ࡱ> RTS5@ 0Abjbj22 7JXXc8lhD)"((((((($*R-x!)i!))L(($V;'@' aVL{' ()0)'T-M-'-' !)!)$MLnonparametric approach to the multiple comparisons problem introduction / advertisement conceptually simple relying only on minimal assumptions => can be applied when assumptions of a parametric approach are untenable ..in some circumstances >> outperforms << parametric approaches -method is neither new nor contentious (origins Fisher 1935, renaissance now, with improved computing technology. "Had R.A. Fisher and hes peers had access to similar resources, possibly, large areas of parametric statistics would have gone undeveloped!") essential concept null hypothesis exchangeability block (eb) permutation distribution randomisation test, mechanics N: number of possible relabellings ti statistic corresponding to relabelling i set of ti for all possible relabellings, constitutes permutation distribution T: value of the statistic for the actual labelling (T is 'random', since under H0 it is chosen from permutation distribution) significance of T/ p-value: counting the proportion of the permutation distribution as or more extreme than T equivalently T must be greater or equal to 100 x (1- ) %-ile of the permutation distribution critical value: c=  N , rounded down, is (c+1)st largest number of permutation distribution single voxel example design: 6 scans 3 active [A] 3 baseline [B] presented alternately, starting with B any statistic can be used; eg. mean difference => T randomisation test: randomisation scheme has twenty outcoms eg. ABABAB, ABABBA, ABBAAB, ABBABA, H0: scans would have been the same whatever the experimental condition away from single voxel; multiple comparison permutation tests per voxel: p-value for H0 statistic summarising the voxel statistic, => MAXIMA presented here are 2 popular types of tests single threshold test suprathreshold cluster test a) single threshold test statistic image is thresholded and voxels with statistic values exceeding thresholds have their null hypotheses rejected => compute permutation distribution of the maximal voxel statistic over the volume of interest mechanics: for each possible relabelling i=1,,N, note timax critical threshold is c+1 largest member of permutation distribution for Tmax where c= [ N], rounded down b) suprathreshold cluster test starts by thresholding statistic image at a predetermined primary threshold then assesses resulting pattern of suprathreshold activity Such suprathreshold cluster tests are more powerful for functional neuroimaging data than the single threshold approach (on the cost of reduced localising power) considerations (only) assumptions: exchangeability additionally: pseudo t-statistics weighted locally pooled voxel variance estimates further constraint: number of possible relabellings smallest p-value largest p-value: limited by computational feasibility, => use a sub-sample of relabellings; approximate permutation test eg. 1 single subject PET with a parametric design ha! design consider: 1 subject, scanned 12 times, 3 randomisation blocks of 4 H0 ? => data would be the same whatever the duration relabelling: 4! ways to permute 4 labels = 24, 24^3 since each block is independently randomisedtotal 13,824 permutationsthis is too much! so we use approximate test; we randomly select 999 of 13,823 plus the T one cluster definition resp. setting of primary threshold; THE QUANDARY the hard bit (for the computer) results eg. 2 multi-subject PET design: n=12, 2 condition presentation orders in balanced randomisation; 6 subj. ABABAB& , 6 subj. BABABABA& H0 for each subject, experiment would have yielded same data were the conditions reversed exchangeability: relabelling enumeration: permute across subj. EB `" units of scans, EB = subjects; 12! / (6! (12-6)! = 924 ways of choosing 6 of 12 to have ABABABA statistic: important aspects: collapsing data within subj., computing statistics across subjects, repeated measures t-statistic eg. 3 multi-subject fMRI activation experiment fMRI data present a special challenge for nonparametric methods design: 12 subj., data: per subject difference image between test- and control condition H0: symmetric distribution of the voxel values of the subjects' difference images have zero mean. exchangeability: :;<Y[n |   1 2 3 C D E Z ` a b z { ϸϦygSgyL hCh@p&hJqhC5CJ(OJQJ\^JaJ(#hL5:CJ(OJQJ\^JaJ(hCCJ aJ mH sH  hLhChLh@p"hyB*hC6CJ$]aJ$mH sH "hyB*hyB*6CJ$]aJ$mH sH hyB*CJ$aJ$mH sH hChCCJ$aJ$hChCCJ$aJ$mH sH hCmH sH hq+mH sH  h@phq+h(T h@ph@p;<Y[o G 2 3 C D E a b { | S^SgdhTgdCS^SgdLgdLgdyB*gdCgdCgdq+gd(TAA{ | }   ; < = ñwgXLX@h$CJ aJ mH sH hJqCJ aJ mH sH hrxhrxCJ aJ mH sH hhTh!CJ H*aJ mH sH h!CJ aJ mH sH hhThrxCJ aJ mH sH hJqhrxCJ$aJ$mH sH hJqhrxCJ,H*aJ,mH sH "hJqhrx5CJ,\aJ,mH sH hrxCJ aJ mH sH "hJqhrx5CJ(\aJ(mH sH hrxmH sH  hCh@ph!hCCJ aJ mH sH  < = p X    .UVgd!gd:gdhTS^SgdJqS^Sgd!S^SgdhTgdrx= >  V  4 t x Ʒ▇{o{_o{[WOGh:mH sH h\smH sH hChJqhJqh$CJ H*aJ mH sH h\sCJ aJ mH sH hJqCJ aJ mH sH hJq5CJ(\aJ(mH sH "hJqh\s5CJ(\aJ(mH sH h$h$CJ aJ mH sH hhTh$CJ aJ mH sH hhTh!CJ H*aJ mH sH h!CJ aJ mH sH h$CJ aJ mH sH "hJqh$5CJ(\aJ(mH sH  VY567tv쬖|ph]UIpIhLqCJ aJ mH sH hLqmH sH hLqhLqmH sH h:mH sH hmCJ aJ mH sH hmCJ H*aJ mH sH h:CJ H*aJ mH sH h!h:0JH*hpuCJ aJ mH sH h!h:0Jh\sCJ aJ mH sH h:h:CJ aJ mH sH h!CJ aJ mH sH h:h:>*CJ aJ mH sH h:CJ aJ mH sH h!mH sH 6uv 'AB[S^Sgd$2S^SgdhTgd"gdhT & Fgd"gdmgdLqgdhTS^SgdmS^Sgd!'(+@B'*LNĸĬġwđaN;$h$25KHOJQJ\^JmH sH $h"5KHOJQJ\^JmH sH *h"h"5KHOJQJ\^JmH sH 2hmh"5CJ KH OJQJ\^JaJ mH sH h"mH sH hhTmH sH h"h"mH sH h^SCJ aJ mH sH hmCJ aJ mH sH h"CJ aJ mH sH hmh"0JhLqCJ aJ mH sH hmCJ H*aJ mH sH hLqCJ H*aJ mH sH NUVYZ[m&(,.d{hhWL;0hhThVrmH sH  hhT5OJQJ\^JmH sH hhThhTmH sH  hVr5OJQJ\^JmH sH $hVr5KHOJQJ\^JmH sH *hVrh"5KHOJQJ\^JmH sH *hVrhVr5KHOJQJ\^JmH sH hhVrmH sH $h$25KHOJQJ\^JmH sH 'h"5H*KHOJQJ\^JmH sH 'h"5H*KHOJQJ\^JmH sH $h"5KHOJQJ\^JmH sH [&fhvxz`pq"#gdrT gda gd`gdagdgdVrgdhTS^Sgd$2S^SgdhTh^hgdVrgdhTS^Sgd$2dfh _`aopqڴڴڨ{ofV?,h5CJ KH OJQJ\^JaJ mH sH haha:CJ&aJ&mH sH hhThhT0Jhg6CJ aJ mH sH hhTh-0J h0Jhg6hVrmH sH hg6h-mH sH h-mH sH h-CJ aJ mH sH hCJ aJ mH sH 2hhVr5CJ KH OJQJ\^JaJ mH sH hVrCJ aJ mH sH hVrhVrmH sH hVrh"CJ0aJ0mH sH !"#35<EYZlnȼuu^RuJB7h\^h\^mH sH h*yumH sH h\^mH sH hCJ aJ mH sH ,h5CJKH OJQJ\^JaJmH sH 2hhrT5CJKH OJQJ\^JaJmH sH hrTCJ aJ mH sH hrThrTCJ0aJ0mH sH hahrT0Jhahg60JhCJ aJ mH sH hhCJaJmH sH hhaCJaJmH sH 2hha5CJKH OJQJ\^JaJmH sH #4523hiBgd$ngdQ$ggdQ$gh^hgdQ$gS^SgdQ$gS^Sgdah^hgd\^S^SgdIwPgdagd\^gd*yugdgdrTgdIwP13457@_bcgiu/ABo^ hahQ$gCJ(KHaJ(mH sH .hah\^5CJ(OJQJ\^JaJ(mH sH h\^h\^CJ aJ mH sH hah\^0JH*hah\^0JhIwPCJ aJ mH sH hu?hCJ aJ mH sH h\^CJ aJ mH sH hamH sH hIwPmH sH h\^mH sH hrTmH sH h\^h\^mH sH h*yumH sH  prtx(2ƾƣwkwawREk<h*yuh$n0Jh$nCJ H*aJ mH sH h$nh$nCJ aJ mH sH hb4h$n0JH*h$nCJ aJ mH sH hb4h$n0Jh$nmH sH hu?hmH sH h{Eh{EmH sH h*yumH sH hQ$gKHmH sH  h?hQ$gCJ(KHaJ(mH sH hQ$gmH sH hQ$gCJ(KHaJ(mH sH  hahQ$gCJ(KHaJ(mH sH hQ$gCJ aJ mH sH hu?hhQ$gCJ0aJ0mH sH np&(:;F-.'@@@@gd?Ogd53gd2tgdgd+Lgd*yu^gdQ$ggdQ$ggd*yugd$n24FHT;EFv|,.5c@&@'@0@O@ּ֕֕։րvpnbbh53CJ aJ mH sH U hS%0Jh53h0JH*hS%h0JhCJ aJ mH sH hS%h+L0Jh+Lh$nmH sH h+Lh+LmH sH h*yumH sH h+LmH sH hQ$gCJ aJ mH sH  hQ$g0Jh+LCJ aJ mH sH h$nCJ aJ mH sH h*yuh$n0Jh*yuh*yu0J%single EB consisting of all subjects. consider subject labels of "+1" and "-1" => there are 212 = 4,096 possible ways of assigning either "+1" or "-1" to each subject. comparison with other methods Final word "non parametric method is very useful, especially if n is low (low degrees of freedom). It is much more powerful (in the context of multiple comparisons)." 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