lueurolmage
I.+, Number
6, 2001, Part 2 of 2 Parts 1 DE
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METHODS
- ANALYSIS
Negative variance problem in fMR1 Didier G. Leibovici Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), University of Oxford, John Radclifse Hospital, Oxford, U.K. How many subjects should be included for an fMRJ study? Besides sample size calculation using power analysis, another constraint is the so-called “negative variance problem” in random subject analysis. This paradoxal value for the subject-variance-component arises by the difference of quantities when using ANOVA variance component estimation [l]. Other methods of variance component estimation do not get this problem only because of a constraint (not to be negative) in the algorithm. This estimation problem is linked to sample sizes (number of subjects: rr, and the number of within-subject measurements: 7) and theoretical ratio of variances (between-subject/wirhin-subject). Intuitively if theoretically this ratio is high, a few subjects will be able to p@ < 0) = p(Rmw/Fm < 1) = FJcF$y .I) > 1 + 0) generate leading a reasonable between-subject variance eswhere$=T7=7i$/o$ timation, but if the ratio is small, more subjects are needed. Sample
sizes and negative
variance
Comparing jixed and random subject analysis one gets for every contrast: Rvar=behveen-subject-variance + Fxvar. Thus, prob(negative estimate)=p(Rvar/FxvarFxvar or RvarFxvar. Influence on sample size for these different maps are seen on fig-l for a verbal study on 12 subjects (courtesy, Jane Adcock, PMRJB). The range of the ratios over regions where the Gaussianised r-map was >2.3 were respectively for 12 and 6 subjects: [0.3-411 and [0.05;18]
tab-l. Uncvrr&& P-value of nega:ive estimate or ratio-var than 1 under Lheoretieal ratios.
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8 d b,
Discussion/conclusion With high ratios of variance (Rvar/Fxvar) we are confident that the Rvar is estimated properly or at least not “wrongly”. With small ratios, either no between-subject variation (quite unbeK@l. Ratii over 6 and 12 subjectli with F tev& (values or lievable) or not enough subjects to estimate it properly can be pvalucs) reject [top] against Rmdom>Pix(unnztad); 8 Irubjeek 10-6 = supposed. If Rvar is too big, one would need a large sample 11.6, Wa = 6.3, 046 = 6.26 and 12 sub&&x 1O-6 = 7, 1O-a = 6.4, size to be able to get significant values on the group map or an 0.06 = 4.5; pottom] nat ~> 1 will produce more acceptable ratios but is the smoothing with these outliers legitimate? References [l] Searle, S.R. Variance Components. John Wiley 1992. [2] Worsley, K. et al. A unified statistical approach for determining significant 4156-73. [3] Worsley, K. et al. A General Statistical Analysis for fMRI Data. HBM
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signals
in images
2OOOS648.
of cerebral
activation.
HBM,
1995,
