In short, I would say that suggestion 1 is the way to go. A key thing to
point out is that when EV's partially overlap it does not mean that
they will be a problem with insufficent independence. The rank deficieny
warning, which kicks in when there is a concern that there is insufficient
information to fully drive the parameter estimation, is a conservative
warning. Hence, there are lots of occasions that this warning appears when
the design is fine, but it is there as a warning to double check your design.
In your case the design appears fine.
All the best.
Oxford University Centre for Functional MRI of the Brain (FMRIB),
John Radcliffe Hospital, Headington, Oxford OX3 9DU, UK.
Tel: (+44)1865-222782 Homepage: http://www.fmrib.ox.ac.uk/~woolrich
On Mon, 16 Feb 2004, Tommi Raij wrote:
> Hi all,
> I am trying to build a fMRI FEAT analysis that would analyze 2x2 factorial
> ANOVA interactions as a first-level analysis in a single subject. In the FSL
> archives this was mainly discussed as a second-level analysis, but I hope
> this can be done as a first-level analysis as well (the FEAT manual
> suggested this possibility but did not describe it much further). A major
> concern in this type of analysis is that the interaction is typically much
> smaller than the main effects. Thus any confounding effects or mistakes in
> the analysis can easily modify or become the whole outcome of a interaction
> study. Any advice how to set this analysis up _really_ correctly in FEAT
> would be much appreciated.
> In the stimuli, there are 2 factors (A and B) at 2 levels (ON and OFF). Thus
> there are 4 types of stimuli: A0B0 = REST, A1B0 = A, A0B1 = B, and A1B1 =
> AB. Physically, the "AB" condition corresponds to a stimulus where both A
> and B are presented simultaneously. Stimulus duration is 0.3 seconds for all
> stimuli A/B/AB.
> We use a sparse sampling design (clustered volume acquisition) with one full
> volume EPI every 10 seconds (taking one full volume of 24 slices takes 1.2
> seconds, which is followed by a pause of 8.8 seconds when no EPI data is
> collected). Each EPI is preceded by a single stimulus (one of the four
> categories REST/A/B/AB) at 4 seconds before EPI onset. The stimuli are
> presented in a random order. We collect about 50 full EPI volumes for each
> of the 4 stimulus categories (hence this is a repeated measures ANOVA with
> 50 values for each cell, for each voxel separately).
> In all the considerations below, EV1=stimulus A, EV2=stimulus B, and
> EV3=stimulus AB (_not_ the interaction). REST is defined implicitly by
> exclusion (as time periods when none of the paradigm files suggest there
> would have been a stimulus) and thus there is no separate paradigm file for
> The FEAT manual (Appendix A) describes "how to model nonlinear interactions
> between two EVs" (factors). I here assume that the "interaction" discussed
> in the FEAT manual at Appendix A is the 2x2 factorial ANOVA interaction,
> please correct me if I am wrong. Below I have three suggestions; Suggestion
> 1 is what the manual would seem to indicate but that does not work.
> *** Suggestion 1. The manual suggests "setting up two original EVs"
> (corresponding to stimuli A and B when presented alone?), and then "an
> interaction term, which will will only be up when both the original EVs are
> up, and down otherwise" (corresponding to stimulus AB?). However for
> Interaction, it is not possible to assign a paradigm file, and paradigm
> files can only be assigned to EV1 and EV2 (A and B). Since calculation of
> interaction requires values from all 4 cells (REST, A, B, and AB), there
> must be some way of telling FEAT how to distinguish between AB and REST
> events. The logical conclusion would be to construct two paradigm files, one
> that is "1" always when stimulus A is present (thus, for A and AB stimuli),
> and another that is "1" always when the stimulus B is present (thus, for B
> and AB stimuli) - thus the AB stimuli occur when both of the two paradigm
> files show "1". Then one would simply generate a third EV (EV3) and select
> that as Interaction between EV1 and EV2. Then one would build the following
> 4 contrasts:
> OC1 [1 0 0] ; for main effect A
> OC2 [0 1 0] ; for main effect B
> OC3 [0 0 1] ; for Interaction (positive)
> OC4 [0 0 -1] ; for Interaction (negative)
> However this leads to that EV1 and EV2 are overlapping (for AB stimuli, both
> are "1" at the same time), and thus they are NOT independent of each other.
> This violates the basic requirement of the GLM that all events be
> independent of each other. Attempts to set up the analysis this way result
> in an error of the type "Problem with processing the model: At least one EV
> is (close to) a linear combination of the others. You should probably alter
> your design. (Design matrix is rand deficient - ratio of min:max eigenvalues
> in SVD matrix is 1.283201e-06)".
> *** Suggestion 2. Given the failure of Suggestion 1, the only way to tell
> all the 4 cells apart (REST, A, B, and AB) would be to have at least 3
> paradigm files (for A, B, and AB; REST would be implicit from the other
> three files).
> Then, then to analyze the interaction, one should have three paradigm files
> (all in 3-column format).
> Paradigm file 1 has "1" every time only the stimulus A is on,
> Paradigm file 2 has "1" every time only the stimulus B is on, and
> Paradigm file 3 has "1" every time the AB stimulus is on.
> Now the three paradigm files are completely non-overlapping and thus
> independent of each other, and GLM should be happy.
> Then I should set the number of original EVs as 4, assign the Custom (3
> column format) paradigm files for EV1 (A), EV2 (B), EV3 (AB), and set
> "Interaction" for EV4, Between EVs [1 2 3] (all three buttons 1-3 selected
> for choosing the interaction components).
> In the contrasts I should select 5 contrasts (for original EVs?) with
> OC1 [1 0 0 0] ;A vs. REST
> OC2 [0 1 0 0] ;B vs. REST
> OC3 [0 0 1 0] ;AB vs. REST
> OC4 [0 0 0 1] ;Interaction (+)
> OC5 [0 0 0 -1] ; Interaction (-)
> No F-tests are selected in this approach.
> With this approach, there is again an error of the type "Problem with
> processing the model: At least one EV is (close to) a linear combination of
> the others. You should probably alter your design. (Design matrix is rand
> deficient - ratio of min:max eigenvalues in SVD matrix is 1.283201e-06)".
> Also according to the FEAT manual [regarding
> Stats/First-level/EVs/Interaction], "this EV [interaction] is produced by
> multiplying together selected EVs". Is this different from a 2x2 factorial
> ANOVA interaction?
> *** Suggestion 3. This is exactly the same as Suggestion 2 (including the
> above settings for EVs and contrasts), but now I add four F-tests in the
> General Linear Model/Contrasts & F-tests:
> F F F F
> 1 2 3 4
> OC1 [X - - -]
> OC2 [- X - -]
> OC3 [- - X -]
> OC4 [- - - X]
> OC5 [- - - X]
> The OC4 and OC5 F-tests should give the 2x2 factorial ANOVA interactions
> (for positive and negative interactions, respectively).
> Which one, if any, of the above would be the correct FEAT setting for
> first-level 2x2 factorial ANOVA w/ repeated measures ?
> Thanks in advance,
> Tommi Raij, M.D., Ph.D.
> Research Fellow
> MGH/MIT/HMS Athinoula A. Martinos Center for Biomedical Imaging
> Building 149, 13th Street, Mailcode 149-2301
> Charlestown, MA 02129 U.S.A.
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> FAX 1-617-726-7422