Shape, relative size, and size???adjustments in morphometrics
Many problems in comparative biology and biological anthropology require meaningful definitions of “relative size” and “shape.” Here we review the distinguishing features of ratios and residuals and their relationships to other methods of “size-adjustment” for continuous data. Eleven statistical techniques are evaluated in reference to one broadly interspecific data set (craniometries of adult Old World monkeys) and one narrowly intraspecific data set (anthropometries of adult Native American males). Three different types of residuals are compared to three versions of shape ratios, and these are contrasted to “cscores,” Penrose shape, and multivariate adjustments based on the first principal component of the logged variance-covariance matrix; all methods are also compared to raw and logged raw data. In order to help us identify appropriate; methods for size-adjustment, geometrically similar or “isometric” versions of the male vervet and the Inuit male were created by scalar multiplication of all variables. The geometric mean of all variables is used as overall “size” throughout this investigation, but our conclusions would be the same for most other size variables. Residual adjustments failed to correctly identify individuals of the same shape in both sampkles. Like residuals, cscores are also sample-specific and incorrectly attribute different shape values to individuals known to be identical in shape. Multivariate “residuals” (e.g., discarding the first principal component and Burnaby's method) are plagued by similar problems. If one of the goals of an analysis is to identify individuals (OTUs) of the same shape after accounting for overalll size differences, then none of these methods can be recommended. We also reject the assertion that size-adjusted variables should be unciorrelated with size of “size-free”; rather, whether or not shape covaries with size is an important empirical determination in any analysis. Without explicit similarity criteria, “lines of subtraction” can be very misleading. Only variables in the Mosimann family of shape rations allowed us to identify sized individuals of the same shape (“Iso-OUTs”). Residuals from isometric lines in logarithmic space, projections of logged data to a plane orthogonal to an isometric vector, and Penrose shape distance based on logged data are also part of this shape family. Shape defined in this manner can be significantly correlated with size in allometric data sets (e.g., guenon craniometrics); ratio shape differences may be largely independent of size in narrowly intraspecific or intrasexual data sets (e.g., Native American anthropometrics). Log-transformations of shape variables are not always necessary or desirable. We hope our findings enciourage other workers to question the assumptions and utility of residuals as size-adjusted data and to explore shape and relative size within Mosimann's explicitly geometric framework. © 1995 Wiley-Liss, Inc.