Hi everyone

A really useful overview of how to correctly specify contrasts when including derivatives 
from Susan Whitfield-Gabrieli.

Cheers

Christopher

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The standard way to test models that include temporal (and/or dispersion)
derivative terms is to use F-tests (for single subject analyses), and
repeated-measures one-way anovas (for second-level analyses). When looking at
the results of a single subject you would specify an F- contrast with two
rows: one with ones in the canonical hrf terms (one per session); and
another with ones in the temporal derivative terms (also one per
session). For example, for a design with four sessions, it would be a
simple F-contrast of the form: [1 0 1 0 1 0 1 0; 0 1 0 1 0 1 0 1]. In
order to perform second-level analyses you would first obtain
individual contrasts for each of the canonical and derivative terms
from each subject (two contrast volumes per subject), and then enter
these volumes into a repeated-measures one-way anova (without
constant) model with one factor that has two levels (one for the
canonical and one for the temporal derivative effects; or three levels
if you also include a dispersion derivative term). This anova should
not assume equal variance across levels of your factor (to account for
different variances between the canonical and derivative terms), and
should not assume sphericity (to account for possible across-subject
correlations between the two terms). After estimating the second-level
anova, the effect of interest is again tested using an F-test of the
form [1 0; 0 1] (which tests whether either the canonical and/or
derivative terms are different from zero).

One caveat of this standard approach using F- tests is that these are
non-directional tests (they will not tell you the directionality of
the effect, e.g. activation / deactivation). An alternative approach
was suggested by Calhoun (2004) in order to be able to
perform directional tests in the context of models that include a
temporal derivative term. The approach is relatively simple, you would
first obtain the individual contrasts for each of the canonical and
derivative terms from each subject (two contrast volumes per subject),
as before. Then, instead of entering these volumes directly into a
second level analysis, you would compute, for each subject, a single
volume estimating the "amplitude" of the effects at each voxel =
sign(V1).*sqrt(V1.^2+V2.^2), where V1 is the canonical effect contrast
volume, and V2 is the temporal derivative effect contrast volume.
These "amplitude" effects estimate the amplitude of the peak response
irrespective of the delay at which it occurrs (within a reasonable
range). Last you would enter these "amplitude" volumes in a simple
second-level t-test for population inferences. This mantains the
advantages of modeling the temporal derivatives (e.g. modeling
differential response delays) while allowing simple directional tests
of the BOLD responses (e.g. looking at activation / deactivation).

Just as a last note, the contrast [.5 .5] that you were mentioning
looking at both the canonical and temporal derivative terms is not
particularly effective because it is simply estimating the average
between the two estimated effects. This average could well be zero
when there is still a significant response, for example when the
actual response is positive (activation) and it preceeds the modeled
response the estimated temporal derivative term will be negative while
the canonical term will be positive, so one can imagine how these
terms could well cancel out leading to an test that is not sensitive
in this scenario. In addition consider that the scaling of the two
terms is different so a simple averaging is again not particularly
meaningful (testing a linear combination of the two terms
is effectively testing a canonical response that is delayed by an
amount given by the particular linear combination used). This is why
in practice instead of using t-contrasts of this form one wants to:
either use F- contrasts (testing the presence of a response at any
given delay within the range modeled by the derivative term), or
compute these "amplitude" effects (effectively computing the size of
the peak response not matter at what delay this peak response occurrs)
to again test for the presence of a response irrespective of the delay
at which it occurs.


REFERENCE: Calhoun, Stevens, Pearson, Kiehl (2004) Neuroimage 22, 252-7