A Nonparametric approach 6 AB randomise 6 BA

A Nonparametric approach…

6 AB… randomise 6 BA… V 5 PET activation experiment… B A B A B A B A B A B A B A B A B A difference 1 2 3 4 5 6 A B A B A B A B A B A B A B A B A B 7 8 9 10 11 12 12 subjects mean difference ¸ variance t-statistic = i

…example H 0: scan would have been same whatever the condition – labelling as active or baseline arbitrary – re-label scans equally likely statistic image • consider all possible relabellings (exchangability) permutation distribution • of each voxel statistic ? • of maximal voxel statistic

mean difference variance smoothed variance t-statistic “pseudo” t-statistic

Sn. PM with “pseudo” t-statistic permutation distribution SPM with standard t-statisic Sn. PM with standard t-statisic? – similar!

Sn. PM • Sn. PM: + minimal assumptions • guaranteed valid + intuitive, flexible, powerful + any statistic: voxel / summary + any summary statistic • maximum pseudo t – restricted volume – cluster size / height / mass – omnibus tests – computational burden – need sufficient relabellings • Uses • low df • dodgy parametric • no parametric results

Non-parametric tests in f. NI… • Classic tests • Wilcoxon rank sum test • Kolmogorov-Smirnov test • Permutation tests • Holmes, Arndt (PET) • Bullmore, Locascio (f. MRI) noise whitening, permutation • Nichols & Holmes (f. MRI) label (re)-randomisation 3 weak distributional assumptions Ôdon’t assume normality äreplace data by ranks Ôlose information äexchangeability Þ independence – f. MRI 3 minimal assumptions Ôexchangeability 3 valid often exact 3 multiple comparisons via maximal statistics 3 flexible ¢computational burden ¢sufficient permutations 3 additional power at low d. f. via “pseudo” t-statistics

Nonparametric approaches… Holmes AP, Blair RC, Watson JDG, Ford I (1996) “Non-Parametric Analysis of Statistic Images from Functional Mapping Experiments” Journal of Cerebral Blood Flow and Metabolism 16: 7 -22 Arndt S, Cizadlo T, Andreasen NC, Heckel D, Gold S, O'Leary DS (1996) “Tests for comparing images based on randomization and permutation methods” Journal of Cerebral Blood Flow and Metabolism 16: 1271 -1279 Nichols TE, Holmes AP (2000) “Nonparametric permutation tests for functional neuroimaging experiments: A primer with examples” Human Brain Mapping (accepted) Bullmore ET, Brammer M, Williams SCR, Rabe-Hesketh S, Janot N, David A, Mellers J, Howard R, Sham P (1995) “Statistical Methods of Estimation and Inference for Functional MR Image Analysis” Magnetic Resonance in Medicine 35: 261 -277 Locascio JJ, Jennings PJ, Moore CI, Corkin S (1997) “Time series analysis in the time domain and resampling methods for studies of functional magnetic resonance brain imaging” Human Brain Mapping 5: 168 -193 Raz J, Zheng H, Turetsky B (1999) “Statistical Tests for f. MRI based on experimental randomisation” (in preparation) Marchini JL, Ripley BD (2000) “A new statistical approach to detecting significant activation in functional MRI” (in preparation)