Biology Reference
In-Depth Information
FIGURE 11.6
Landmarks sampled on S. gouldingi,
and the traditional morphometric measurement scheme
based on those landmarks.
The relationship between X and Y often fits a model, the power law (
Huxley, 1932
):
bX
k
Y
(11.1)
5
where k is the growth rate of part Y relative to X, and b is the size of Y when X is at unit
size. To ease fitting the model to data, it is often rewritten in a linear form:
log
ð
Y
Þ
5
log
ð
b
Þ
1
k log
ð
X
Þ
(11.2)
Expressed in this form, we can use linear regression to estimate the parameters b and k;
they are the intercept (because log(1)
0) and slope, respectively, of a linear regression of
log(Y) on log(X).
Table 11.1
gives the regression coefficients, b and k, of the variables
shown in
Figure 11.6
regressed on SL. We should note that the literature is inconsistent on
the symbols used for these two coefficients, but because b and k are widely used in the
literature on allometry, we follow that convention.
Usually, the coefficients are estimated by simple bivariate regression, but multivariate
regression yields the same estimates as obtained from bivariate analysis. We can therefore
treat the bivariate estimates of k as components of the vector {k
1
, k
2
, k
3
,
5
k
P
}, where P is
the number of measurements. The estimates of log(b) are then components of the vector
{log(b
1
), log(b
2
), log(b
3
),
...
log(b
P
)}. In studies of traditional size measurements, allometric
coefficients are often estimated by principal components analysis (PCA), following
Jolicoeur (1963)
who first proposed that PC1 is a multivariate allometry vector when PC1
is extracted from a variance
covariance matrix of log-transformed measurements.
Conceptually, multivariate regression and PCA differ in that PCA does not single out one
variable as the independent size variable. Instead, size is a linear combination of the
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