Bivariate versus multivariate allometry A note on a paper by Jungers and German.код для вставкиСкачать
AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 59:321-322(1982) Bivariate Versus Multivariate Allometry: A Note on a Paper by Jungers and German M. HILLS British Museum (Natural History), London S W7 SBD?England Allometry, Bivariate regression, Multivariate KEY WORDS analysis, Principal components ABSTRACT Jungers and German (1981) found differences when they compared 1)coefficients of allometry from bivariate plots of log measurements versus log body weight with 2) those coefficients from the first principal component of the log measurements excluding body weight. It is argued here that an arbitrary choice of unit for “internal size” is all that separates these coefficients. When the unit is chosen to make internal size isometric with body weight the coefficients agree rather well. Jungers and German (1981) compared two sorts of coefficients of allometry: those obtained from separate bivariate analysis of linear measurements versus body weight or some other measure of total body size, and those obtained from a multivariate analysis of all measurements excluding body weight. They concluded, from a study of several sets of data, that the two sorts of coefficients differ, and hence that the coefficients obtained from the first principal component of the multivariate analysis should not be used in place of those relative to actual size. This is, of course, quite correct, but in fact, all that really separates the two sorts of coefficients is the choice of a unit for measuring the multivariate measure of “internal size.”In the absence of any knowledge of body weight relationships, the unit must be chosen arbitrarily, and there is then no reason why the two sorts of coefficients should agree. They are simply not comparable. When body weight relationships are known, however, the unit can be chosen to make the coefficients comparable, and they then agree rather well. Let xl,..., xp refer to a number of linear measurements (on a log scale) and let Z refer to log (body weight) or some other standard measurement of total body size on a log scale. The bivariate analyses are all of the form braically as the linear combination of measurements, m,x,+ ...+mpxp,which has the greatest variability. Geometrically, if the observations in the sample are plotted as points in a p-dimensional space defined by the p measurements, then the vector m=(m,,..., mJ is the direction in which the sample is most variable. This vector is only determined as far as direction is concerned. Its length, defined to be (Em*)%,is arbitrary. Jungers and German argued that since the isometric value of the bi is 1the length of m should be made equal top”. Then, if all the elements of m were equal, their common value would also be 1. The numbers (m,,...,mp)may be interpreted as coefficients of allometry relative to an internal measure of size based on xl,..., 5 but excluding Z. In the example using seven linear measurements from the growth of Cebus albifrons which Jungers and German quote, b=(1.50,1.71,1.07,1.73,1.79,1.26,1.51) m=(0.99,1.17,0.68,1.15,1.19,0.79,0.89), showing that two sorts of allometry coefficient are not comparable. Suppose the linear relationships xi=a,+biZ are good ones, i.e., the points in the scatter plot are close to the straight line, so that we may describe the data by a simple statistical model, xi=ai+biZ x,=a,+ biZ+ei where the blf are the usual (bivariate) coefficients of allometry. In the multivariate where ei is an error to account for the fact that analysis, the first principal component of the covariance matrix of (x, , xp)is defined algeReceived March 15.1982; accepted June 29, 1982 0002-948318215903-0321$01.00 0 1982 ALAN R. LISS, INC. M. HILLS 322 the points do not lie exactly on a straight line. Then the first principal component of the covariances of (%..., 3) will be approximately (bl,..., b,) provided the variability of the errors are small compared to the variability of Z. In other words, if b and m are scaled to have the same length then they should be close. Scaling b from the example to have length 7%gives, internalsizeonZisa(m,b,+ ...+mpbP)sothat (Y should be chosen equal to (Cqbi)-'. For the Cebus example quoted by Jungers and German, Cmibi=10.67 and p=7.0 so mil(pa) becomes 1.52 mi. Scaling m by a factor 1.52 gives, (1.50,1.78,1.03,1.75,1.81,1.20,1.35) (0.97,1.15,0.71,1.14,1.16,0.80,0.97) which is, indeed, very close to m. The same is true in the other examples quoted by Jungers and German and suggests that, in practice as well as theory, the way in which each x,scales with Z provides a good approximation of the direction of the first principal component based on (xi,..., xp)alone. Why then should the data suggest that the first principal component is a very poor indicator of how the measurements scale separately with Z? The answer is that the units for the internal measure of size, relative to which (m,,...,mJ are coefficients of allometry, will affect the way internal size scales with Z. The internal measure of size for a specimen with measurements (X,,..., 5)is measured in the direction of the first principal component, qx,. i.e., it is proportional to m,x,+ Choosing a unit is equivalent to choosing the constant of proportionality (a).The regression coefficient of xi on size defined in this fashion is mJ(pa) so that, for the purposes of this argument, we may think of this as a coefficient of allometrv relative to internal size. Let us now choose a so that internal size scales isometrically with Z. The regression coefficient of ...+ which is now quite close to b. For the other example in Jungers and German (Lemuridae), b=(0.98,0.76,0.87,0.73,0.63,0.68) m=(1.20,0.99,1.09,0.93,0.83,0.88) and mi/(pa) becomes 0.78 mi so that, after scaling, m becomes, (0.94,0.77,0.85,0.73,0.65,0.69) which is again in close agreement with b. I t is only in this very special situation where the values of b and m are to be compared that the choice of a matters. In general, it is sensible to scale m to have length p s , as Jungers and German did. This preserves the value of 1 as the isometric value, and corresponds to measuring internal size as +...+m,x,)/p. (mlxl LITERATURE CITED Jungers, WL, and German RZ (1981)Ontogenetic and interspecific skeletal allometry in non-human primates: bivariate versus multivariate analysis. Am. J. Phys. AnthroDol. 55:195-202.