# [Ifeffit] limits for second shell

Wayne Lukens wwlukens at lbl.gov
Fri Aug 21 13:21:55 CDT 2009

```Dear Eugenio,

There is a standard statistical test to answer just this problem, the
F-test.  To use the test, you first do the refinement including the
contribution from the second shell and record chi-squared, which we will
call c1, and the number of parameters, p1.  Then you redo the fit
without the second shell but keeping everything else the same including
the k-weighting and the fit range; record the chi-squared, which we will
call c2, and the number of parameters, p2. Finally, you will need the
number of independent data points, which is the number given by ifeffit
plus 2 (Stern’s rule), idp. Then you need to calculate F, which is given
by F=(c2-c1)*(idp-p2)/[(p1-p2)*c1]. Using Excel, you can calculate the
probability of F using FDIST(F,(p1-p2),(idp-p1)). This gives the
probability that the improvement in chi-squared due to adding the second
shell to the fit is due to noise in the data. The usual criterion for
the F-test is that FDIST()<0.05, which means that the improvement in the
fit due to including that shell is two sigma over the noise. I think it
is OK to include the shell even if FDIST()>0.05, but you should report
the probability that the improvement in the fit is due to random noise.

This is a standard test in crystallography, where it is known as the
Hamilton test.

In short, the F-test tells you the probability that the improvement in
the fit due to including a given shell is due to random error, or can be
considered “real.” The F-test has one advantage of chi-squared tests in
that it is a ratio of chi-squared of two fits, so the standard deviation
of the data cancels. As with everything else, the F-test is model
dependent and can give the wrong answer due to non-random errors such
as problems with Feff.  If you have "noisy" data, the F-test is probably
pretty good; if you have data with little noise that goes high-k, I
would be more careful applying it.

The wikipedia entry on the F-test is OK, but used to have the formula
wrong. There is a paper on using this test in EXAFS analysis:

“A Variation of the F-Test for Determining Statistical Relevance of
Particular Parameters in EXAFS Fits” Downward, L.; Booth, C. H.; Lukens,
W. W.; Bridges, F. X-RAY ABSORPTION FINE STRUCTURE - XAFS13: 13th
International Conference. AIP Conference Proceedings, Volume 882, pp.
129-131 (2007).

Sincerely,

Wayne
--
Wayne Lukens
Staff Scientist
Lawrence Berkeley National Laboratory
email: wwlukens at lbl.gov
phone: (510) 486-4305
FAX: (510) 486-5596

Eugenio Otal wrote:
> Hi,
> I have a sample of a pure Er2O3 (blue line in the attached graph) and a
> sample of doped ZnO with erbium that has segregated the same oxide (red
> line) by thermal treatmen.
> The signal for de segregated oxide gets noisy around k=9 because the
> sample is so diluted, but the radial distribution shows second shell
> signal, smaller than the pre oxide, but still a signal. My doubt is
> about how to know if the second shell is real and if that second shell
> can be useful to obtain information. Is there a criteria to know that?
> Some limit in k-space?
> Thanks, euG
>
>
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--
Wayne Lukens
Staff Scientist
Lawrence Berkeley National Laboratory
email: wwlukens at lbl.gov
phone: (510) 486-4305
FAX: (510) 486-5596

```