Ilker, Philipp already defined well why - I think - this rationale is incorrect, if not dangerous, especially along the lines of statistical power. As he indicated, using Procrustes residuals as data means a covariance matrix will never be full rank, owing to the invariance in size, orientation, and position of landmark configurations following GPA. At most, the dimensions of the data space can be kp - g, where k is the number of landmark dimensions (2 or 3), p is the number of landmarks, and g is the number of invariant dimensions due to GPA (with or without sliding landmarks) or n - 1 if (kp - g) > n - 1. As he also pointed out, increasing landmarks can increase the spatial resolution, meaning that if n - 1 is the limited number of dimensions, the distances between specimens can increase in the n - 1 dimensional space that results from increasing p. If by “statistical power” one means an increased probability to reject a null hypothesis that population centroids (mean configurations) are the same, then increasing resolution should enhance one’s ability to reject a null hypothesis.

I think tying the dimensionality of the space where the hypothesis is tested to the number of landmarks precludes appreciating Philipp’s comment about spatial resolution. I do not wish to necessarily advocate using a limited number of PCs as shape data, as a rule, but one can appreciate that given a choice between two configurations - one with seven fixed 2D landmarks (10 PCs after GPA) and one with the first 10 PCs obtained from configurations with hundreds of landmarks - the separation of groups in the latter case might be more prominent than in the former, hence increasing statistical power. Whether hundreds of landmarks are needed, or 50, or 20, or 10, or even only 7, or whether increasing statistical power is important, is a question that must be answered case by case with empirical results. However, placing an a priori limit on the number of landmarks one can define because of the size of samples one can collect is certain way to limit statistical power, especially when small samples are all that’s available. Cheers! Mike > On Jun 3, 2017, at 5:31 AM, Ilker ERCAN <ier...@msn.com> wrote: > > when we perform multivariate analysis, It must be n>p otherwise determinant > of Generalized variance equals to zero therefore it must be 2*l<n for 2D or > 3*l<n for 3D l: landmark number > Best wishes > Ilker ERCAN > > > Gönderen: Norman MacLeod <n.macl...@nhm.ac.uk <mailto:n.macl...@nhm.ac.uk>> > Gönderildi: 3 Haziran 2017 Cumartesi 11:18 > Kime: MORPHMET > Konu: Re: [MORPHMET] Re: number of landmarks and sample size > > In discussions like these it would be helpful if the writer could clarify > whether they are referring to the concepts of biological homology, > topological homology or "semantic homology". These aren't the same things and > the whole issue of “homology” in geometric morphometrics has always seemed, > at least to me, to be very confused. For example, refer to the definitions of > “homology” and “landmark” in the Glossary on the SB Morphometrics web site. > Because it means different things to different specialists homology isn't a > term to be thrown around as lightly as morphometricians seem prone to do. > Imprecise and/or ambiguous usage renders the meaning of sentences difficult > or impossible to understand for me and I suspect confuses others as well. > > Norm MacLeod > > > > On 3 Jun 2017, at 08:53, alcardini <alcard...@gmail.com> wrote: > > > > Hi Philipp, > > I am not worried about the number of variables (although I am not sure > > one needs thousands of highly correlated points on a relatively simple > > structure and seem to remember that Gunz and you suggest to start with > > many and then reduce as appropriate). > > > > Regardless of whether point homology makes sense, I am worried that > > many users believe that semilandmarks (maybe after sliding according > > to purely mathematical principles) are the same as "traditional > > landmarks" with a clear one-to-one correspondence. Even saying that > > what's "homologous" is the curve or surface is tricky, because at the > > end of the day that curve/surface is discretized using points, shape > > distances are based on those points and there are many ways of placing > > points with no clear "homology" (figure 7 of Oxnard & O'Higgins, > > 2009); indeed, in a ontogenetic study of the cranial vault, for > > instance, where sutures may become invisible in adults and therefore > > cannot be used as a "boundary", semilandmarks close to the sutures may > > end up on different bones in different stages/individuals. > > > > Semilandmarks are a fantastic tool, which I am happy to use when > > needed, but they have their own limitations, which one should be aware > > of. > > Cheers > > > > Andrea > > > > > > > > On 03/06/2017, mitte...@univie.ac.at <mitte...@univie.ac.at> wrote: > >> I think a few topics get mixed up here. > >> > >> Of course, a sample can be too small to be representative (as in Andrea's > >> example), and one should think carefully about the measures to take. It is > >> also clear that an increase in sample size reduces standard errors of > >> statistical estimates, including that of a covariance matrix and its > >> eigenvalues. But, as mentioned by Dean, the standard errors of the > >> eigenvalues are of secondary interest in PCA. > >> > >> If one has a clear expectation about the signal in the data - and if one > >> does not aim at new discoveries - a few specific measurements may suffice, > >> perhaps even a few distance measurements. But effective exploratory > >> analyses have always been a major strength of geometric morphometrics, > >> enabled by the powerful visualization methods together with the large > >> number of measured variables. > >> > >> Andrea, I am actually curious what worries you if one "collects between > >> 2700 and 10 400 homologous landmarks from each rib" (whatever the term > >> "homologous" is supposed to mean here)? > >> > >> Compared to many other disciplines in contemporary biology and biomedicine, > >> > >> a few thousand variables are not particularly many. Consider, for instance, > >> > >> 2D and 3D image analysis, FEA, and all the "omics", with millions and > >> billions of variables. In my opinion, the challenge with these "big data" > >> is not statistical power in testing a signal, but finding the signal - the > >> low-dimensional subspace of interest - in the fist place. But this applies > >> to 50 or 100 variables as well, not only to thousands or millions. If no > >> prior expectation about this signal existed (which the mere presence of so > >> many variables usually implies), no hypothesis test should be performed at > >> all. The ignorance of this rule is one of the main reasons why so many GWAS > >> > >> and voxel-based morphometry studies fail to be replicable. > >> > >> Best wishes, > >> > >> Philipp > >> > >> -- > >> MORPHMET may be accessed via its webpage at http://www.morphometrics.org > >> <http://www.morphometrics.org/> > www.morphometrics.org <http://www.morphometrics.org/> > www.morphometrics.org <http://www.morphometrics.org/> > This is an independent site for maintaining various services related to > (mostly biological) shape analysis. Primarily, this supports the current > incarnation of the ... > > > >> --- > >> You received this message because you are subscribed to the Google Groups > >> "MORPHMET" group. > >> To unsubscribe from this group and stop receiving emails from it, send an > >> email to morphmet+unsubscr...@morphometrics.org. > >> > > > > > > -- > > > > Dr. Andrea Cardini > > Researcher, Dipartimento di Scienze Chimiche e Geologiche, Università > > di Modena e Reggio Emilia, Via Campi, 103 - 41125 Modena - Italy > > tel. 0039 059 2058472 > > > > Adjunct Associate Professor, School of Anatomy, Physiology and Human > > Biology, The University of Western Australia, 35 Stirling Highway, > > Crawley WA 6009, Australia > > > > E-mail address: alcard...@gmail.com, andrea.card...@unimore.it > > WEBPAGE: https://sites.google.com/site/alcardini/home/main > > <https://sites.google.com/site/alcardini/home/main> > <https://sites.google.com/site/alcardini/home/main> > main - Andrea Cardini - Google Sites > <https://sites.google.com/site/alcardini/home/main> > sites.google.com <http://sites.google.com/> > Dr Andrea Cardini. Researcher, Dipartimento di Scienze Chimiche e Geologiche, > Università di Modena e Reggio Emilia, Via Campi, 103 - 41125 Modena - Italy > > > > > > FREE Yellow BOOK on Geometric Morphometrics: > > http://www.italian-journal-of-mammalogy.it/public/journals/3/issue_241_complete_100.pdf > > > > <http://www.italian-journal-of-mammalogy.it/public/journals/3/issue_241_complete_100.pdf> > HYSTRIX - italian-journal-of-mammalogy.it > <http://www.italian-journal-of-mammalogy.it/public/journals/3/issue_241_complete_100.pdf> > www.italian-journal-of-mammalogy.it > <http://www.italian-journal-of-mammalogy.it/> > HYSTRIX the Italian Journal of Mammalogy Volume 24(1) • 2013 Edited and > published by Associazione Teriologica Italiana Editor in Chief Giovanni A > > > > > > ESTIMATE YOUR GLOBAL FOOTPRINT: > > http://www.footprintnetwork.org/en/index.php/GFN/page/calculators/ > > <http://www.footprintnetwork.org/en/index.php/GFN/page/calculators/> > Footprint Calculator - Global Footprint Network > <http://www.footprintnetwork.org/en/index.php/GFN/page/calculators/> > www.footprintnetwork.org <http://www.footprintnetwork.org/> > How big is your Footprint? Take this quiz to find out your biggest areas of > resource consumption and learn how to tread more lightly on the Earth. > > > > > > -- > > MORPHMET may be accessed via its webpage at http://www.morphometrics.org > > <http://www.morphometrics.org/> > www.morphometrics.org <http://www.morphometrics.org/> > www.morphometrics.org <http://www.morphometrics.org/> > This is an independent site for maintaining various services related to > (mostly biological) shape analysis. Primarily, this supports the current > incarnation of the ... > > > > --- > > You received this message because you are subscribed to the Google Groups > > "MORPHMET" group. > > To unsubscribe from this group and stop receiving emails from it, send an > > email to morphmet+unsubscr...@morphometrics.org. > > > > > _____________________________________________________ > > Professor Norman MacLeod > The Natural History Museum, Cromwell Road, London, SW7 5BD > (0)207 942-5204 (Office Landline) > (0)785 017-1787 (Mobile) > http://paleonet.org/MacLeod// <http://paleonet.org/MacLeod//> > > Department of Earth Sciences, University College > London, Gower Street, London WC1E 6BT, UK > > Nanjing Institute of Geology & Palaeontology, > Chinese Academy of Sciences, 39 Beijing, Donglu, Nanjing, China > _____________________________________________________ > > > > > > > > > > > > > > > -- > MORPHMET may be accessed via its webpage at http://www.morphometrics.org > <http://www.morphometrics.org/> > --- > You received this message because you are subscribed to the Google Groups > "MORPHMET" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to morphmet+unsubscr...@morphometrics.org. > > -- > MORPHMET may be accessed via its webpage at http://www.morphometrics.org > <http://www.morphometrics.org/> > --- > You received this message because you are subscribed to the Google Groups > "MORPHMET" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to morphmet+unsubscr...@morphometrics.org > <mailto:morphmet+unsubscr...@morphometrics.org>. -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org --- You received this message because you are subscribed to the Google Groups "MORPHMET" group. 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