Biology Reference
In-Depth Information
variables that explains the correlations among them. When the variables are very highly
correlated, as they usually are in ontogenetic studies of size measurements, least squares
regression and PCA tend to give very similar results. For example,
Table 11.2
shows the
estimates of the slope for the measurements of S. gouldingi obtained by regression and
PCA. The numbers may appear to be quite different, but the differences disappear when
the coefficients are rescaled to make each one the ratio between k for a dependent variable
and k for SL, the independent variable in the regression. Because SL is the independent
variable, k
SL
5
1. Rescaling the coefficients by dividing each by the one for SL gives the
values shown in
Table 11.3
. The estimates obtained by multivariate regression and PCA
are identical. The 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 the measured part
relative to that of a standard (X), such as overall body size. When k is 1.0, the growth of
the measured part keeps pace with X. Their proportions are constant throughout growth.
Such measurements are termed “isometric”. When k is greater than 1.0, the measured part
grows more rapidly than X so the relative size of that part increases; these measurements
are termed “positively allometric”. When k is less than 1.0, the measured part grows more
slowly than X, so its relative size decreases even though its absolute size increases; these
measurements are termed “negatively allometric”. The problem with classifying coefficients
solely in terms of ratios between the part-specific growth rate (k) and the growth rate of X
is that all ks are relative growth rates. When several part-specific ks equal each other, their
proportions (relative to each other) are constant. Several measurements have nearly equal
allometric coefficients, 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; v15,
v16, v21 and v22). 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.
To look at a second example, this one of mammalian craniofacial growth, we consider
the ontogenetic allometries of the two rodents mentioned above, the cotton rat (Sigmodon
fulviventer) and house mouse (Mus musculus domesticus). These two species differ strikingly
in life-history; cotton rats are precocial, meaning that they are relatively mature at birth.
Cotton rats open their eyes within a day of birth, and can also hear and even walk at that
time. In contrast, house mice (like most muroid and sciurid rodents) are altricial; their
eyes do not open for 10 days, when their ears also open and they begin to walk. Despite
that striking difference in maturity at birth, both species wean at approximately 21 days.
The ontogenetic allometries for 12 craniofacial measurements, relative to skull length, v1
