Sorry for the confusion. I was going for brevity! And failed.
I know that the multiplicity correction is applied on a per-hkl basis in
the calculation of Rmeas. However, the average multiplicity over the
whole calculation is most likely not an integer. Some hkls may be
observed twice while others only once, or perhaps 3-4 times in the same
scaling run.
Allow me to do the error propagation properly. Consider the scenario:
Your outer resolution bin has a true I/sigma = 1.00 and average
multiplicity of 2.0. Let's say there are 100 hkl indices in this bin. I
choose the "true" intensities of each hkl from an exponential (aka
Wilson) distribution. Further assume the background is high, so the
error in each observation after background subtraction may be taken from
a Gaussian distribution. Let's further choose the per-hkl multiplicity
from a Poisson distribution with expectation value 2.0, so 0 is
possible, but the long-term average multiplicity is 2.0. For R
calculation, when multiplicity of any given hkl is less than 2 it is
skipped. What I end up with after 120,000 trials is a distribution of
values for each R factor. See attached graph.
What I hope is readily apparent is that the distribution of Rmerge
values is taller and sharper than that of the Rmeas values. The most
likely Rmeas is 80% and that of Rmerge is 64.6%. This is expected, of
course. But what I hope to impress upon you is that the most likely
value is not generally the one that you will get! The distribution has a
width. Specifically, Rmeas could be as low as 40%, or as high as 209%,
depending on the trial. Half of the trial results falling between 71.4%
and 90.3%, a range of 19 percentage points. Rmerge has a middle-half
range from 57.6% to 72.9% (15.3 percentage points). This range of
possible values of Rmerge or Rmeas from data with the same intrinsic
quality is what I mean when I say "numerical instability". Each and
every trial had the same true I/sigma and multiplicity, and yet the R
factors I get vary depending on the trial. Unfortunately for most of us
with real data, you only ever get one trial, and you can't predict which
Rmeas or Rmerge you'll get.
My point here is that R statistics in general are not comparable from
experiment to experiment when you are looking at data with low average
intensity and low multiplicity, and it appears that Rmeas is less stable
than Rmerge. Not by much, mind you, but still jumps around more.
Hope that is clearer?
Note that in no way am I suggesting that low-multiplicity is the right
way to collect data. Far from it. Especially with modern detectors
that have negligible read-out noise. But when micro crystals only give
off a handful of photons each before they die, low multiplicity might be
all you have.
-James Holton
MAD Scientist
On 7/7/2017 2:33 PM, Edward A. Berry wrote:
> I think the confusion here is that the "multiplicity correction" is
> applied
> on each reflection, where it will be an integer 2 or greater (can't
> estimate
> variance with only one measurement). You can only correct in an
> approximate
> way using using the average multiplicity of the dataset, since it
> would depend
> on the distribution of multiplicity over the reflections.
>
> And the correction is for r-merge. You don't need to apply a correction
> to R-meas.
> R-meas is a redundancy-independent best estimate of the variance.
> Whatever you would have used R-merge for (hopefully taking allowance
> for the multiplicity) you can use R-meas and not worry about
> multiplicity.
> Again, what information does R-merge provide that R-meas does not provide
> in a more accurate way?
>
> According to the denso manual, one way to artificially reduce
> R-merge is to include reflections with only one measure (averaging
> in a lot of zero's always helps bring an average down), and they say
> there were actually some programs that did that. However I'm
> quite sure none of the ones we rely on today do that.
>
> On 07/07/2017 03:12 PM, Kay Diederichs wrote:
>> James,
>>
>> I cannot follow you. "n approaches 1" can only mean n = 2 because n
>> is integer. And for n=2 the sqrt(n/(n-1)) factor is well-defined. For
>> n=1, neither contributions to Rmeas nor Rmerge nor to any other
>> precision indicator can be calculated anyway, because there's nothing
>> this measurement can be compared against.
>>
>> just my 2 cents,
>>
>> Kay
>>
>> On Fri, 7 Jul 2017 10:57:17 -0700, James Holton
>> <[log in to unmask]> wrote:
>>
>>> I happen to be one of those people who think Rmerge is a very useful
>>> statistic. Not as a method of evaluating the resolution limit,
>>> which is
>>> mathematically ridiculous, but for a host of other important things,
>>> like evaluating the performance of data collection equipment, and
>>> evaluating the isomorphism of different crystals, to name a few.
>>>
>>> I like Rmerge because it is a simple statistic that has a simple
>>> formula
>>> and has not undergone any "corrections". Corrections increase
>>> complexity, and complexity opens the door to manipulation by the
>>> desperate and/or misguided. For example, overzealous outlier rejection
>>> is a common way to abuse R factors, and it is far too often swept under
>>> the rug, sometimes without the user even knowing about it. This is
>>> especially problematic when working in a regime where the statistic of
>>> interest is unstable, and for R factors this is low intensity data.
>>> Rejecting just the right "outliers" can make any R factor look a lot
>>> better. Why would Rmeas be any more unstable than Rmerge? Look at the
>>> formula. There is an "n-1" in the denominator, where n is the
>>> multiplicity. So, what happens when n approaches 1 ? What happens
>>> when
>>> n=1? This is not to say Rmerge is better than Rmeas. In fact, I believe
>>> the latter is generally superior to the first, unless you are working
>>> near n = 1. The sqrt(n/(n-1)) is trying to correct for bias in the R
>>> statistic, but fighting one infinity with another infinity is a
>>> dangerous game.
>>>
>>> My point is that neither Rmerge nor Rmeas are easily interpreted
>>> without
>>> knowing the multiplicity. If you see Rmeas = 10% and the multiplicity
>>> is 10, then you know what that means. Same for Rmerge, since at n=10
>>> both stats have nearly the same value. But if you have Rmeas = 45% and
>>> multiplicity = 1.05, what does that mean? Rmeas will be only 33% if
>>> the
>>> multiplicity is rounded up to 1.1. This is what I mean by "numerical
>>> instability", the value of the R statistic itself becomes sensitive to
>>> small amounts of noise, and behaves more and more like a random number
>>> generator. And if you have Rmeas = 33% and no indication of
>>> multiplicity, it is hard to know what is going on. I personally am a
>>> lot more comfortable seeing qualitative agreement between Rmerge and
>>> Rmeas, because that means the numerical instability of the multiplicity
>>> correction didn't mess anything up.
>>>
>>> Of course, when the intensity is weak R statistics in general are not
>>> useful. Both Rmeas and Rmerge have the sum of all intensities in the
>>> denominator, so when the bin-wide sum approaches zero you have another
>>> infinity to contend with. This one starts to rear its ugly head once
>>> I/sigma drops below about 3, and this is why our ancestors always
>>> applied a sigma cutoff before computing an R factor. Our
>>> small-molecule
>>> colleagues still do this! They call it "R1". And it is an excellent
>>> indicator of the overall relative error. The relative error in the
>>> outermost bin is not meaningful, and strangely enough nobody ever
>>> reported the outer-resolution Rmerge before 1995.
>>>
>>> For weak signals, Correlation Coefficients are better, but for strong
>>> signals CC pegs out at >95%, making it harder to see relative errors.
>>> I/sigma is what we'd like to know, but the value of "sigma" is still
>>> prone to manipulation by not just outlier rejection, but massaging the
>>> so-called "error model". Suffice it to say, crystallographic data
>>> contain more than one type of error. Some sources are important for
>>> weak spots, others are important for strong spots, and still others are
>>> only apparent in the mid-range. Some sources of error are only
>>> important at low multiplicity, and others only manifest at high
>>> multiplicity. There is no single number that can be used to evaluate
>>> all
>>> aspects of data quality.
>>>
>>> So, I remain a champion of reporting Rmerge. Not in the high-angle
>>> bin,
>>> because that is essentially a random number, but overall Rmerge and
>>> low-angle-bin Rmerge next to multiplicity, Rmeas, CC1/2 and other
>>> statistics is the only way you can glean enough information about where
>>> the errors are coming from in the data. Rmeas is a useful addition
>>> because it helps us correct for multiplicity without having to do math
>>> in our head. Users generally thank you for that. Rmerge, however, has
>>> served us well for more than half a century, and I believe Uli Arndt
>>> knew what he was doing. I hope we all know enough about history to
>>> realize that future generations seldom thank their ancestors for
>>> "protecting" them from information.
>>>
>>> -James Holton
>>> MAD Scientist
>>>
>>>
>>> On 7/5/2017 10:36 AM, Graeme Winter wrote:
>>>> Frank,
>>>>
>>>> you are asking me to remove features that I like, so I would feel
>>>> that the challenge is for you to prove that this is harmful however:
>>>>
>>>> - at the minimum, I find it a useful check sum that the stats
>>>> are internally consistent (though I interpret it for lots of other
>>>> reasons too)
>>>> - it is faulty I agree, but (with caveats) still useful IMHO
>>>>
>>>> Sorry for being terse, but I remain to be convinced that removing
>>>> it increases the amount of information
>>>>
>>>> CC’ing BB as requested
>>>>
>>>> Best wishes Graeme
>>>>
>>>>
>>>>> On 5 Jul 2017, at 17:17, Frank von Delft
>>>>> <[log in to unmask]> wrote:
>>>>>
>>>>> You keep not answering the challenge.
>>>>>
>>>>> It's really simple: what information does Rmerge provide that
>>>>> Rmeas doesn't.
>>>>>
>>>>> (If you answer, email to the BB.)
>>>>>
>>>>>
>>>>> On 05/07/2017 16:04, [log in to unmask] wrote:
>>>>>> Dear Frank,
>>>>>>
>>>>>> You are forcefully arguing essentially that others are wrong if
>>>>>> we feel an existing statistic continues to be useful, and instead
>>>>>> insist that it be outlawed so that we may not make use of it,
>>>>>> just in case someone misinterprets it.
>>>>>>
>>>>>> Very well
>>>>>>
>>>>>> I do however express disquiet that we as software developers feel
>>>>>> browbeaten to remove the output we find useful because “the
>>>>>> community” feel that it is obsolete.
>>>>>>
>>>>>> I feel that Jacob’s short story on this thread illustrates that
>>>>>> educating the next generation of crystallographers to understand
>>>>>> what all of the numbers mean is critical, and that a
>>>>>> numerological approach of trying to optimise any one statistic is
>>>>>> essentially doomed. Precisely the same argument could be made for
>>>>>> people cutting the “resolution” at the wrong place in order to
>>>>>> improve the average I/sig(I) of the data set.
>>>>>>
>>>>>> Denying access to information is not a solution to
>>>>>> misinterpretation, from where I am sat, however I acknowledge
>>>>>> that other points of view exist.
>>>>>>
>>>>>> Best wishes Graeme
>>>>>>
>>>>>>
>>>>>> On 5 Jul 2017, at 12:11, Frank von Delft
>>>>>> <[log in to unmask]<mailto:[log in to unmask]>>
>>>>>> wrote:
>>>>>>
>>>>>>
>>>>>> Graeme, Andrew
>>>>>>
>>>>>> Jacob is not arguing against an R-based statistic; he's pointing
>>>>>> out that leaving out the multiplicity-weighting is prehistoric
>>>>>> (Diederichs & Karplus published it 20 years ago!).
>>>>>>
>>>>>> So indeed: Rmerge, Rpim and I/sigI give different information.
>>>>>> As you say.
>>>>>>
>>>>>> But no: Rmerge and Rmeas and Rcryst do NOT give different
>>>>>> information. Except:
>>>>>>
>>>>>> * Rmerge is a (potentially) misleading version of Rmeas.
>>>>>>
>>>>>> * Rcryst and Rmerge and Rsym are terms that no longer have
>>>>>> significance in the single cryo-dataset world.
>>>>>>
>>>>>> phx.
>>>>>>
>>>>>>
>>>>>>
>>>>>> On 05/07/2017 09:43, Andrew Leslie wrote:
>>>>>>
>>>>>> I would like to support Graeme in his wish to retain Rmerge in
>>>>>> Table 1, essentially for exactly the same reasons.
>>>>>>
>>>>>> I also strongly support Francis Reyes comment about the
>>>>>> usefulness of Rmerge at low resolution, and I would add to his
>>>>>> list that it can also, in some circumstances, be more indicative
>>>>>> of the wrong choice of symmetry (too high) than the statistics
>>>>>> that come from POINTLESS (excellent though that program is!).
>>>>>>
>>>>>> Andrew
>>>>>> On 5 Jul 2017, at 05:44, Graeme Winter
>>>>>> <[log in to unmask]<mailto:[log in to unmask]>> wrote:
>>>>>>
>>>>>> HI Jacob
>>>>>>
>>>>>> Yes, I got this - and I appreciate the benefit of Rmeas for
>>>>>> dealing with measuring agreement for small-multiplicity
>>>>>> observations. Having this *as well* is very useful and I agree
>>>>>> Rmeas / Rpim / CC-half should be the primary “quality” statistics.
>>>>>>
>>>>>> However, you asked if there is any reason to *keep* rather than
>>>>>> *eliminate* Rmerge, and I offered one :o)
>>>>>>
>>>>>> I do not see what harm there is reporting Rmerge, even if it is
>>>>>> just used in the inner shell or just used to capture a flavour of
>>>>>> the data set overall. I also appreciate that Rmeas converges to
>>>>>> the same value for large multiplicity i.e.:
>>>>>>
>>>>>> Overall InnerShell
>>>>>> OuterShell
>>>>>> Low resolution limit 39.02 39.02 1.39
>>>>>> High resolution limit 1.35 6.04 1.35
>>>>>>
>>>>>> Rmerge (within I+/I-) 0.080 0.057 2.871
>>>>>> Rmerge (all I+ and I-) 0.081 0.059 2.922
>>>>>> Rmeas (within I+/I-) 0.081 0.058 2.940
>>>>>> Rmeas (all I+ & I-) 0.082 0.059 2.958
>>>>>> Rpim (within I+/I-) 0.013 0.009 0.628
>>>>>> Rpim (all I+ & I-) 0.009 0.007 0.453
>>>>>> Rmerge in top intensity bin 0.050 - -
>>>>>> Total number of observations 1265512 16212 53490
>>>>>> Total number unique 17515 224 1280
>>>>>> Mean((I)/sd(I)) 29.7 104.3 1.5
>>>>>> Mn(I) half-set correlation CC(1/2) 1.000 1.000 0.778
>>>>>> Completeness 100.0 99.7 100.0
>>>>>> Multiplicity 72.3 72.4 41.8
>>>>>>
>>>>>> Anomalous completeness 100.0 100.0 100.0
>>>>>> Anomalous multiplicity 37.2 42.7 21.0
>>>>>> DelAnom correlation between half-sets 0.497 0.766 -0.026
>>>>>> Mid-Slope of Anom Normal Probability 1.039 - -
>>>>>>
>>>>>> (this is a good case for Rpim & CC-half as resolution limit
>>>>>> criteria)
>>>>>>
>>>>>> If the statistics you want to use are there & some others also,
>>>>>> what is the pressure to remove them? Surely we want to educate on
>>>>>> how best to interpret the entire table above to get a fuller
>>>>>> picture of the overall quality of the data? My 0th-order request
>>>>>> would be to publish the three shells as above ;o)
>>>>>>
>>>>>> Cheers Graeme
>>>>>>
>>>>>>
>>>>>>
>>>>>> On 4 Jul 2017, at 22:09, Keller, Jacob
>>>>>> <[log in to unmask]<mailto:[log in to unmask]>> wrote:
>>>>>>
>>>>>> I suggested replacing Rmerge/sym/cryst with Rmeas, not Rpim.
>>>>>> Rmeas is simply (Rmerge * sqrt(n/n-1)) where n is the number of
>>>>>> measurements of that reflection. It's merely a way of correcting
>>>>>> for the multiplicity-related artifact of Rmerge, which is
>>>>>> becoming even more of a problem with data sets of increasing
>>>>>> variability in multiplicity. Consider the case of comparing a
>>>>>> data set with a multiplicity of 2 versus one of 100: equivalent
>>>>>> data quality would yield Rmerges diverging by a factor of ~1.4.
>>>>>> But this has all been covered before in several papers. It can be
>>>>>> and is reported in resolution bins, so can used exactly as you
>>>>>> say. So, why not "disappear" Rmerge from the software?
>>>>>>
>>>>>> The only reason I could come up with for keeping it is historical
>>>>>> reasons or comparisons to previous datasets, but anyway those
>>>>>> comparisons would be confounded by variabities in multiplicity
>>>>>> and a hundred other things, so come on, developers, just comment
>>>>>> it out!
>>>>>>
>>>>>> JPK
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> -----Original Message-----
>>>>>> From:
>>>>>> [log in to unmask]<mailto:[log in to unmask]>
>>>>>> [mailto:[log in to unmask]]
>>>>>> Sent: Tuesday, July 04, 2017 4:37 PM
>>>>>> To: Keller, Jacob
>>>>>> <[log in to unmask]<mailto:[log in to unmask]>>
>>>>>> Cc: [log in to unmask]<mailto:[log in to unmask]>
>>>>>> Subject: Re: [ccp4bb] Rmergicide Through Programming
>>>>>>
>>>>>> HI Jacob
>>>>>>
>>>>>> Unbiased estimate of the true unmerged I/sig(I) of your data (I
>>>>>> find this particularly useful at low resolution) i.e. if your
>>>>>> inner shell Rmerge is 10% your data agree very poorly; if 2% says
>>>>>> your data agree very well provided you have sensible
>>>>>> multiplicity… obviously depends on sensible interpretation. Rpim
>>>>>> hides this (though tells you more about the quality of average
>>>>>> measurement)
>>>>>>
>>>>>> Essentially, for I/sig(I) you can (by and large) adjust your
>>>>>> sig(I) values however you like if you were so inclined. You can
>>>>>> only adjust Rmerge by excluding measurements.
>>>>>>
>>>>>> I would therefore defend that - amongst the other stats you
>>>>>> enumerate below - it still has a place
>>>>>>
>>>>>> Cheers Graeme
>>>>>>
>>>>>> On 4 Jul 2017, at 14:10, Keller, Jacob
>>>>>> <[log in to unmask]<mailto:[log in to unmask]>> wrote:
>>>>>>
>>>>>> Rmerge does contain information which complements the others.
>>>>>>
>>>>>> What information? I was trying to think of a counterargument to
>>>>>> what I proposed, but could not think of a reason in the world to
>>>>>> keep reporting it.
>>>>>>
>>>>>> JPK
>>>>>>
>>>>>>
>>>>>> On 4 Jul 2017, at 12:00, Keller, Jacob
>>>>>> <[log in to unmask]<mailto:[log in to unmask]><mailto:[log in to unmask]>>
>>>>>> wrote:
>>>>>>
>>>>>> Dear Crystallographers,
>>>>>>
>>>>>> Having been repeatedly chagrinned about the continued use and
>>>>>> reporting of Rmerge rather than Rmeas or similar, I thought of a
>>>>>> potential way to promote the change: what if merging programs
>>>>>> would completely omit Rmerge/cryst/sym? Is there some reason to
>>>>>> continue to report these stats, or are they just grandfathered
>>>>>> into the software? I doubt that any journal or crystallographer
>>>>>> would insist on reporting Rmerge per se. So, I wonder what
>>>>>> developers would think about commenting out a few lines of their
>>>>>> code, seeing what happens? Maybe a comment to the effect of
>>>>>> "Rmerge is now deprecated; use Rmeas" would be useful as well.
>>>>>> Would something catastrophic happen?
>>>>>>
>>>>>> All the best,
>>>>>>
>>>>>> Jacob Keller
>>>>>>
>>>>>> *******************************************
>>>>>> Jacob Pearson Keller, PhD
>>>>>> Research Scientist
>>>>>> HHMI Janelia Research Campus / Looger lab
>>>>>> Phone: (571)209-4000 x3159
>>>>>> Email:
>>>>>> [log in to unmask]<mailto:[log in to unmask]><mailto:[log in to unmask]>
>>>>>> *******************************************
>>>>>>
>>>>>>
>>>>>> --
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>>>>>>
>>>>>
>>
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