Dear All,
I'd like to add a few comments on sampling (landmarks but also specimens). I hope that some of the other subscribers, who know much more than I do about morphometrics, will refine and correct my points.

A very short one on my two papers. They make a very simple point: if one is landmarking just one side of a structure with object symmetry simply to speed up data collection, then mirror-reconstructing the missing side will make a nicer visualization and probably make shape data which are closer to those obtained by landmarking both sides. The difference may be tiny and I said "probably" because I am reporting results of empirical studies: out of 11-12 datasets, all but one had shape distances closer to those of the full bilateral landmark data after mirror-reconstructing the missing side. This did not work in one dataset which happened to have a very large amount of fluctuating asymmetry. To what extent these results are generalizable, I can't say but everyone can plan a small preliminary analysis to check it in her/his own data.

I fully agree with Aki that, if time, money etc. are not a constraint, even when one is not interested in asymmetry, it is better to measure both sides. That's in fact true also for structures with matching symmetry.

In terms of the choice of landmarks, I wish to stress (once more!) that quality may be more important than quantity: first one should think well about what she/he wants to measure, which will relate to the specific question being asked, and then decide about where and how many landmarks to use. There are at least two wonderful papers I suggested several times on this issue:
Oxnard & O'Higgins, 2009, Biological Theory 4(1), 84–97.
Klingenberg, 2008, Evol Biol 35:186–190

Then, especially for semilandmarks, I guess that as Aki (and others before) suggested, one can see what a good compromise is between information and the number of points (maybe considering also, but not principally, the visualization).

For sample size, one should consider whether differences are presumably big (and a small sample might be OK...ish) or small (as in most microevolutionary studies, which generally require large N). I believe that Rohlf, already in the early days of geometric morphometrics, had written a software for exploring statistical power in shape data (TPSPower) but I am not sure if he kept developing it. In any case, power and sensitivity (to sampling) analyeses are certainly available in R. With small differences, although resampling methods may allow to perform tests even with tiny samples, power will be low and estimates (say, mean size and shape, variance and covariance etc.) will be likely inaccurate. Unfortunately, often, the most interesting taxa are rare populations (or fossils) for which specimens are difficult to find.

A couple of people told me that there's an important paper coming out soon on sampling error in geometric morphometrics and it might suggest that one really needs huge samples. I would not be surprised and suspect that the few empirical studies we did (a couple of papers in Zoomorphology) were overoptimistic despite already suggesting (more or less) that one might need several dozens of specimens even when differences are relatively large and the number of landmarks was not particularly large. Again, they were empirical studies and one cannot say how generalizable they are. Anyway, I look forward to this new paper and hope it will be announced in MORPHMET, as well as I look forward to Aki's paper.



On 29/05/17 18:35, Aki Watanabe wrote:
Dear Lea,

Unfortunately, there isn't (yet) a magic mathematical formula to determine whether you've sampled enough landmarks, but there are some exploratory approaches you can take to see if you're landmark sampling is converging to the "true" shape variation. One simple thing you can do is sample as many landmarks as you can on a representative sampling of specimens, then create a PC morphospace. Then, subsample the landmarks (e.g., 75%, 50%, 25% of the landmarks) and see if the PC morphospace from these subsampled datasets mirror the distribution of shapes of the full dataset. If the morphospaces begin deviating from the PC morphospace of the full dataset, then you have a visual cue that the subsampling is not adequately characterizing the shape variation of your specimens. In terms of a statistically significant test for landmark sampling, I suppose one can test for correlation between subsampled and full dataset, but because the subsampled and full dataset will be auto-correlated to some extent, the null would have to reflect this.

Alternatively, I have a script that automatically subsamples the landmarks of a given dataset and creates a plot to see how well the subsampled datasets converge to the point distribution of the full dataset. If you are interested, I would be happy to describe the technique in more detail and/or run the analysis on your dataset if you don't mind sending me the data. The script is currently under review for a journal, so it's not available yet to the public.

Also, as you mention, having more shape variables (i.e., number of landmarks x 2 or 3 depending on 2-D or 3-D landmarks) than the number of specimens will generally reduce the power of statistical tests. There are ways to counter this issue (e.g., Q-mode approach recently proposed by Dean Adams).

Now, concerning the sampling of bilateral landmarks, Andrea Cardini has recently written a nice pair of papers on the subject:

Cardini, A. 2016. Left, right or both? Estimating and improving accuracy of one-side-only geometric morphometric analyses of cranial variation. J Zool Syst Evol Res.

Cardini, A. 2016. Lost in the other half: improving accuracy in geometric morphometric analyses of one side of bilaterally symmetric structures. Syst Biol.

These papers highlight the artifact that originates from performing Procrustes alignment on "one-side-only" datasets. At least for alignment purposes, I suggest sampling both sides of bilaterally symmetric structures.

Hope this helps.

All the best,

On Tuesday, May 9, 2017 at 12:26:04 PM UTC+1, Lea Wolter wrote:

    Hello everyone,

    I am new in the field of geometric morphometrics and have a question
    for my bachelor thesis.

    I am not sure how many landmarks I should use at most in regard to
    the sample size. I have a sample of about 22 individuals per
    population or maybe a bit less (using sternum and epigyne of
    spiders) with 5 populations.
    I have read a paper in which they use 18 landmarks with an even
    lower sample size (3 populations with 20 individuals, 1 with 10).
    But I have also heard that I should use twice as much individuals
    per population as land marks...

    Maybe there is some mathematical formula for it to know if it would
    be statistically significant? Could you recommend some paper?

    Because of the symmetry of the epigyne I am now thinking of using
    just one half of it for setting landmarks (so I get 5 instead of 9
    landmarks). For the sternum I thought about 7 or 9 landmarks, so at
    most I would also get 18 landmarks like in the paper.

    I would also like to use two type specimens in the analysis, but I
    have just this one individual per population... would it be totally
    nonesens in a statistical point of view?

    Thanks very much for your help!

    Best regards

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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

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