Abstract
One of the most basic but problematic issues in modern morphometrics is how many specimens one needs to achieve accuracy in samples. Indeed, this is one of the most regularly posed questions in introductory courses. There is no simple and certainly no absolute answer to this question. However, there are a number of techniques for exploring the effect of sampling, and our aim is to provide an example of how this might function in a simplified but informative way. Thus, using resampling methods and sensitivity analyses based on randomized subsamples, we assessed sampling error in horse teeth from several modern and fossil populations. Centroid size and shape of an upper premolar (PM2) were captured using Procrustes geometric morphometrics. Means and variances (using three different statistics for shape variance) were estimated, as well as their confidence intervals. Also, the largest population sample was randomly split into progressively smaller subsamples to assess how reducing sample size affects statistical parameters. Results indicate that mean centroid size is highly accurate; even when sample size is small, errors are generally considerably smaller than differences among populations. In contrast, mean shape estimation requires large samples of tens of specimens (ca. >20), although this requirement may be less stringent when variance in a population is very small (e.g. populations that underwent strong genetic bottlenecks). Variance in either centroid size or shape can be highly inaccurate in small samples, to the point that sampling error makes it as variable as differences among spatially and chronologically well-separated populations, including two which are highly distinctive as a consequence of strong artificial selection. Likely, centroid size and shape variance require no <15–20 specimens to achieve a reasonable degree of accuracy. Results from the simplified sensitivity analysis were largely congruent with the pattern suggested by bootstrapped confidence intervals, as well as with the observations of a previous study on African monkeys. The analyses we performed, especially the sensitivity assessment, are simple and do not require much time or computational effort; however, they do necessitate that at least one sample is large (50 or more specimens). If this type of analyses became more common in geometric morphometrics, it could provide an effective tool for the preliminarily exploration of the effect of sampling on results and therefore assist in assessing their robustness. Finally, as the use of sensitivity studies increases, the present case could form part of a set of examples that allow us to better understand and estimate what a desirable sample size might be, depending on the scientific question, type of data and taxonomic level under investigation.
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Acknowledgments
Funding was provided to G.B. from the Leverhulme Trust project grant scheme (F/09 757/B) and to K.S. and A.C. from the Lang Fund for Human-Environmental Anthropology, Department of Anthropology, Stanford. We are extremely grateful to the following for facilitating access to materials and providing essential input for specific samples: ICE and PRZ: F. Mayer (Museum für Naturkunde, Berlin); THB and PRZ: R. Sabin and L. Tomsett (Natural History Museum, London); CHI: T. Yaqi and H. Songmei (Shaanxi Archaeological Research Institute, Xi’an), KAZ: Z. Samashev and K. Kashkanbajev (Institute of Oriental Studies, Almaty), RUS: P. Kotsinsev (Institute of Ecology of Plants and Animals, Yekaterinburg) and B. Hanks (Department of Archaeology, Pittsburgh); CRO: D. Brajkovic (Institute for Quaternary Palaeontology and Geology, Zagreb) and P. Miracle (Department of Archaeology, Cambridge) and HUN: A. Choyke (Department of Medieval Studies, Budapest) and A. Endrődi (Budapest History Museum, Budapest). We thank N. Vibla and J. Li for assistance in recovering samples and translation; A. Evin (Natural History Museum, Paris) for providing a set of duplicate samples from Berlin. We are deeply grateful to: I. Dryden (University of Nottingham), P. Gunz (Max Planck Institute for Evolutionary Anthropology, Leipzig), L. Monteiro (Universidade Estadual do Norte Fluminense), S. Schlager (University of Freiburg) and other MORPHMET subscribers for help and suggestions with R scripts and packages, as well as with references; P. Mitteroecker (University of Vienna) for his important feedback on how to estimate multivariate variance; and P.D. Polly (Indiana University, Bloomington) for the most useful discussions on teeth evolution and shape analysis. Also, we would like to thank a lot two anonymous reviewers who provided truly interesting and most constructive comments that greatly improved the original version of this manuscript.
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Communicated by Andreas Schmidt-Rhaesa.
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Cardini, A., Seetah, K. & Barker, G. How many specimens do I need? Sampling error in geometric morphometrics: testing the sensitivity of means and variances in simple randomized selection experiments. Zoomorphology 134, 149–163 (2015). https://doi.org/10.1007/s00435-015-0253-z
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DOI: https://doi.org/10.1007/s00435-015-0253-z