> I have several questions concerning an er-fMRI design with parametric
> modulation. I have a learning task and I want to look for modulations of
> activity in time, using both 1st and 2nd order polynomial basis functions. I
> am confused about the contrasts to use.
> Say I have one condition A with a 2nd order parametric modulation: the first
> three regressors in the design matrix will include the convolved onset
> vector, the linear term, and the nonlinear term, respectively.
>
> 1. Is it correct to use a [0 1 1] F-contrast to look for areas that show
> either linear OR nonlinear modulation of activity with time? Will the
> t-contrast [0 1 0] show linear modulations alone, or would I have to set up a
> separate model with only 1st order parametric modulation?

To test a linear and/or quadratic modulation, use the F-contrast:

    [0 1 0
     0 0 1]

(your contrast [0 1 1] is not appropriate, since it only tests for the mean
of linear and nonlinear effects, which is not particularly meaningful).

A [0 1 0] contrast tests for significant linear component, but does not prevent
there also being a quadratic component.


> 2. As far as I know, in a design with only 1st order parametric modulation and
> one condition A, the t-contrast [0 1] will give increases and the t-contrast
> [0 -1] will give decreases of BOLD activity in time.
> In a design with 2nd order parametric modulation, the rationale for using a +1
> rather than a -1 weight are not clear to me. Looking at plots of parametric
> responses, I have noticed that a +1 t-contrast gives significant areas that
> show an upright U-shaped curve, whereas a -1 t-contrast gives significant
> areas that shown an inverse U-shaped curve.
> Is this something that should be expected?
> More in general, is there any way, with 2nd order parametric modulations, to
> assess the directionality (shape) of the effects, in the same way as one does
> with linear parametric modulations (+1 = signal increase, as opposed to -1 =
> signal decrease)?

It is just as you say: a +1 weight on the quadratic (2nd-order) column tests for
a U-shape; a -1 weight for an inverted U.

Note that the overall pattern in a voxel depends on both the 1st and 2nd order
estimates and could be monotonic or non-monotonic (depending on the relative
weight of the two estimates). If monotonic, you could, I suppose, label as an
overall increase or decrease.


> 3. In my actual experiment I have two conditions A and B. I want to look for
> areas whose activity is significantly more modulated in time in condition A
> than in B. I set up a RFX model with a group of subjects in which I compare
> the contrasts testing for [task A x time] interactions with contrasts testing
> for [task B x time] interactions. Using a F-test to contrast the interactions
> of task A with those of task B, I find a significant area X.  The plot of
> contrast estimates shows that the effect has a negative sign.
> If I set up a separate RFX model including only [task A x time] interactions,
> I find the same area X to be significantly modulated and, again, the plot of
> contrast estimates shows that the effect has a negative sign (indicating an
> inverse U shaped signal time course).
> How are these results to be interpreted? As an evidence that:
> a) the activity in area X is significantly more modulated in time in task B
> than in task A.
> b) the signal of area X in condition A has an inverse U shape and the effect
> of condition A is more significant than condition B.
> Extracting the plot of parametric response in area X averaged over all
> subjects seem to support interpretation b), as condition A presents a
> relatively pronounced inverse U-shaped response, whereas condition B presents
> an 'almost flat' upright U-shaped response.

It depends on 1) what F-test you did in the model with both A and B modulations,
and 2) what contrasts you used to create the con*.img files that comprise the data for
the model.

Assuming the data were con*.imgs created from T-contrasts [0 0 1] on quadratic
modulations of A and B separately (or simply the beta*.img files for the relevant columns)...

...and the F-contrast in your model with A and B was [1 -1]...

then a negative value for this F-contrast at voxel X means that B shows more
of a U-shaped modulation than does A.

As you realised, you cannot tell from this alone whether A and B both show
U-shaped modulations (just larger for B), both show inverted-U-shaped
modulations (just smaller for B), or A shows an inverted-U-shape and B a U-shape.

Your second model with A only goes some way towards determining the precise
pattern, though you would need to do the same with B only to be sure.

Or more simply, you could plot the F-contrast [1 0; 0 1] - assuming you didn't
model subject effects (or if you did, I think ploting the predicted data might
give you an idea of the sign of both modulations - or refit without the subject effects).

Note that you could also take both linear and quadratic modulation contrasts
to a 2nd-level model (an ANOVA with 4 conditions) and evaluate the F-contrast
[1 0 -1 0; 0 1 0 -1] in order to test for any difference in modulations (linear
and/or quadratic) - but remember to use nonsphericity correction to allow
for different scalings of the linear and quadratic effects (and possibly for
correlations among errors if A and B are repeated measures).

Rik


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DR RICHARD HENSON

MRC Cognition & Brain Sciences Unit
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URL:    http://www.mrc-cbu.cam.ac.uk/~rik.henson

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