials with incorrect responses in 3 rd column

33. ER Design Matrix Example includes transformation parameters from motion correction as nuisance regressors –Useful for removing noise due to head motion –However, if motion is correlated with task then this will reduce statistical significance motion parameters

34. Nuisance Regressors Other examples of nuisance regressors –Physiologic measurements (eg. HR & respiration) –Indicator of runs (aka sessions) to account for global differences between runs run indicators

35. Contrasts Describes the tests that you want to perform on the data Specified as a linear combination of your regressors Graphically displayed above design matrix For basic experimental condition vs control task comparison weights would be 1’s for experimental regressors and -1 for control task regressors: H o : 1* exp + -1* ctrl = 0

36. Analysis Outputs Output from analysis software will typically include some visual representation of the results and tables of areas of activation Variety of tools available to interrogate and visualize results

37. Local Maxima Reporting Utility that generates a list of coordinates that correspond to the highest values in the statistical map grouped by ‘cluster’ Clusters are defined by spatially contiguous set of voxels above a statistical threshold (p-value) SPM reports the maxima within each cluster and up to 2 sub-maxima at least 8 mm from the other maxima

38. SPM Local Maxima Report

39. Voxel Surfing/Plotting Used to examine how well the changes in the MRI signal follow the on-off characteristics of the task

40. Basic Display of Results Simply display all of the t- statistics or other statistic in gray scale or with color coding Useful for getting an overall sense of the results Can more readily see if there is “ringing” Can see the data in its most basic form Use threshold of 0.1 or higher in SPM to get similar type of display

41. Glass Brain (SPM) Glass brain is a maximum intensity projection (MIP) generated for all three orthogonal planes Quick way to see what is activated when there are only a few areas

42. Fusion of Functional Results and Anatomical Image Use co-register to overlay onto a subject’s own anatomical scan SPM has option to overlay onto standard brain in Talairach space

43. Orthogonal Viewer (aka Sections in SPM) Method of exploring spatial relationship of activations relative to the anatomy Useful for precisely visualizing and localizing areas of activity

44. Volume Rendering Image processing method to generate 3D view of the anatomy and areas of activation Activations are displayed on the surface of the brain based on t-statistic & distance from surface SPM5 - areas deeper than 10mm have 1/2 the intensity of voxels closer to surface SPM5 has options to “brighten” activation to increase range of colors

45. Flat Map Algorithm that “irons out wrinkles” in the cortex to create a flat sheet representation of the cortex Freesurfer (Fischl & Dale) and Caret (Van Essen & Drury) are two main programs Brain is inflated and then cut to flatten

46. Flat Maps of Visual and Orbital-Frontal Areas

47. Inflated Brain

48. Gyri and sulci are denoted using different shades of gray when combined with functional data Useful for looking at borders of areas of activation Example includes retinotopic mapping

49. Surface Based Analysis

50. Writing up fMRI Data - Methods Describe subjects - how many, demographics (M/F, handedness, IQ, etc), inclusion/exclusion criteria, any scanned subjects thrown-out if so why Describe task (what were subjects asked to do) and performance (what did they actually do) Describe data acquisition protocol - scanner model, head-coil used, image sequence parameters, etc Describe analysis methods - give details; “We used SPM” is not sufficient, version of SPM used, processing steps used, parameters for each

51. Writing up fMRI Data - Results Location of results - typically use cluster peak though not always, specify using coordinates (be specific about MNI vs Talairach, conversion used if any) Location of results based on anatomical location (gyrus or other landmarks) Must have statistical analysis to support all claims and specify whether corrected for multiple comparisons and if so what method Visual depiction of results - nice to see overlay of stats on high resolution image for all slices not just those that are you are most interested in reporting about though not always possible due to space (consider inclusion in supplementary material section) Plots especially for interactions - Group A > Group B for Task-Control