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Variable Threshold Methods for Functional Neuroimaging. Thomas Nichols, Ph.D. Department of Biostatistics U. of Michigan Randy Buckner, Ph.D. Department of Psychology, Washington U., St. Louis. Introduction.
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Variable Threshold Methods for Functional Neuroimaging Thomas Nichols, Ph.D. Department of Biostatistics U. of Michigan Randy Buckner, Ph.D. Department of Psychology, Washington U., St. Louis
Introduction Functional neuroimaging analyses usually result in a statistic image which assesses voxelwise evidence for an experimental effect. Conventionally this image is thresholded with a single value, and any voxels greater than the threshold are declared significant. While unconventional, a spatially varying threshold can be used as long as one controls the Familywise Error Rate (FWER), the chance of one or more false positives. In this work we show how variable thresholding of statistical maps can be used to focus power. We present a specific rule for Volume of Interest (VOI) analyses, such that the remainder of the brain is assessed yet power is still focused on the VOIs. We demonstrate the general method on a PET dataset.
Motivation • Several possible strategies for variable thresholds • Focus power • Lower thresholds in regions of expected activation,while raising thresholds elsewhere • Equalize power • Lower threshold in regions of high variability • Optimize arbitrary criterion • E.g. Allocate 80% power to voxels with high gray matter content. • This work uses variable thresholds to focus power within a VOI
FWE FWE Problem Illustration:VOI and Whole Brain Analyses • 10 simulated 2D null datasets • First analyze VOI only, control FWER at 10% • Then analyze whole brain, control FWE at 10% Null Data 2,300 voxels in brain 300 voxels in VOI VOI =0.01/300, u=3.41 Rest of Brain =0.01/2000, u=3.89 Total FWE FWE FWE FWER > %10
FWER controlled for each analysis, but • Using both analyses allows excess false positives • Can show FWER always exceeds nominal 0 and is almost (J+1)0, where J is the number of VOIs. • Two possible solutions, one using a variable-threshold Bonferroni rule, another using an empirical distribution of weighted p-values
Preliminaries • Testing voxels 1,...,i,...,V in the brain • Want to find set of -thresholds {i}i=1,...,V to control FWER overall at 0 (say 5%) • Brain divided into J+1 VOIs • VOIs 1 through J are of interest • VOI J+1 is the rest of the brain • VOI j contains Vj voxels, V = jVj • Voxel indices of VOI j are Vj • Voxel i significant if p-value pii
Solution I: Bonferroni • Any set thresholds {i} that satisfy ii = 0control the FWER • Note that i = 0/V, the usual constant-threshold Bonferroni rule satisfies this definition • FWER-Corrected Bonferroni p-values • piB = (0/i)pi = wipi • A weighted p-value, with weight wi = (0/i) • VOI j has FWER iVji,
Apportioning FWER • Constant-threshold • VOI j has (Vj/V)100% of the FWER 0 • Arbitrary apportioning 0 • Dedicate qj100% of 0 to VOI j • j qj = 1 • The -thresholds are then i = 0qj/Vj, for all iVj • The corrected/weighted p-values are wipi , wi = Vj/qj
Solution I: Special Case, VOI Averaged Data • Simple result for data summarized by VOI • Have J VOI’s of interest, but also want to look at whole brain data • Use threshold 0/(J+1) on each VOI test • Use FWER-corrected threshold of 0/(J+1) on whole brain data • By Bonferroni, this will control FWER overall • Not so conservative since dependencies usually not so strong between VOIs and between each VOI and brain data as a whole.
Example • Have 9 averaged VOIs, one t-test for each • Use -threshold of 0.05/(9+1) = 0.005 on each • Use random field/permutation/Bonferroni method to threshold whole brain at 0.005 FWER-corrected. • Equally “spends” 0/(J+1) = 0.005 on each VOI and remainder of the brain
Solution II:Permutation Test • Previous result (create corrected p-values with wipi, wi = Vj/qj) is conservative, as it is based on Bonferroni • Can improve sensitivity by building empirical distribution of minimum weighted p-values (Westfall & Young, pp 184-187) • Weighting accounts for differential sensitivity • Taking minimum accounts for multiple testing problem, result then controls FWER
FWER-Corrected weighted p-values are • piw = P( mini*wi*pi*wipi ) • Estimate this with the permutation distribution of {mini*wi*pi* } • For each relabeling of data, compute p-value image, apply weights, then note minimum.
Real Data Example • PET finger opposition task • One subject, 12 scans (Kinahan & Noll, 1999) • A priori VOI including motor strip • x>0,-40<y<0,z>0, Talairach • 76,023 voxel in the brain, 9.4% in VOI • Analyzed data three ways • Single threshold (usual whole brain approach) • Variable threshold • Dedicate q1=75% of 0 to the VOI, 25% to rest • Single threshold on VOI only
Results • Variable threshold method • Has lower threshold in VOI, higher outside • VOI-only method • More sensitive, but can’t make inferences outside VOI VOI t Mean
Conclusion • Variable thresholding methods are used implicitly with VOI studies, but can be applied more subtly; power can be focused on regions, yet some 0 can be reserved to spread thinly over the remainder of the brain. • Simple Result: If Bonferroni-correcting for J VOIs, instead correct for J+1 regions, and examine whole brain at 0/(J+1) corrected.
