chap-13 4/6/2004 17: 28 page 330
330 GEOMETRIC MORPHOMETRICS FOR BIOLOGISTS
are identical. Perhaps the most important distinction between regression and PCA is that
PCA does not provide estimates of b.
Interpreting allometric coefficients
The interpretation of k is straightforward – it is the growth rate of one measurement
relative to that of a standard, the growth rate of X. When k is 1.0, the growth of the part
keeps pace with that of X, which we will take as the whole body – i.e. their proportions
are constant throughout growth. Such measurements are termed “isometric.” When k
is greater than 1.0, the part increases its size relative to overall body size; these parts
are termed “positively allometric.” When k is less than 1.0, the part decreases in its size
relative to body size; these measurements are termed “negatively allometric.” Only one
measurement in Table 13.1 is isometric, but many coefficients are equal to each other, so
we could view them as isometric relative to each other. We are not constrained to think of
k
i
solely in terms of growth rates of each part relative to body size – all ratios among the k
s
are relative growth rates as well. Several measurements are isometric relative to each other,
including the four measurements of body depth (measured from landmarks 4 and 5, which
are at the anterior and posterior bases of the dorsal fin). These four (v15, v16, v21 and
v22) grow at equal rates relative to each other, so relative to each other their proportions
do not change over ontogeny. All four are positively allometric relative to body length, so
the body (in that region) deepens relative to its length. Among the negatively allometric
measurements are the most anterior lengths (v2, v3, v4, v8, v28, v29, v30) and the two
most posterior ones (v24, v27). This means that measurements in the anterior head and
caudal regions shorten relative to the whole body (of course they do not actually shorten –
they lengthen in an absolute sense, it is just that they shorten relative to the length of the
body). Consequently, the head and caudal region form a relatively smaller fraction of body
length in adults than in juveniles.
The interpretation of b is less straightforward, and there has been some controversy
about its biological meaning. One reason for doubting that b has any general biological
significance is that its value depends on the units of measurement; unlike k, b is not a
dimensionless quantity. However, a more important one is that log(b) is the value of
log(Y) when log(X) is zero, a size at which Y might not yet exist. For example, when
the body is 1 mm long, the dorsal fin might not have developed yet so it cannot have a
meaningful size. Additionally, log(b) is estimated under the assumption that k is constant
from log(X) =0, not just that it is constant over the range of values actually sampled.
Under one condition, b does have a simple interpretation. When populations do not
differ in k, a difference in b does have a meaning because the difference in b will persist
throughout the entire ontogeny. Even if we hesitate to infer a value for log(Y) when
log(X) =0, under the condition that both species have the same value of k, the regression
lines would differ only in elevation. So at any point in ontogeny, they will differ in elevation
according to their difference in b. We might reasonably hesitate to claim that they have
diverged by the time that log(X) =0, but we could claim that the difference between the
species arose prior to the stage when we first observe them, and it is invariant throughout
the rest of ontogeny. That difference in b says how those populations will differ at any
given value of X. For example, if we are comparing the brains of several species that do
not differ in k, we can determine their relative brain size at all body sizes by comparing