Let's say you have decided that you want to know if the CA-CB bond
of residue 123 in your favorite protein differs from the expected value
for that type of bond. You solve the structure and refine a model
against your crystallographic data, then look at residue's 123 CA-CB
bond and find that it is 3 sigma from the expected value. Is this
observation unlikely given the uncertainties in the parameters of the model?
Now, let's look at a different case. You have solved and refined a
model of your favorite protein. After examining all of 1000 bond
lengths in your model you notice that the CA-CB bond of residue 123 is 3
sigma from its expected value. Is this observation unlikely given the
uncertainties in the parameters of the model?
Even though you are looking at the same bond in the same model and
see exactly the same thing, the calculation of the probability that this
bond is actually different than is usual it very different. The
calculation that you want to perform - the classic p test based on a
Normal distribution - is valid for the first case but is quite
inappropriate for the second.
It is clearly much more likely that, among 1000 bonds, one of them
will have a deviation of 3 sigma. In fact I would say it is a near
certainty.
This twist of statistical analysis was never discussed in the basic
classes on stats that I took and most scientists tend to ignore it. To
avoid the apparent paradox that you are confronting you have to include
in your calculations the consequences of the actual question you have asked.
There are huge problems with calculating this sort of "significance"
because it is quite tempting to change your question after the fact and
conclude that something is significant when it is not. TNT always
produced a list of the geometry outliers after refinement. If you
notice that a residue in the active site is present in that list, you
will be tempted to forget that this residue was brought to your
attention by a search over all geometry restraints and not a prior
interest in the active site.
This is a problem that many other fields of research are contending
with. One solution is to publish the questions you hope your model will
answer before you perform the research. That is certainly difficult
with our sort of research.
An example from another area might be helpful. A researcher
performs a survey of a lot of people asking questions about their diet
and about their medical history. Very often the published conclusion
will be that, say, dietary item number 5 is correlated with medical
condition number 12. These studies tend to assess the significance of
this result by just comparing the odds of these two items having the
observed magnitude of correlation.
This ignores the fact that a host of correlations were calculated
and only this one was "significant". If the survey had 20 dietary
factors and 20 conditions then 400 comparisons were made and it was a
virtual certainty that one of them would be "significant" unless the
proper correction made to the probability calculations.
Dale E. Tronrud
On 11/8/2022 3:25 PM, James Holton wrote:
Thank you Ian for your quick response!
I suppose what I'm really trying to do is put a p-value on the
"geometry" of a given PDB file. As in: what are the odds the deviations
from ideality of this model are due to chance?
I am leaning toward the need to take all the deviations in the structure
together as a set, but, as Joao just noted, that it just "feels wrong"
to tolerate a 3-sigma deviate. Even more wrong to tolerate 4 sigma, 5
sigma. And 6 sigma deviates are really difficult to swallow unless your
have trillions of data points.
To put it down in equations, is the p-value of a structure with 1000
bonds in it with one 3-sigma deviate given by:
a) p = 1-erf(3/sqrt(2))
or
b) p = 1-erf(3/sqrt(2))**1000
or
c) something else?
On 11/8/2022 2:56 PM, Ian Tickle wrote:
Hi James
I don't think it's meaningful to ask whether the deviation of a single
bond length (or anything else that's single) from its expected value
is significant, since as you say there's always some finite
probability that it occurred purely by chance. Statistics can only
meaningfully be applied to samples of a 'reasonable' size. I know
there are statistics designed for small samples but not for samples of
size 1 ! It's more meaningful to talk about distributions. For
example if 1% of the sample contained deviations > 3 sigma when you
expected there to be only 0.3 %, that is probably significant (but it
still has a finite probability of occurring by chance), as would be
finding no deviations > 3 sigma (for a reasonably large sample to
avoid sampling errors).
Cheers
-- Ian
On Tue, Nov 8, 2022, 22:22 James Holton <jmhol...@lbl.gov> wrote:
OK, so lets suppose there is this bond in your structure that is
stretched a bit. Is that for real? Or just a random fluke? Let's
say
for example its a CA-CB bond that is supposed to be 1.529 A long,
but in
your model its 1.579 A. This is 0.05 A too long. Doesn't seem like
much, right? But the "sigma" given to such a bond in our geometry
libraries is 0.016 A. These sigmas are typically derived from a
database of observed bonds of similar type found in highly accurate
structures, like small molecules. So, that makes this a 3-sigma
outlier.
Assuming the distribution of deviations is Gaussian, that's a pretty
unlikely thing to happen. You expect 3-sigma deviates to appear less
than 0.3% of the time. So, is that significant?
But, then again, there are lots of other bonds in the structure. Lets
say there are 1000. With that many samplings from a Gaussian
distribution you generally expect to see a 3-sigma deviate at least
once. That is, do an "experiment" where you pick 1000
Gaussian-random
numbers from a distribution with a standard deviation of 1.0.
Then, look
for the maximum over all 1000 trials. Is that one > 3 sigma? It
probably
is. If you do this "experiment" millions of times it turns out
seeing at
least one 3-sigma deviate in 1000 tries is very common. Specifically,
about 93% of the time. It is rare indeed to have every member of a
1000-deviate set all lie within 3 sigmas. So, we have gone from one
3-sigma deviate being highly unlikely to being a virtual certainty if
you look at enough samples.
So, my question is: is a 3-sigma deviate significant? Is it
significant
only if you have one bond in the structure? What about angles?
What if
you have 500 bonds and 500 angles? Do they count as 1000 deviates
together? Or separately?
I'm sure the more mathematically inclined out there will have some
intelligent answers for the rest of us, however, if you are not a
mathematician, how about a vote? Is a 3-sigma bond length deviation
significant? Or not?
Looking forward to both kinds of responses,
-James Holton
MAD Scientist
########################################################################
To unsubscribe from the CCP4BB list, click the following link:
https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1
<https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1>
This message was issued to members of www.jiscmail.ac.uk/CCP4BB
<http://www.jiscmail.ac.uk/CCP4BB>, a mailing list hosted by
www.jiscmail.ac.uk <http://www.jiscmail.ac.uk>, terms & conditions
are available at https://www.jiscmail.ac.uk/policyandsecurity/
------------------------------------------------------------------------
To unsubscribe from the CCP4BB list, click the following link:
https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1
<https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1>
########################################################################
To unsubscribe from the CCP4BB list, click the following link:
https://www.jiscmail.ac.uk/cgi-bin/WA-JISC.exe?SUBED1=CCP4BB&A=1
This message was issued to members of www.jiscmail.ac.uk/CCP4BB, a mailing list
hosted by www.jiscmail.ac.uk, terms & conditions are available at
https://www.jiscmail.ac.uk/policyandsecurity/
