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Dear Hans, However, I do have the following concerns about this approach. > > Suppose XN is the nuisance-only model and rN are the nuisance-only > residuals formed after fitting XN to data. Then > > (1) Elements of vector rN - rN(1),rN(2),..etc. are not independent. > > (2) Further, elements of rN have non-constant variance under the nuisance-only model (null hypothesis). > > Given (1) and (2), how do you justify the permutation of "raw" residuals > rN? > Well spotted! Indeed, if XN is anything other than a column of ones, the resulting residuals are not exchangeable. Hence the 'permutation GLM' approach that randomise takes is only approximately exact outside of the standard, simple models (i.e. one-sample t, two-sample t, simple correlation, etc). However, in a 2008 HBM poster<http://www.fmrib.ox.ac.uk/%7Enichols/NicholsRidgway_OHBM2008.pdf>, Ged Ridgway and I found that the approximate methods are actually quite accurate. See the poster for more details. It seems like (2) is easily corrected as follows: > > If, H = XN * inv(XN^T XN) * XN^T then if we transform the residuals > > rN_modified(k) = rN(k) / sqrt(1 - H(k,k)) > > then elements of rN_modified will have the same variance under the null > hypothesis. ( Elements of rN_modified are still not independent. ) > > I would have thought that rN_modified would be used for permutation. What > are your thoughts on this? > This could help, but note that fixing hetereogeneous variance doesn't address the dependence issue (i.e. that H isn't diagonal). It turns out that there is actually a fairly large literature on ways to do permutation with nuisance variables; we reference some papers in the poster, and Ged Ridgway's thesis has even more. I think what you're suggesting here is reminience of a paper that Ged found by a Korean group, but I'll have to defer to Ged on the details. So, again, to summarize, the method that randomise uses is indeed only approximate, but in our evaluations it appeared to be quite accurate, in particular in the most challenging (low n) cases. -Tom On Thu, Jul 2, 2009 at 3:39 PM, Thomas Nichols <[log in to unmask]>wrote: > Dear Hans, > > Yes, I can confirm, the "raw" residual from the nuisance-only model is not > modified in any way... it is simply permuted before having the nuisance > effect added back on. > > Does this clarify things? > > -Tom > > > On Thu, Jul 2, 2009 at 4:18 PM, Hans Tissot <[log in to unmask]> wrote: > >> Dear Jesper, >> Thanks for the reply. It does seem pretty straightforward. However, the >> devil is often in the details. >> >> So to summarize your point - >> >> (1) Let the residual vector for a voxel be r = [r_1,r_2,...r_n] >> (after fitting the null only model, n = timepoints in the model) >> >> (2) This "raw" residual vector r is permuted. It is *not* modified or >> standardized in any fashion before permuting. >> >> Is that accurate? >> >> Thanks, >> Hans. >> >> >> On Thu, Jul 2, 2009 at 11:02 AM, Jesper Andersson <[log in to unmask]>wrote: >> >>> Dear Hans, >>> >>> Thanks, but I am interested in a more detailed answer about exactly how >>> the residuals are permuted :). Here's my question again for reference: >>> >>> --- >>> I have a question regarding randomise residuals. As per my understanding >>> randomise fits the null model only to the data and calculates null only >>> residuals. Then it permutes these null only residuals and adds them back >>> onto the fitted null model to create realizations of null data. My question >>> is: >>> >>> Are these null only residuals modified (or standardized) in any way >>> before permuting them? If so, exactly how? >>> >>> >>> I'm no expert on randomise, but is seems pretty straightforward to me. >>> >>> Let's say you have a model with two groups and age as a covariate, and >>> that your contrast happens to be [1 0], i.e. you are interested in effects >>> of group after affects of age have been removed. >>> >>> By virtue of you contrast not spanning the age regressor randomise can >>> identify it as a "confound" and regress out all effects of age. What is left >>> is the residuals, i.e. that which in our model can be explained either by >>> group or not at all. randomise will the permute these residuals (equivalent >>> to permuting the group indicators), for each permutation fitting the GLM to >>> all voxels and calculating the t-statistic. Depending on your inference it >>> may then save away maximum voxel, maximum cluster size etc, thus building an >>> empirical distribution of that statistic. >>> >>> I hope this is clear? >>> >>> Good Luck Jesper >>> >>> --- >>> >>> Thanks, >>> Hans. >>> >>> >>> >>> On Thu, Jul 2, 2009 at 10:48 AM, Matthew Webster < >>> [log in to unmask]> wrote: >>> >>>> Hello Hans, >>>> Randomise separates the input model into tested and >>>> nuisance effects, the input data is adjusted for the nuisance effects and >>>> this adjusted data is then fitted to the full permuted model.. >>>> >>>> Many Regards >>>> >>>> Matthew >>>> >>>> Hi FSL experts, >>>>> >>>>> I have a question regarding randomise residuals. As per my >>>>> understanding randomise fits the null model only to the data and calculates >>>>> null only residuals. Then it permutes these null only residuals and adds >>>>> them back onto the fitted null model to create realizations of null data. My >>>>> question is: >>>>> >>>>> Are these null only residuals modified (or standardized) in any way >>>>> before permuting them? If so, exactly how? >>>>> >>>>> Thanks, >>>>> >>>>> Hans Tissot. >>>>> >>>> >>> >>> >> > > > -- > ____________________________________________ > Thomas Nichols, PhD > Director, Modelling & Genetics > GlaxoSmithKline Clinical Imaging Centre > > Senior Research Fellow > Oxford University FMRIB Centre > -- ____________________________________________ Thomas Nichols, PhD Director, Modelling & Genetics GlaxoSmithKline Clinical Imaging Centre Senior Research Fellow Oxford University FMRIB Centre