Dear Kalina,

it appears that your concern is with regard to the inequivalence between
betas estimated from 'good data' segments vs. 'bad data' segments, and not
so much with the validity of beta as a representative of 'percent change'.
This is a valid concern, and if we consider the different betas (from
different pixels and across subjects) as independent measurements with
associated measurement errors (estimation variance) we can easily derive the
optimal weights. Assuming indepence between the parameter estimation errors
the MSE estimator for BETA (the 'true' beta underlying all the measured
betas) is given by the formula derived by Gauss:

VAR=1/((1/sigma1^2)+(1/sigma2^2)+...)
BETA=((beta1/sigma1^2)+(beta2/sigma2^2)+...)*VAR

where sigma1, sigma2... are the estimation variances of beta1, beta2...
Intuitively, this formula downweighs voxels with high variance.

As far as I understand, the sigmas can be calculated using: ResMS*xX.Bcov
(a little more tinkering is needed in order to get those variables).

I hope this is correct...   Does anyone have any comments or a simpler way
in mind?
                                                                                        Shy

> -----Original Message-----
> From: Kalina Christoff [mailto:[log in to unmask]]
> Sent: Monday, April 16, 2001 1:47 AM
> To: Shy Shoham
> Cc: [log in to unmask]
> Subject: RE: percent signal change questions (a simple hack)
>
>
>
> 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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