From: "Rory M Wilson (CDH) 13-7938" [[email protected]]
Organization: MDS QMW
To: [email protected]
Date: Fri, 6 Apr 2001 15:39:15 BST
Subject: Re: Hamilton's test

  Dear Nagesh
         The R-factor is described in:
Volume 4 of International Tables for Crystallography section 4.2
pages 288 to 310.
    if you have a look at Walter Hamilton's original paper:
Acta Crystallographica (1965), vol. 18, P502-510
you will see that it is based on the F-test which is covered in 
most reasonable statistics text books.
                 Yours
                     Rory.

Date: Fri, 6 Apr 2001 10:37:40 -0700 (PDT)
Subject: Re: Hamilton's test
From: "Holger Kohlmann" [[email protected]]
To: [email protected]


> I have obtained different R factors. Now I want to decide which is the
> most correct model. I have been suggested to use Hamilton R-factor
> ratio test.

I would be careful with the Hamilton test in the case of powder 
diffraction, as your observations are not really independent from each 
other!


-- 
Dr. Holger Kohlmann * High Pressure Science and Engineering Center * 
Department of Physics * University of Nevada, Las Vegas * 4505 South 
Maryland Parkway, Box 4002 * Las Vegas, Nevada 89154-4002, USA * phone [+1]
(702)895 1716 * fax [+1](702)895 0804 * mailto:[email protected]

Date: Fri, 06 Apr 2001 16:08:50 -0400
From: "Brian H. Toby" [[email protected]]
Organization: NIST Center for Neutron Research
To: [email protected]
Subject: Re: Hamilton's test


> I would be careful with the Hamilton test in the case of powder
> diffraction, as your observations are not really independent from each
> other!

This is a common misconception. (If not common, at least it was my
misconception until I had several long conversations with Ted Prince.)

The Hamilton R-factor test is just a Student's t-Test. It tests if a
model is significantly improved by the addition of more adjustable
parameters compared to the more highly constrained model. One can use
ratios of Rwp for this. Personally, I find it easier to compute the
appropriate F distribution then to use the tables in Hamilton's paper.
The discussion on p128-9 in Ted's book (Mathematical Techniques in
Crystallography and Materials Science) is rather terse, but does derive
this.

The observations must be statistically independent, but need not be
independent in the sense of what they physically measure. If you measure
a full sphere of single crystal data, you will get the same R-factor
test result with that full data set as you would get by merging the data
to the unique subset, provided that the uncertainties are handled
correctly.

One additional note. Properly, the test cannot be used to compare
different models, rather it must be used where one model is a subset of
the other with respect to the varied parameters. If you fit data to "y =
mx + b", you can compare that to a fit of "y = mx", but you cannot
compare a fit of "y = mx + b" to a fit using "y = m cos(x)".

Brian

********************************************************************
Brian H. Toby, Ph.D.                    Leader, Crystallography Team
[email protected]      NIST Center for Neutron Research, Stop 8562
voice: 301-975-4297     National Institute of Standards & Technology
FAX: 301-921-9847                        Gaithersburg, MD 20899-8562
                http://www.ncnr.nist.gov/xtal
********************************************************************

Date: Mon, 09 Apr 2001 10:32:04 +0200
To: [email protected]
From: Jonathan WRIGHT [[email protected]]
Subject: Re: Hamilton's test

> The observations must be statistically independent, but need not be
> independent in the sense of what they physically measure.

It seems implicit from that sentence that the datapoints must be physically
measuring some aspect of the model, is that the case? For example: Does a
Hamilton test on two crystallographic models give the same results if you
throw out the datapoints which only have background contributions? The
"reducio ad absurdum" would be to use Rwp from a multipattern fit where
only one pattern contains the phase of interest. Maybe that's statistically
valid if some of the least squares parameters depend on all of the patterns? 

Bill David's work on deriving the number of independent peaks in a powder
pattern seems to offer a route to significance testing for different
crytallographic models (J.Appl.Cryst.(1999) 32:654). Perhaps it just adds
to the confusion.

Cheers,

Jon Wright

Date: Mon, 09 Apr 2001 08:11:07 -0600
To: [email protected]
From: [email protected] (Bob Von Dreele)
Subject: Re: Hamilton's test

>I would be careful with the Hamilton test in the case of powder
>diffraction, as your observations are not really independent from each
>other!

Strictly speaking this is not true. The individual measurements of powder 
diffraction profile intensities are independent measurements. They do not 
depend on the order of their measurement, for example. The fact that a 
string of observations proceed over some feature of the diffraction profile 
(i.e. a Bragg peak) is not evidence of their "dependence". The only 
exception to this is profile measurements taken on a film or image plate 
where one observation may "bleed over" onto neighboring ones. Only in that 
case are the profile points correlated with each other in a statistical sense.

Bob Von Dreele

Date: Mon, 09 Apr 2001 17:08:34 +0200
From: Radovan Cerny [[email protected]]
Subject: Re: Hamilton's test
To: [email protected]
Organization: University of Geneva

Bob Von Dreele wrote:

>The only exception to this is profile measurements taken on a film or 
>image plate where one observation may "bleed over" onto neighboring 
>ones. Only in that case are the profile points correlated with each 
>other in a statistical sense.
>
> Bob Von Dreele

In that case even the single crystal data resulting from the integration
of images registred by an image plate are not strictly independent!!??
Information read from one pixel can depend on the information registred
in a neigbouring pixel.

-- 
Radovan Cerny              
Laboratoire de Cristallographie    
24, quai Ernest-Ansermet              
CH-1211 Geneva 4, Switzerland                            
Phone  : [+[41] 22] 702 64 50, FAX : [+[41] 22] 702 61 08
mailto : [email protected]
URL    : http://www.unige.ch/crystal/cerny/rcerny.htm

Date: Mon, 09 Apr 2001 09:24:58 -0600
To: [email protected]
From: [email protected] (Bob Von Dreele)
Subject: Re: Hamilton's test

Dear Radovan,
At 05:08 PM 4/9/01 +0200, you wrote:
>Bob Von Dreele wrote:
>
> >The only exception to this is profile measurements taken on a film 
> or >image plate where one observation may "bleed over" onto 
> neighboring >ones. Only in that case are the profile points correlated 
> with each >other in a statistical sense.
>
> > Bob Von Dreele
>
>In that case even the single crystal data resulting from the integration
>of images registred by an image plate are not strictly independent!!??
>Information read from one pixel can depend on the information registred
>in a neigbouring pixel.

This would be true if the spots are close enough to each other so that they 
overlap (i.e. in protein patterns). Another case is that of 2-D area 
neutron detectors where one strong reflection can interfere with the 
measured intensities of other reflections seen in the same crystal setting 
because it can "blind" the whole detector. The main point is that one 
observation must somehow be correlated in the measuring process with other 
observations for them to not be statistically independent. The fact that 
the suite of observations are of the same object whether it be Bragg peaks 
in a powder pattern or some other experimentally observed feature does not 
make these observations  "correlated". Bottom line is that Hamilton's test 
is just as valid for powder data as it is for single crystal data.

Bob Von Dreele

Date: Mon, 09 Apr 2001 18:10:01 +0200
From: Radovan Cerny [[email protected]]
Subject: Re: Hamilton's test
To: [email protected]
Organization: University of Geneva


Bob Von Dreele wrote:

> >In that case even the single crystal data resulting from the integration
> >of images registred by an image plate are not strictly independent!!??
> >Information read from one pixel can depend on the information registred
> >in a neigbouring pixel.
> 
> This would be true if the spots are close enough to each other so that they
> overlap (i.e. in protein patterns)

No Bob, that was not my point. When you integrate one single crystal
reflection, well separated from the others, what data do you use :
information coded in individual pixels. Have they "bled over" into the
neighboring pixels during the reading procedure? The pixels do not know
whether it was powder or single crystal who sent the photons to them :-)
I do not think that the powder data integrated from an image plate are
more correlated than those measured with a point detector.

Anybody who has developped the image plate software?

Best regards
-- 
Radovan Cerny              
Laboratoire de Cristallographie    
24, quai Ernest-Ansermet              
CH-1211 Geneva 4, Switzerland                            
Phone  : [+[41] 22] 702 64 50, FAX : [+[41] 22] 702 61 08
mailto : [email protected]
URL    : http://www.unige.ch/crystal/cerny/rcerny.htm