Hi Shy and Kalina,
My understanding is that beta equates to percent change regardless of
fit. If noise is high in the system, then the betas may be
inaccurate, but any measure of percent change would be equally
inaccurate.
Let's take the example of a block design where our regressor is equal
to 0 or 1 (eg. no hemodynamic smoothing). In this case, if the fit is
good, then all the data points at condition 0 would center around
that mean, and all the data points at condition 1 center around that
mean, and the difference in mean would be exactly equal to the
percent change.
If the fit is poor, the above logic is unchanged. The percent change
between conditions will equal the slope.
Now, we make the regressor vary from 0 to 1 by modeling the
hemodynamic response. The analogy gets a little more complicated: for
a change of say 0.2 in our regressor, the percent change we measure
is actually 20 % of the actual percent change. This is exactly what
the regression automatically does.
This is how I understand the equivalence between percent change and slope.
Your point is well taken, that is, if there is a lot of noise, then
how meaningful is the beta? I guess what I am saying is that this
question is equivalent to asking if there is a lot of noise, how
meaningful is the % change. For that reason, it is important to
understand and apply the statistics, be they fixed-effects, random
effects, and conjunction analyses, especially in points 1 + 2 that
you made.
I hope this is accurate and helpful.
Gosh, I've got a thesis to write. But this was too much fun to ignore.
Michael
At 12:47 AM -0700 4/16/01, Kalina Christoff wrote:
>Dear Shy and everyone,
>
> > I'm not sure if I understand why beta only represents the percent change
> > under perfect fit conditions. My understanding is that Y and beta are the
> > (data) fit and the fitting parameter respectively, and so should be
> > representative of the percent change always. Perhaps you can provide an
> > example? This old message by Karl Friston appears to agree with me:
> >
> > > > There is no automatic facility but the percent (of whole brain signal)
> > > > activation of a voxel is easily calculated from the parameter estimates
> > > > - the variable 'beta' in working memory following a plot. These values
> > > > correspond to a VOI defined by the spatial smoothing kernel, centred
> > > > on the selected voxel.
> > > >
> > > > I hope this helps - Karl
> >
> > I do agree however that perhaps instead of calculating the mean
>beta for the
> > whole VOI fitting the model to a "collective time series" (e.g. the
> > eigenseries that spm_regions returns), may give a more representative
> > number. Perhaps someone else can weigh in...
>
>
>I do hope someone else with more experience than me would comment further
>on this and weigh in.
>
>I might very well be missing an obvious thing, or I might just plainly be
>wrong. But - at the risk of exposing my ignorance further - let me try to
>express better, and in less extreme terms, what I meant in my previous
>email.
>
>The beta value in linear regression corresponds to the slope of the line
>fitted through the data points. If the data points fall on, or closely
>around, the regression line, there would be a good fit and the beta value
>would be descriptive of the average percent signal change. But let's also
>imagine a case where the data are spread so that they form a cloud that
>looks more like a circle or a square, or just a cloud with no particular
>shape. Now there would be many lines we can fit through these data, and
>some of them might have a steep slope (high beta), while some may have
>almost no slope (low beta). This situation would correspond to a poor fit
>and strictly speaking, the beta values would not be interpretable.
>
>In an activation map (T-values), we should see only voxels associated with
>betas coming from well fitted regression lines - because beta values
>from poorly fitted regression lines are associated with high error
>variance and therefore their corresponding T-statistics would be low.
>
>So one can argue that when an ROI is defined based on an activated
>cluster, the corresponding beta values would be indicative of the percent
>signal change. However, it seems to me this would only be true in a very
>specific case: when beta values corresponding to significantly activated
>voxels only are extracted. Furthermore, these voxels would have to be
>significantly activated during all conditions for which beta values are
>extracted.
>
>Such a specific case - in which beta values would correspond to percent
>signal change - would be, indeed, an on-off blocked design, where the beta
>values for a particular subject are extracted, from a cluster that is
>activated for this subject.
>
>However, any further complications in design, or question of interest,
>would pose dangers to using beta values as indicating percent signal
>change. The typical cases would be:
>
>1) when an ROI is defined anatomically, or as a cluster of activation at
>the group level. If it is now applied to individual subjects in order to
>extract beta values, it may lead to extracting beta values associated with
>poorly fit regression lines.
>
>2) when there are more than 2 conditions in the design. Let's say there
>are 3 conditions, A, B, and C. We define an ROI as a cluster activated in
>the A-B comparison, and we plot 3 bargraphs, with the average beta for all
>three conditions. The problem now would be that the average beta value
>for condition C may come from a poorly fit regression line, and may be
>very different than the percent signal change in the raw intensities.
>
>Sorry if I'm complicating things more than necessary - maybe your
>situation does not involve any of the two "complications" above. And maybe
>the short answer to your question is that, beta values would be indeed
>indicative of the percent signal change only if the variance associated
>with each of these average beta values is relatively small (though after
>averaging from many voxels in an ROI the variance may become quite small,
>without being originally so).
>
>I look forward to any comments on this,
>
>Kalina
>
>
>_____________________________________________________________________________
>Kalina Christoff Email: [log in to unmask]
> Office: Rm.430; (650) 725-0797
>Department of Psychology Home: (408) 245-2579
>Jordan Hall, Main Quad Fax: (650) 725-5699
>Stanford, CA 94305-2130 http://www-psych.stanford.edu/~kalina/
>_____________________________________________________________________________
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