Hi Pavel

Phew!  Lots of questions - this could take a while:

> - where this formula come from and what are the grounds for this?

It's just the RMSD of the density divided by its standard uncertainty
(sigma), which we're assuming is the same for all grid points: this
isn't quite true, sigma is higher on or near rotation axes, but the
effect is sufficiently small that we can ignore it.  It comes from
taking the negative log of the likelihood function of the density
values assuming a normal error distribution, which gives you
chi-squared, i.e. sum(delta_rho^2)/sigma(rho)^2.  The likelihood is
the standard measure of the consistency of a model, in this case an
atomic model which gives you rho_calc, with the data (rho_obs), and
delta_rho = rho_obs - rho_calc.

> - how to make sense of the numbers. Say I used this formula and I got a
> number X; how can I tell if it is good or not good?

There's a standard procedure for significance testing which is
explained in all statistics textbooks (or go here:
http://en.wikipedia.org/wiki/Significance_test). You decide on a level
of significance ('critical p-value') which represents the probability
of getting the observed or a more extreme result purely by chance,
assuming that the 'null hypothesis' is true, i.e. that the difference
density in question doesn't represent any real signal, only random
error ('noise').  You can use p=1% or even p=0.1% if you want to be
even more confident that what you see isn't just random error: you are
trying to avoid the situation where you reject the null hypothesis and
conclude that there's real difference density present when there
really isn't ('Type 1 error').  Of course making the p-value too small
might mean that you miss real difference density ('Type 2 error').
Then you look up your chi-squared value and the 'number of degrees of
freedom' in the relevant published statistical table of upper critical
values of the chi-square distribution (e.g. go here:
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm).
This is just the cumulative distribution function of the chi-square
distribution, so can be readily computed using the appropriate
continued fraction expansion (see 'upper incomplete gamma function':
http://en.wikipedia.org/wiki/Incomplete_Gamma_function).  This means
that non-tabulated values can be used, for example for the normal
distribution the p-value corresponding to the usual '3 sigma'
threshold is 0.27%, so it makes sense to use the same p-value here
(p=0.1% corresponds to 3.3 sigma for the normal distribution).

One very important point that I glossed over in my previous e-mail is
the role of Npoints: this is the number of *independent* density
values in the sum above.  This is slightly tricky because of course
normally we over-sample the maps which means the density values are no
longer independent.  However we can get round this because we know
that at the Shannon limit where the grid spacing in the map = Dmin/2
the density values become statistically independent, so that if we
over-sample with a grid spacing of say Dmin/4 (which is what I always
use for this), the over-sampling factor is 2 in each direction so Ndof
= Npoints/8.

Let's say Npoints = 400, Ndof = 50, then for p=0.1%, chi-squared =
86.661, so RMS-Z-score = sqrt(86.661/50) = 1.32 so that's your
threshold, i.e. a value bigger than this probably means that the
difference density is real.  However if it's less note that it
*doesn't* prove there's nothing there, it just means that the data
isn't good enough to come to a firm conclusion - you might find
stronger evidence if you were to obtain better data - remember always
that absence of evidence isn't evidence of absence!

> - do you think it is better than looking at three values {map CC, 2mFo-DFc,
> mFo-DFc} and why?

Yes, because all the information you need is encapsulated in 1 number
per region of interest!  But I don't understand what you mean by
2mFo-DFc & mFo-DFc being counted each as 1 number.  Surely you have 1
value of each of these at every grid point, or at least 1 value per
maximum in the case say of an extended ligand?

Note that I'm not proposing anything new, this is all explained in
standard statistics textbooks (Kendall's Advanced Theory of Statistics
by Stuart & Ord is probably the best).  In fact this is exactly my
point: why re-invent the wheel (and likely end up with a square one!)
when the appropriate statistics is all there in the textbooks and has
been for ~ 80 years?

> - why 2(mFo-DFc)?

Randy Read (AC 1986, A42, 140-149) showed that for a partial structure
with errors the expected values of the true Fs for the complete
structure (FN) for which Fo's have been obtained experimentally, and
for a partial structure model (FP) respectively are:

              FN = (2mFo-DFc)exp(i phi_calc)    ... for acentric reflections,
              FN = mFo exp(i phi_calc)              ... for centric reflections,
              FP = DFc exp(i phi_calc)              ... for both
acentric and centric.

Hence the difference map coefficients DF=FN-FP are respectively:

              DF = 2(mFo-DFc)exp(i phi_calc)    ... for acentrics,
              DF = (mFo-DFc)exp(i phi_calc)      ... for centrics,

This is consistent with the observation that difference map peaks in
non-centrosymmetric structures appear at half the theoretical height
(assuming the phase errors are small), so you need to multiply the
coefficients by 2 to get the right value, whereas peaks in
centrosymmetric structures appear at the true height so don't need to
be corrected.  This has been known for a long time (e.g. see Blundell
& Johnson, 1976, p.408).

> - how the "region of interest is defined"?

You define it!  - exactly the same as you do for RSCC & RSR.  Note
that although there is a significance test for the CC, none exists for
the R-factor.  The basic problem with the R-factor is that it
conflates 2 effects: because of the sum over the data in the
denominator, R is a function both of the absolute values of the errors
and of the values of the data relative to the errors, so weak data
always has a high R-factor, hence a high value of the R-factor for
weak data really tells you nothing about the errors.  This is apparent
when you look at Rmerge values for intensity data: the appropriate
statistic which quantifies the data quality in that case is not Rmerge
at all but the average I/sigma(I).

> - how you compute sigma(rho)?

See my reply to George Sheldrick's post.

> By suggesting to use {map CC, 2mFo-DFc, mFo-DFc} I was assuming that:
> - map CC will tell you about similarities of shapes and it will not tell you
> about how strong the density is, indeed.  So, using map CC alone is clearly
> insufficient. Also, we more or less have feeling about the values, which is
> helpful.
> - 2mFo-DFc will tell you about the strength of the density. I mean, if you
> get 2.5sigma at the center of atom A -  it's good (provided that map CC is
> good), and if it is 0.3sigma you should get puzzled.
> - Having excess of +/- mFo-DFc density will tell you something too.

The problem is how is all this information quantified in an objective
and statistically justifiable way in order to arrive at a firm
conclusion?

Cheers

-- Ian