How many specimens do I need? Sampling error in geometric morphometrics: testing the sensitivity of means and variances in simple randomized selection experiments

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

Arabidopsis phenotyping through geometric morphometrics

“…Although sampling problems arising from scarcity of specimens is certainly not a problem in Arabidopsis studies, having a large number of plants could indeed pose a problem because of the limited room in expensive growth chambers that is needed to perform physiological experiments in a highly controlled environment. Minimum sample size estimation is not trivial, because it may vary depending on the natural shape variation within the assessed population and the kind of scientific question being addressed [85]. Future work should investigate the effect of sample size onto statistical parameters, since this parameter affects shape estimates more than size [85].…”

“…Minimum sample size estimation is not trivial, because it may vary depending on the natural shape variation within the assessed population and the kind of scientific question being addressed [85]. Future work should investigate the effect of sample size onto statistical parameters, since this parameter affects shape estimates more than size [85]. …”

Anyone with a Long-Face? Craniofacial Evolutionary Allometry (CREA) in a Family of Short-Faced Mammals, the Felidae

“…The analysis was done using species means regardless of sex, to increase power by including all species, and repeated within subfamilies using both a simple 'non-comparative' MA regression, as well as one which takes phylogeny into account using PICs and the same range of evolutionary models as in the PGMM analyses. The MA regression was selected as the relationship between the two variables is symmetrical, they use the same unit of measure and are both likely to have similar amount of errors (unlike geometric morphometric data, where size and shape have different units of measures and centroid size generally is more accurate than shape both in terms of sample mean estimates and measurement error -Cardini et al 2015b). …”

“…Estimates of species mean shapes can be strongly affected by sampling error (Cardini & Elton 2007;Cardini et al 2015b). However, as for size, in a macroevolutionary analysis at familial/subfamilial levels, with 13 different genera out of a total of 27 species available for the study, and a range of sizes spanning two orders of magnitude, interspecific shape differences are likely to be mostly so large to be fairly accurately described by using just one or a few specimens per species.…”

Mammalian distal humerus fossils from eastern Montana, USA with implications for the Cretaceous-Paleogene mass extinction and the adaptive radiation of placentals

“…We digitized six LMs (Figure 3.1; Table 5) on each specimen image using tpsDig version 2.17 (Rohlf, 2013a). We acknowledge that plotting landmarks on a two-dimensional (2D) picture of a threedimensional structure may result in a loss of information and a degree of inaccuracy (Cardini, 2014); however, 2D approximation should be sufficient to capture morphological variation because the landmarks are approximately coplanar (Cardini et al, 2015). All of the digitizing was done by one of us (L.B.D.…”