Dear Shy and everyone,
First, let me thank everyone for their comments on this issue so far --
On Mon, 16 Apr 2001, Shy Shoham wrote:
> 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 indeed my main concern (and thanks, by the way, for formulating it
quite clearly!).
But if I could raise one more side of this issue - there is also the
situation in which we are looking at event-related data. In an
event-related experiment, it is conceivable that there may be a poor fit
even in the presence of 'good data'. That could be a case in which the
data behaves quite consistently, and exhibits significant percent signal
change, but our model for how it behaves (e.g., the HRF) is not
perfect in describing it - which would (but please correct me here if I'm
wrong) result in low betas associated with high variance. If we looked at
these betas now, this may lead us to an underestimate of the true
percent signal change.
Given this, I wonder if our discussion about the validity of using beta
values as indicative of percent signal change, is also applicable to
event-related studies? My first reaction to this is "not quite", but
what do others think about this?
A very useful tool for dealing with this issue, but also with other
issues, would be a routine allowing easy report of the goodness-of-fit
index associated with each voxel. I wonder if anyone is thinking along
these lines or working on an SPM compatible facility for that (or for
regression diagnostics in general)?
> 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).
>
This seems a very interesting approach, and I would also be quite curious
to hear further comments --
All the best,
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/
_____________________________________________________________________________
|