James Holton suggested a reason why the "forefathers" used a 3-sigma cutoff.
I'll give another reason provided to me years ago by one of those guys, Lyle Jensen. In
the 70s we were interested in the effects of data-set thresholds on refinement (Acta
Cryst., B31, 1507-1509 (1975)) so he explained his view of the history of
"less-than" cutoffs for me. It was a very Seattle-centric explanation.
In the 50s and 60s, Lyle collected intensity data using an integrating Weissenberg camera and a
film densitometer. Some reflections had intensities below the fog or background level of the film
and were labeled "unobserved". Sometimes they were used in refinement, but only if the
calculated Fc values were above the "unobserved" value.
When diffractometers came along with their scintillation counters, there were measured quantities
for each reflection (sometimes negative), and Lyle needed some way to compare structures refined
with diffractometer data with those obtained using film methods. Through some method he never
explained, a value of 2-sigma(I) defining "less-thans" was deemed comparable to the
"unobserved" criterion used for the earlier structures. His justification for the
2-sigma cutoff was that it allowed him to understand the refinement behavior and R values of these
data sets collected with newer technology.
I don't know who all contributed to the idea of a 2-sigma cutoff, nor whether
there were theoretical arguments for it. I suspect the idea of some type of
cutoff was discussed at ACA meetings and other places. And a 2-sigma cutoff
might have sprung up independently in many labs.
I think the gradual shift to a 3-sigma cutoff was akin to "grade inflation".
If you could improve your R values with a 2-sigma cutoff, 3-sigma would probably be
better. So people tried it. It might be interesting to figure out how that was brought
under control. I suspect a few troublesome structures and some persistent editors and
referees gradually raised our group consciousness to avoid the use of 3-sigma cutoffs.
Ron
On Mon, 30 Jan 2012, James Holton wrote:
Once upon a time, it was customary to apply a 3-sigma cutoff to each and
every spot observation, and I believe this was the era when the "~35% Rmerge
in the outermost bin" rule was conceived, alongside the "80% completeness"
rule. Together, these actually do make a " reasonable" two-pronged criterion
for the resolution limit.
Now, by "reasonable" I don't mean "true", just that there is "reasoning"
behind it. If you are applying a 3-sigma cutoff to spots, then the expected
error per spot is not more than ~33%, so if Rmerge is much bigger than that,
then there is something "funny" going on. Perhaps a violation of the chosen
space group symmetry (which may only show up at high resolution), radiation
damage, non-isomorphism, bad absorption corrections, crystal slippage or a
myriad of other "scaling problems" could do this. Rmerge became a popular
statistic because it proved a good way of detecting problems like these in
data processing. Fundamentally, if you have done the scaling properly, then
Rmerge/Rmeas should not be worse than the expected error of a single spot
measurement. This is either the error expected from counting statistics (33%
if you are using a 3-sigma cutoff), or the calibration error of the
instrument (~5% on a bad day, ~2% on a good one), whichever is bigger.
As for completeness, 80% overall is about the bare minimum of what you can
get away with before the map starts to change noticeably. See my movie here:
http://bl831.als.lbl.gov/~jamesh/movies/index.html#completeness
so I imagine this "80% rule" just got extended to the outermost bin. After
all, claiming a given resolution when you've only got 50% of the spots at
that resolution seems unwarranted, but requiring 100% completeness seems a
little too strict.
Where did these rules come from? As I recall, I first read about them in the
manual for the "PROCESS" program that came with our R-axis IIc x-ray system
when I was in graduate school (ca 1996). This program was conveniently
integrated into the data collection software on the detector control
computers: one was running VMS, and the "new" one was an SGI. I imagine a
few readers of this BB may have never heard of "PROCESS", but it is listed as
the "intensity integration software" for at least a thousand PDB entries. Is
there a reference for "PROCESS"? Yes. In the literature it is almost always
cited with: (Molecular Structure Corporation, The Woodlands, TX). Do I still
have a copy of the manual? Uhh. No. In fact, the building that once
contained it has since been torn down. Good thing I kept my images!
Is this "35% Rmerge with a 3-sigma cutoff" method of determining the
resolution limit statistically valid? Yes! There are actually very sound
statistical reasons for it. Is the resolution cutoff obtained the best one
for maximum-likelihood refinement? Definitely not! Modern refinement
programs do benefit from weak data, and tossing it all out messes up a number
of things. Does including weak data make Rmerge/Rmeas/Rpim and R/Rfree go
up? Yes. Does this make them more "honest"? No. It actually makes them
less useful.
Remember all R factors are measures of _relative_ error, so it is important
to remember to ask the question: "Relative to what?". For Rmerge, the "what"
is the sum of all the spot intensities (Blundell and Johnson, 1976). Where
you run into problems is when you restrict the Rmerge calculation to a single
resolution bin. If the sum of all intensities in the bin is actually zero,
then Rmerge is undefined (dividing by zero). If the signal-to-noise ratio is
~1, then the Rmerge equation doesn't "blow up" mathematically, but it does
give essentially random results. This is because Rmerge values for data this
weak take on a Cauchy distribution, and no matter how much averaging you do,
Cauchy-distributed values have a random mean. You can see in the classic
Weiss & Hilgenfeld (1997) paper that they had to use "outlier rejection" with
their fake data to get Rmerge to behave even with a signal-to-noise ratio of
2. The "turn over point" where the Rmerge equation becomes mathematically
well-behaved (Gaussian rather than Cauchy distribution) is when the
signal-to-noise ratio is about 3. I believe this is why our forefathers used
a 3-sigma cutoff.
Now, a 3-sigma cutoff on the raw observation data may sound like heresy
today, and I do NOT recommend you feed such data to refinement or other
downstream programs. But, it is important to remember what you are trying to
measure! If you are trying to detect scaling errors, then you should be
looking at spots where scaling errors are not masked by other kinds of error.
For example, a spot with only one photon in it is not going to tell you very
much about the accuracy of your scales, but its average
|delta-intensity|/intensity is going to be huge. That is, the pre-R-factor
sigma cutoff isolates the R factor calculation to spots dominated by scaling
errors. Including weaker data with their Cauchy-distributed R factor simply
adds noise to the value of the R factor itself.
So, I'd say if you have a reviewer complaining that your Rmerge in the
outermost bin is too high, simply tell the editor that you did not use a
3-sigma cutoff on the raw data for the Rmerge calculation, and ask if he/she
would prefer that you did.
-James Holton
MAD Scientist
On 1/27/2012 9:55 AM, Jacob Keller wrote:
Clarification: I did not mean I/sigma of 2 per se, I just meant
I/sigma is more directly a measure of signal than R values.
JPK
On Fri, Jan 27, 2012 at 11:47 AM, Jacob Keller
<j-kell...@fsm.northwestern.edu> wrote:
Dear Crystallographers,
I cannot think why any of the various flavors of Rmerge/meas/pim
should be used as a data cutoff and not simply I/sigma--can somebody
make a good argument or point me to a good reference? My thinking is
that signal:noise of>2 is definitely still signal, no matter what the
R values are. Am I wrong? I was thinking also possibly the R value
cutoff was a historical accident/expedient from when one tried to
limit the amount of data in the face of limited computational
power--true? So perhaps now, when the computers are so much more
powerful, we have the luxury of including more weak data?
JPK
--
*******************************************
Jacob Pearson Keller
Northwestern University
Medical Scientist Training Program
email:j-kell...@northwestern.edu
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