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Bivariate versus multivariate allometry A note on a paper by Jungers and German.

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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.
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