Hi,

I think to start delving into the origins of this, we can recap high
school algebra of simultaneous equations (or data fitting) as a simple
start. For a system of 2 variables x,y, you will need at least 2
equations to solve for the 2 variables, giving you a data to parameter
ratio of 1. Similarly for 3 unknowns x,y,z, you will need 3 equations
at least to fully solve the system of equations.

If you look through some of the original small
molecule/crystallography books, you can follow up on it. One of the
more recent texts that talks about it is Bernhard Rupp's "Biomolecular
Crystallography": "The basic idea in refinement is that a system of p
independent simultaneous equations is solved." Refinement involves the
fitting of an excess n available experimental data to p model
parameters. So just as in the case of the above simultaneous
equations, this system of simultaneous equations can be solved only if
n >= p (https://books.google.com/books?id=gTAWBAAAQBAJ&lpg=PA622&ots=xZFfmIyGeV&dq=data%20to%20parameter%20ratio%20in%20simultaneous%20equation&pg=PA622#v=onepage&q&f=false).

So the higher the resolution of your diffraction data set, the higher
your 'n', so more detailed the fitting that can be done. When
resolution is low, we use restraints and constraints to get a good
data/parameter ratio.

I assume this is what you were looking for and I hope this helps to
understand the origins of the data/parameter ratio.

Yes, having few thousand more data at I/sig ~2 may help, but really
the only way to tell is if you get better maps and R-values keeping
good statistics.

Regards,
Debanu.


On Tue, Aug 11, 2015 at 8:54 PM, Pavel Afonine <[log in to unmask]> wrote:
> Hi Jacob,
>
> making sense of data-to-parameters ratio is trickier than it may seem. It is
> not as simple as comparing the number of reflections Nreflections with the
> number of parameters, which typically but not always is Natoms * (3 xyz + 1
> Biso or 6 Baniso + some occupancies + some other parameters).
>
> In refinement we use restraints and/or constraints. While constraints reduce
> number of parameters explicitly and can be counted, restraints do so in a
> less obvious way. When restraints are used the a priori information is added
> as a weighted term to the total target that is optimized: T = Tdata + weight
> * Trestraints. The "weight" prescribes the dose of extra information to be
> added (keep in mind: the weight changes during refinement!). This is exactly
> why we can still refine individual coordinates or isotropic B-factors at
> typical "macromolecular" resolutions such as 2-4A or so.
>
> Having said this, and being unable to estimate this number even
> approximately, personally, I could not care less about it for practical
> purposes.
>
> And to answer your very question: no, I do not know where this is discussed
> in great details, apart from being perennially said on mailing lists.
> Perhaps I should have started with this first -;)
>
> All the best,
> Pavel
>
>
> On Tue, Aug 11, 2015 at 8:17 PM, Keller, Jacob <[log in to unmask]>
> wrote:
>>
>> Dear Crystallographers,
>>
>> I've long heard second-hand about the need for favorable
>> observation-to-parameter ratios, but have never really delved into the
>> original literature. Does anyone know of a good source explaining and/or
>> demonstrating this requirement, and perhaps showing how aspects of
>> crystallography per se color the relationship? Ideally this would be an
>> original source (I've found this seems to be the only way to really get to
>> the bottom of things, and is much more efficient than reading digests or
>> reviews, although the "James" might be an exception.)
>>
>> For a crystallographic example, some observations are I/sig of 2, some 50,
>> and it does not seem right to weigh them equally for these purposes. More
>> specifically, I have a case of a truncated dataset which is cut off at 1.7
>> Ang due to the detector, but I think the dataset would have gone to 1.4-1.5
>> with a different setup. Does having a few thousand more measurements at
>> I/sig = ~2 make that much of a difference? How should one think about this?
>>
>> Thanks,
>>
>> Jacob Keller
>>
>> *******************************************
>> Jacob Pearson Keller, PhD
>> Looger Lab/HHMI Janelia Research Campus
>> 19700 Helix Dr, Ashburn, VA 20147
>> email: [log in to unmask]
>> *******************************************
>
>