Eric,

We don't actually use the selective-averaging paradigm per se.  I
was just using the term because it seems to have entered common
usage.  As Karl noted, selective averaging is just a special case
of a linear model.  If we assume that the responses are linear and
the noise is white and Gaussian, the estimates of the time-courses will
be unbiased as long as the design matrix has full rank.  We would 
also like the variance to be roughly equal across all samples of the time-
courses. To do this, we usually jitter the SOA over more intervals than 
are required just to get a full-rank design matrix.

Some people in our group use counter-balancing because it is intuitively
appealing and can be incorporated into most protocols, but is certainly
not a requirement.  We have done one study (without counterbalancing)
that showed no significant difference between timecourses estimated 
from rapidly presented trials and those estimated from widely spaced trials.  

I would expect the residual errors from a linear model doing point
estimation (I will use that term instead of selective averaging) to
be closer to iid than those from a model using temporal basis functions
because the basis set (i.e, kronecker delta functions) spans the space of
all possible response functions, so the null space is empty. For temporal 
basis functions, the null space is nonempty and can contribute 
correlation to the residuals.  I wouldn't lose any sleep over this one
though.  The residuals arent iid anyway because we don't have perfect
methods to model the non-whiteness of the noise.



					John

--------------------------------------------------------------

John Ollinger
Washington University  
Neuro-imaging Laboratory
Campus Box 8225	        
St. Louis, MO 63110			 
http://imaging.wustl.edu/Ollinger

On Fri, 5 Mar 1999, ERIC  ZARAHN wrote:

> There is a question or two that I have about selective averaging that
> no one seems to have mentioned (so it is probably a non-issue,
> but I'll ask anyway). I'd appreciate any comments from John, Richard,
> or anyone else who feels comfortable with the theory of this method:
> 
> 1) Can one show that the estimates of the response that one obtains for
> a particular event are unbiased as in least-squares solutions of the GLM?
> 
> The reason my intuition raises this question is because when
> averaging in a fashion time-locked to some event, the method 
> is ignorant of other events (both those of the same class as well
> as those of other classes) in the paradigm. Now perhaps with a
> counterbalancing scheme, you could try and cancel the effect of the
> other events, but this would then be an additional constraint
> in the use of selective averaging that is not present for the use
> of least-squares-GLM. In addition, would not such counterbalancing
> be only approximate in practice since to be perfect one would have to
> counterbalance for every position relative to each event?
> 
> 2) Relatedly, what are the properties of the residual errors one obtains
> with selective averaging? That is, are they iid even in the absence
> of counterbalancing; what about with counterbalancing? In what way can
> this be proved?
> 
> I'd greatly appreciate any help here.
> 
> Sincerely,
> Eric 
> 
> 
> Eric Zarahn
> University of Pennsylvania
> 



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