Analysis of ratios in multivariate morphometry

Abstract

The analysis of ratios of body measurements is deeply ingrained in the taxonomic literature. Whether for plants or animals, certain ratios are commonly indicated in identification keys, diagnoses, and descriptions. They often provide the only means for separation of cryptic species that mostly lack distinguishing qualitative characters. Additionally, they provide an obvious way to study differences in body proportions, as ratios reflect geometric shape differences. However, when it comes to multivariate analysis of body measurements, for instance, with linear discriminant analysis (LDA) or principal component analysis (PCA), interpretation using body ratios is difficult. Both techniques are commonly applied for separating similar taxa or for exploring the structure of variation, respectively, and require standardized raw or log-transformed variables as input. Here, we develop statistical procedures for the analysis of body ratios in a consistent multivariate statistical framework. In particular, we present algorithms adapted to LDA and PCA that allow the interpretation of numerical results in terms of body proportions. We first introduce a method called the "LDA ratio extractor," which reveals the best ratios for separation of two or more groups with the help of discriminant analysis. We also provide measures for deciding how much of the total differences between individuals or groups of individuals is due to size and how much is due to shape. The second method, a graphical tool called the "PCA ratio spectrum," aims at the interpretation of principal components in terms of body ratios. Based on a similar idea, the "allometry ratio spectrum" is developed which can be used for studying the allometric behavior of ratios. Because size can be defined in different ways, we discuss several concepts of size. Central to this discussion is Jolicoeur's multivariate generalization of the allometry equation, a concept that was derived only with a heuristic argument. Here we present a statistical derivation of the allometric size vector using the method of least squares. The application of the above methods is extensively demonstrated using published data sets from parasitic wasps and rock crabs.

FIGURE 1.

Scatter plots of the four most discriminating ratios for Pteromalus albipennis (dots) and P. solidaginis (triangles). Plot (a) shows first versus second ratio, plot (b) third versus fourth ratio.

FIGURE 2.

Application of the PCA ratio spectrum using the Pteromalus data, with Pteromalus albipennis (dots) and P. solidaginis (triangles). (a) Scatterplot of a principal component analysis (PCA) in shape space. (b) PCA ratio spectrum of the first principal component. The ratio formed from the extremal points (i.e., gaster breadth:tergum 7 length) explains a large part of the variation of the first component. In contrast, ratios formed from characters lying close to each other in the spectrum (e.g., marginal vein:postmarginal vein) explain very little. This is apparent in the scatterplot (c). Confidence intervals (horizontal bars in (b), see Methodology section) were estimated with a bootstrap.

FIGURE 3.

Scatterplot of isometric size versus first principal component in shape space for the Pteromalus data set, with Pteromalus albipennis (dots) and P. solidaginis (triangles). The mean size of P. solidaginis is obviously smaller but it still lies within the range of Pteromalus albipennis.

FIGURE 4.

The allometry ratio spectrum for the Leptograpsus variegatus data set for blue type males (a) and for orange type males (b) respectively. The characters shown are carapace length (CL) and width (CW), width of frontal lobe (FL), rear width (RW), and body depth (BD) (see Results section). The bars do not represent confidence intervals here.

FIGURE 5.

Scatter plots of isometric size versus log-ratios body depth:rear width (a) and carapace length:width (b) for the orange type males in the Leptograpsus variegatus data set.

FIGURE 6.

Scatterplots of principal component analyses (PCA) of a single species of Pteromalus (n = 32 specimens of Pteromalus albipennis, p = 23 variables of body measurements; data from Baur 2002), showing the effect of different definitions of size and shape. Specimen labeled y is a clone of specimen x but with all variables scaled by a factor of 1.4. The two specimens have therefore equal values for all their ratios and are only separated along the isometric size axis, as indicated by the line connecting x with y. (a) Scatterplot of first against second and (b) of second against third component respectively of a standard PCA on the covariance matrix of log-transformed data. The first component is considered as a general size measure because its coefficients have the same sign and are of similar magnitude for all variables. However, they are not exactly the same, thus the first component of a standard PCA is usually considered as the allometric size axis (Jolicoeur 1963; Claude 2008). The remaining components define the shape space in this analysis. Note that the line of isometry is not parallel to the first component, and, thus, reflects the different size measures. As a result, specimens x and y are also widely separated points in the shape space, although viewed from their body proportions they are identical. For (c) and (d) the same data were used, but here they were subjected to a PCA after removal of isometric size (for details of computation, see the Methodology section). Now, the line of isometry connecting x with y lies of course parallel to the isometric size axis (c). In the shape space (d) the two specimens form a single point, because only those specimens appear distinct which also differ in body proportions.