Dear Greig,
> I've just read the Buchel et al. (1998) paper on zero, first and second
> order terms in analyses of parametric fMRI studies. I'd like to conduct a
> similar analysis/series of analyses on some data that I have that I
> mentioned in a previous post regarding a parametric study design. To recap,
> conditions run in the order 1412131 where conditions 2, 3 and 4 represent
> increases in level of difficulty and condition 1 a control. If I understand
> the Buchel et al. (1998) paper correctly, the approach I adopt should
> proceed something like this:
>
> Analysis (1): model the simple box-car between task and control
> irrespective of the manipulation of difficulty (conditions 1212121, boxcar
> -1 1) [zero-order]
Correct.
> Analysis (2): linear term as single covariate of interest (e.g., -3 -1 1 3
> for conditions now entered as 1412131) with confound/covariate of no
> interest being the simple boxcar function from (1) above [first-order]
> Analysis (3): quadratic term as single covariate of interest (1 -3 -3 1)
> now with 2 confounds/covariates of no interest entered, namely the boxcar
> and linear terms from (1) and (2) above [second-order expansion]
> With covariates (inc. confounds) convolved with a hrf.
> Is this correct? And returning to the issue of interpretation raised
> earlier - compare the SPM{F} maps from (1) to (3) above?
Seems ok. Be aware of the fact that you have only 4 levels for the
parameter. My guess is that the second order model will only reveal
very strong (inverted e.g. U-shape) non-linearities. That is similar to
the paper where we had 5 levels. Anything that is only slightly
curvilinear will be sufficiently explained by the linear term in
analysis 2.
Another thing to keep in mind is that the control condition might not
reflect level one of the parameter. This can lead to spurious
non-linearites, where the control condition shows very low signal and
all other conditions show high signal. In this case the second order
gives a nice approximation of this step function. BUT this is not a
non-linearity related to the parameter of interest. In our paper we did
not include the baseline condition.
The approach described in Buchel et al. (1998) is a way of model
selection. This is particularly problematic in fMRI, where one tries to
explain the variance in all voxels with the same design matrix. Andrew
is thinking about proper model selection in the context of functional
neuroimaging. So for the time being, the technique proposed in the
paper might indicate that a linear model is sufficient to explain the
variance in region B and that a non-linear model explains more variance
in region A. That also means that if a voxel is significant in analysis
3, that was not significant in analysis 2 there might be a difference
in the response function for these two voxels. BUT, you haven't tested
for that yet. To do that you have to directly compare both voxels
outside SPM. If you want to compare two groups of subjects you can use
the technique presented in Buchel et al. 1996 (Neuroimage). Although
this was applied to PET data, it is conceptually identical to the
approach presented here.
-Christian
Dr Christian Buechel
Wellcome Dept of Cognitive Neurology
London, UK
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