MULTIVARIATE ALLOMETRY AND AUSTRALOPITHECINE VARIATION.

Allometry is differential relative growth, or simply a change in shape with increasing size. The concept of allometry is invoked by some students to explain the variation among early hominid fossils which others attribute to taxonomic variation. Brace (1963) and most recently Wolpoff (1973a) explain morphological shape differences between the "robust" and "gracile" forms of australopithecines as a result of the greater body size of the former. Brace (1963) and Brace and Montagu (1965) state that the supposedly more advanced gracile form would actually appear more primitive if allometrically "blown up" to the size of a large pongid. These arguments have not involved any empirical quantitative investigations of allometry, although there is a steadily increasing battery of numerical methods designed to detect and describe allometric relationships. Attention in relative growth studies traditionally centers on Huxley's (1932) power formula y =ax" where the allometry coefficient k describes the differential relation between increase in variates y and x. As it is based on bivariate logarithmic regression, Huxley's model is limited to two dimensions. This is an inefficient approach if many variables are simultaneously under consideration. As one solution to this limitation, Jolicoeur (1963a) recommends computing the prin-cipal components of a logarithmic covariance matrix for all the variables. The first (major) axis for principal components computed from linear measurements almost always represents a vector or path along which larger specimens have grown more in all variables than smaller specimens have. In the logarithmic state, the directional cosines of the major component are independent of order of magnitude in individual measurements and therefore indicate relative growth rates. Jolicoeur (1963b) proves that the ratio of directional cosines for any two variables is proportional to the k computed from the bivariate allometry equation. If proportions remain constant over different sized cases in a sample, all angles described by the major axis from the origin will be equal. This multivariate generalization of the allometry equation "is the most efficient way to describe complex shape changes" (Jolicoeur, 1963b). A statistical test of the hypothesis that a given vector is that described by the major axis of a random sample is provided by Anderson (1963). A hypothesis of multivariate isometric growth, where all dimensions increase at the same rate, can be formulated as equality of all p directional cosines on the major axis to a value of (p)-Y2. Hopkins (1966) shows that values for the p-variate analogue of the allometry equation are proportional to k if (and only if) the logarithmic covariance matrix is of rank one. Otherwise the major axis cannot adequately describe overall relative growth since there are other, independent component axes that must be considered meaningful. Hopkins advocates use of rotated component axes (factor analysis) to deal with unequal lesser roots and give a better estimate of the actual parameters of the functional relation. Sprent (1972) indicates that Hopkins' factor analytic approach does not solve the problem, for if the sample logarithmic covariance matrix is not of rank one, the multidimensional data lie not on a line but on a plane or in many dimensions. Different factors can be derived from