The Truss: Body Form Reconstructions in Morphometrics" 2

Strauss, R. E., and F. L. Bookstein (Museum of Zoology, The University of Michigan, Ann Arbor, Michigan 48109) 1982. The truss: body form reconstruction in morphometrics. Syst. Zool., 31:113-135.-In principle, any measured distances between landmarks of a form may serve as characters for morphometric analyses. Systematic studies typically are based on a highly biased and repetitious sample of these. But collections of landmarks and the distances among them must be homologous from form to form for comparisons to be meaningful, and an adequate character set should at least permit the full reconstruction of the original configuration of landmarks. We describe a geometric protocol for character selection, the truss network, which enforces systematic coverage of the form and which exhaustively and redundantly archives the landmark configuration. Reconstruction of the form from truss measures provides Cartesian coordinates for landmarks and allows estimation of, and compensation for, measurement error. Samples of forms may be averaged and standardized to one or more common reference sizes by regression of measured distances on a composite measure of body size, followed by reconstruction of the form using distance values predicted by the regression functions at some standard body size. Principal component loadings of distance measures may be indicated directly on the truss network to display patterns of within-group allometry or between-group shape differences. Because the truss enforces use of cross measurements, discrimination among groups may be enhanced. Composite mapped forms are useful in biorthogonal analyses of differences in shape because they allow the comparison of averaged forms among samples. Certain patterns of principal component loadings are concordant with, and provide an initial sampling of, the biorthogonal grids for these deformations. [Allometry; biorthogonal analysis; discriminant analysis; fishes; morphometrics; multivariate analysis; principal components; triangulation; truss.] Systematists are often interested in quantifying differences in form among different species, conspecific populations, or ontogenetic stages. Customarily, morphometric data are taken without regard for allometry or its variations among populations or growth stages. It has been common to study growth, for example, by analysis of body length or height or weight only; to describe the shapes of bones by measures of their lengths and 1 A contribution of the Morphometrics Study Group, University of Michigan: Fred Bookstein, Barry Chernoff, Ruth Elder, Julian Humphries, Gerald Smith, and Richard Strauss. 2 An earlier version of this material was presented at the Symposium "Morphometric Studies of Fishes, Amphibians, and Reptiles," American Society of Ichthyologists and Herpetologists, Corvallis Oregon, June 1981. widths; to characterize entire forms by relative lengths and breadths of head, trunk, tail, and appendages. Although such measures are deeply entrenched in the methodology of systematics, their usefulness in solving real biological problems may be limited. There are far more homologous measures on biological forms than are used in typical multivariate data sets (Humphries et al., 1981), and results of morphometric analyses can depend upon the particular set of measurements chosen. If the selection of distance measures does not correspond by accident or design to the principal directions of shape difference, the resulting descriptions of the differences between forms will be inadequate. There are several biases and weaknesses inherent in traditional character sets such as those in our Figure 1 or in