In response to Peter Das's request to summarise to the list, I got 13 replies in
total.
Most pointed to Aitchison, either The statistical analysis of compositional
data, JRSS B, 44, p139 or "The Statistical Analysis of Compositional Data",
Chapman and Hall, 1986. I dug out the JRSS B article, which relates to
proportional composition when variables are continuous (with examples of
geology), which wasn't what I needed.
For my purposes, the most helpful observation was by Tim Cole
<[log in to unmask]>:
Anthony Edwards talked about this in his book "Likelihood" (CUP 1972). On page
140 there is such a triangle with a confidence interval ellipse for the
frequencies p, q and r of the A, B and O blood group genes.
Other responses:
I have just finished reading one of our Master's student's disserattaion who
uses this method. He cites Vacher, Journal of Geoscience 53(3) 324-333. He has
also written a SAS macro to make these plots that I expect he would be happy to
let you have access to. I am copying this to him.
David Ramsay & Gillian Raab <[log in to unmask]>
Soil studies use triangular graphs for the proportions of clay, silt & sand,
see:
http://www.teachingkate.org/dirt.htm
While doing a web search for this I found a reference to the following which
must be applying a similar technique to socio-demographic data: DORLING, D.,
Johnston, R.J. and Pattie, C.J. (1996), Representing, exploring and analysing
electoral change using triangular graphs, Environment and Planning A, 28,
979-998. I'd be interested to find (free!) software for these graphs as I
would like to plot tenure
for areas by owner occupiers, public & private renters.
<[log in to unmask]>
It's called various things - 'ternary diagram' 'triangular diagram' etc. There
is a STATA ado file to draw them created by Nicolas Cox at Durham (not the
confidence region)
Denise Howel <[log in to unmask]>
There were a couple of references in the ASA magazine "Chance". Wainer, H.
(1995), 8,1,48-54 and Allen, T. (2002), 15,3, 29-35. I believe you can now
access at least some of the papers online. Perhaps you might find what you're
looking for in the references listed there.
<[log in to unmask]>
According to Michael Friendly, this can be traced to the 13th century. Nothing
much then seems to have happened until the m 19th century, when Moebius picked
up the idea. LLull, R. (1274-1283). Artifitium electionis personarum. Biblioteca
Apostolica Vaticana, Cod. Vat. lat. 9332, f. 11r-12v. There are many accessible
references using names such as trilinear plots. There is a rich vein of
literature in the journal
_Mathematical Geology_. Closer to you, they are often used in genetics.
Nick Cox <[log in to unmask]>
David Clayton, almost certainly in Applied Statistics. Have been used to plot %
of votes for 3 party elections.
<[log in to unmask]>
----------------------------------------------------------------------------
Robert G. Newcombe PhD CStat FFPH
Professor of Medical Statistics
Wales College of Medicine
Cardiff University
Heath Park
Cardiff CF14 4XN
Phone 029 2074 2329
Fax 029 2074 2898
http://www.cardiff.ac.uk/medicine/epidemiology_statistics/research/statistics/newcombe.htm
>>> P Das <[log in to unmask]> 27/09/05 21:02:46 >>>
I first saw this representation of a trinomial used in geology
(description of composition of petroleum from different sources) in a
Dutch-language book "De methodes der graphische voorstelling" [Methods
of Graphical Representation] by B.G. Escher, 1934.
I used it in presenting statistical results, often to the bewilderment
of outsiders. So I switched back to just plotting p1 and p2 on
rectangular axes, which is difficult enough to grasp for many.
The confidence region is new for me. Can't think how this could be
elliptical. I would expect it to be symmetrical in p1, p2 and p3. (That
is not necessarily circular.)
Please summarize to the list or to me!
My original query 26/9/2005:
> I vaguely recall seeing some reference to the use of an equilateral
> triangle as
> a geometrcal representation for any three proportions p1, p2 and p3
> adding to 1,
> and perhaps also a vaguely (won't be exactly) elliptical confidence
> region for
> the point representing (p1, p2, p3). Could someone please send me the
> reference
> for this?
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