Multivariate Discrimination by Shape in Relation to Size

Abstract

Humphries, J. M., F. L. Bookstein, B. Chernoff, G. R. Smith, R. L. Elder, and S. G. Poss (Museum of Zoology, Centerfor Human Growth and Development, Museum of Paleontology, and Division of Biological Sciences, The University of Michigan, Ann Arbor, Michigan 48109) 1981. Multivariate discrimination by shape in relation to size. Syst. Zool., 30:291-308.-The diverse methods for analyzing size-free shape differences tend to be guided by computational expediency rather than geometric principles. We question the use of ratios and ad hoc combinations of spatially unrelated measures. Neither are linear discriminant functions or series of independent regressions helpful to the visualization of shape differences. A bridge is needed between traditional quantitative methods and the geometrical analysis of shape. In principle any measured transects between landmarks of a form can serve as characters in a morphometric analysis. Systematic studies use a highly non-random sample of these, particularly biased regarding geometrical information. We suggest defining size and shape in terms of factors-estimates of common to a universe of measured distances. The model presented here calculates a linear combination of variables that quantifies shape differences among populations, independent of size. In analyses in which the first two principal components confound size and shape, size is removed from one axis with shear coefficients derived from regression of general size on principal components centered by group. The general size factor is estimated by the principal axis of the within-group covariance matrix of the log-transformed data. Residuals from the regression of general size on the transformed axes approximate a shape-discriminating factor that is uncorrelated with size within group and displays the interpopulation shape differences borne by the first two principal components. The results bear a direct and interpretable correspondence to biorthogonal analysis of shape difference. [Multivariate analysis; principal components; discriminant functions; morphometrics; size-free shape; allometry; fishes.] Systematists need procedures that allow them to discriminate among groups of organisms that vary in size. The groups included in a study can be chosen a priori (e.g., several species or geographic populations within a species) or a posteriori (as a conclusion resulting from some method of analysis). However the groups are chosen, it has long been considered desirable to discriminate among them on the basis of size-free shape derived from distance measures. The terms shape and size have been used in various and sometimes conflicting ways (Huxley, 1932; Thompson, 1942; Simpson, Roe and Lewontin, 1960; Gould, 1966; Mosimann, 1970; Sprent, 1972; Bookstein, 1978). We construe size and shape not as measured variables, but as general factors, linear combinations most parsimoniously accounting for the associations among the distance measures. Size, in particular, is not a single variable such as biomass or a standard length, but a factor which, when called upon to predict all the distance measures within a population, leaves the smallest mean squared residual. We prefer a factor whose algebraic form acknowledges the allometric relationship (Jolicoeur, 1963). Our shape discriminators need to be independent of size (Flessa and Bray, 1977; Mosimann and James, 1979) in order to partition out the effects of growth (e.g., individuals of differing age and size). In general, shape can be defined as the geometry of the organism after information about position, scale, and orientation has been removed (Bookstein, 1978:8). There is then an endless variety of shape remaining. While the quantification of size as a general factor de-