Growth and shape: Measurements and metrics

Abstract

To the extent that growth markers provide more information about developmental processes than is typically available in ontogenetic shape analyses, analyses of them are richer than those relying on size and shape data alone. Most workers will not have access to that kind of data because most species do not naturally record growth processes, and few workers have the resources to make those processes visible by marking growing individuals. Visible markers of growth permit more detailed and precise descriptions of the process, although they can detract from shape analyses when the shape at the outset of the process cannot be recovered from the coordinates of the markers. That problem applies to the analyses of growing scutes; the shape of the plastron at to cannot be inferred from the location of the landmarks recording the growth markers because they no longer abut (as they necessarily did at to). Growth markers are not the only source of information about process. Such information can also be obtained by measuring causal factors (such as levels of growth hormones) and analyzing their impact on shape. The covariances between those factors and shape, which can be analyzed by regression (Monteiro, 1999) or partial least squares (Rohlf and Corti, 2000) will almost invariably yield more information than can be obtained from records of shape change over time. Magwene overstated the defects of analyses based purely on size and shape data. There is more information available when the same individuals are repeatedly measured at visible growth markers, and asymmetric growth rates at sutures cannot be documented without landmarks that reveal those asymmetries, just as any other process of asymmetric growth cannot be documented in the absence of necessary data. However, most of his arguments about shape data apply to all analyses of landmarks, including his. In particular, no landmark-based method offers information about changes between landmarks; those changes are necessarily matters of inference. When it comes to the specifically mathematical differences between conventional geometric methods and those recommended by Magwene, i.e., complex number notation and the PCA of shape coordinates obtained by the two-point registration, the methods offered by Magwene do not offer a richer, more powerful, or even equally as general an approach. © Oxford University Press 2001.