Biology Reference
In-Depth Information
(
McKinney, 1988; McKinney and McNamara, 1991
). Skipping that translation step rede-
fines all the terms. For example, “neoteny” no longer means a decrease in developmental
rate. Instead it means a decrease in growth rate of any measurement regressed on size
(
McKinney, 1986, 1988; McKinney and McNamara, 1991
). To be consistent with Gould's
and Alberch et al.'s definition of the term (as well as with much of the preceding litera-
ture), the hypothesis of neoteny must be translated as predicting that positively allometric
coefficients will decrease but negatively allometric coefficients will increase. If this seems
counterintuitive, consider what it means to be pedomorphic (i.e. “childlike”). A pedo-
morphic descendant resembles the ancestral juvenile. The extreme case would be an adult
that does not depart at all from the juvenile shape as it grows. For that to be the case,
growth must be isometric (shape does not change over ontogeny). In less extreme cases,
the descendant's ontogeny is more nearly isometric than the ancestor's. Based on that rea-
soning, positively allometric coefficients will decrease in slope, in the direction of isometry,
and negatively allometric coefficients will increase in slope, also in the direction of isome-
try. Positively and negatively allometric coefficients approach isometry from opposing
directions. Of course, the coefficients must all change by the appropriate amount, not just
in the appropriate direction.
Considering that pedomorphosis results from truncating the ancestral ontogeny, and
peramorphosis from extending it, we would anticipate that the vectors of allometric coeffi-
cients would point in the same direction
the two vectors differ only by an extension or
truncation, and thus in length, not in direction. From geometry, it should be obvious that
when the regression vectors actually are the same line, they point in the same direction
and therefore the angle between them is 0
. Because the correlation between the two vec-
tors is the cosine of the angle, many studies have measured the correlation between the
vectors to determine if they differ by much. They often are very highly correlated, for
example, in the case of sigmodontine rodents, the correlation between the ontogenetic tra-
jectories of post-weaning trajectories are, on average 0.981 in comparisons between conge-
neric species and 0.962 in comparisons between genera, corresponding to angles of 11.2
and 15.84
(
Voss and Marcus, 1992
). Similarly, comparisons between pygmy chimpanzee
(Pan paniscus), common chimpanzee (Pan troglodytes) and the gorilla (Gorilla gorilla) yield
correlations that range from 0.964 to 0.977, corresponding to angles from 12.31
to 15.42
.
When angles are that small, testing them for a significant deviation from 0
may not seem
necessary, and they typically were not tested. However, it is worth examining even such
obviously interpretable results with a jaundiced eye because explicit statistical testing can
yield surprising outcomes.
Calculating the Angle Between Two Vectors
The angle between any two vectors A and B, each with P components, may be com-
puted by taking the dot product (also called the “inner product”) of the two vectors. The
dot product is calculated by multiplying the corresponding components of the two vectors
together, then summing those products. For example, if we have two vectors, A and B,
with A
[A
1
, A
2
, A
3
,...A
P
] and B
[B
1
, B
2
, B
3
,...B
P
], the dot product is:
5
5
A
B
A
1
B
1
1
A
2
B
2
1
A
3
B
3
1 ...
A
P
B
P
(11.5)
5
