Hmm,
Scalepack uses a subtle twist on the 1/sig^2 weighting that has some interesting consequences.

According to the manual, weights for individual observations are calculted as such:

W= 1/((sigma*sscale*fscale)^2 + (<I>*error)^2)

where
"sigma" is taken directly from integration
"sscale" is the scaling factor for these sigmas
"fscale" is the film scaling factor
"<I>" is the expected intensity
"error" is the estimated error of each measurement

Making some simplifications, we can say that the film scale and sigma scale (fscale and sscale) are both 1.0.  If that is the case and the estimated error is zero, than the weights reduce to 1/sigma^2 as per a previous response.  However, because error is never zero, the weights by which sigmas are scaled change notably.  I've attached a plot of weight factors as determined from the scalepack model that shows normalized weights as a function of I/Sigma for several different error models.

The models included are 
1/S weighting
1/S^2 weighting (scalepack case with estimated error = 0.0)
1/S^2 + estimated error of 2,5,10,25,and 50 percent.

Error models that include a non-zero estimated error differ from straight ahead inverse sigma weighting.  Basically, when I/sigma is roughly equal to the inverse of the estimated error, (e.g. I/sigma = 50 when estimated error = 0.02), the weights reach a plateau and do not change significantly as signal to noise increases.  However, below this cutoff, the weights shift with signal to noise ratio as they do for 1/S^2 weighting.  The net effect is that weighting becomes more balanced overall with less emphasis on the strongest data and more on the weaker.  In detail, the estimated error defines a point at which the data are considered "error free", that is, a single large weight is applied to all data above a certain signal to noise cutoff.  Below this cutoff, weights behave as normal 1/S^2 weights but are proportionally higher than they would be without the estimated error factor.

This phenomenon is part of the reason I don't like to vary estimated error with resolution in scalepack.  To me, the estimated error reflects an overall error of measurement associated with a given experimental setup (crystal, image plate, scanner, CCD, etc.) and not a property of a crystal that changes with resolution.  Basically, the estimated error establishes a threshold above which data are considered trustworthy enough not to be down-weighted and either a reflection is strong enough not be down-weighted or it isn't, resolution shouldn't be a factor.  Obviously the program authors disagree with me or else they would not have make resolution dependent weighting an option.

Of course, other programs such as Scala do different things. 

Anyway, that's my $0.02

--Paul

Scalepack uses

--- On Tue, 3/16/10, Jacob Keller <[log in to unmask]> wrote:

> From: Jacob Keller <[log in to unmask]>
> Subject: [ccp4bb] Equation for combining measurements
> To: [log in to unmask]
> Date: Tuesday, March 16, 2010, 7:00 PM
> Dear Crystallographers,
> 
> A basic question: what is the equation currently used to
> combine/merge multiple measurements of I and sigI? (Or a
> reference would be fine as well (or maybe better.))
> 
> Thanks,
> 
> Jacob Keller
> 
> *******************************************
> Jacob Pearson Keller
> Northwestern University
> Medical Scientist Training Program
> Dallos Laboratory
> F. Searle 1-240
> 2240 Campus Drive
> Evanston IL 60208
> lab: 847.491.2438
> cel: 773.608.9185
> email: [log in to unmask]
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>