Biology Reference
In-Depth Information
and the standard deviation of a normal distribution fitted to the data. He uses normal the-
ory (citing
Feller, 1968
) to predict the mean and standard deviation:
ðZ
YÞ
2
X
mean
Y
(10.9)
5
1
2
2
ðZ
YÞ
2
SD
X
5
(10.10)
12
and he uses those to estimate the range parameters:
3
1
Y
X
mean
2
SD
X
(10.11)
5
3
1
Z
X
mean
2
SD
X
(10.12)
5
1
Extending nearest-neighbor analysis to geometric data is straightforward. Distances
D
i
and
R
i
are measured by Procrustes distance; estimates of means, standard deviations or
ranges used in the Monte Carlo simulation are obtained by calculating the statistics from
the coordinates of each landmark. The rest is straightforward: a Monte Carlo data set is
generated and
R
i
is calculated for each specimen, and these are used to estimate
P
mean
. The
simulation is reiterated numerous times, yielding the distribution of
P
mean
values over the
Monte Carlo sets. It is then possible to carry out all the usual statistical tests using this
distribution.
We illustrate nearest-neighbor analysis by testing two hypotheses about piranha
disparity:
1.
Piranha body shapes, both juvenile and adult, are further apart than expected.
2.
Those shapes are more clumped than expected.
The reason for testing these hypotheses separately is that a conservative test of one is a
liberal test of the other. For the hypothesis of over-dispersion, the conservative approach
uses the Strauss and Sadler estimator of the range
the estimator enlarges the range
so that large distances between points will not necessarily be further apart than expected.
However, that expansion of the range can lead to a liberal test of clumping (under-
dispersion) because, within that expanded range, observations may be closer than
expected. To be conservative, we would test the hypothesis of over-dispersion using the
enlarged range, but we would use parameters of the observed range to test a hypothesis of
clustering. Each hypothesis will be tested using two null models, one uniform and the other
Gaussian, because we have no good reason to view one as a more plausible random model.
Using the uniform model, the average
P
mean
of the juveniles is
0.2810 and the 95%
2
range of
P
mean
is from
0.1792, an interval that excludes zero. This result sug-
gests a non-random distribution, with distances being smaller than expected under a ran-
dom uniform model. Using the Gaussian model, the average
P
mean
52
0.3551 to
2
2
0.2758 and its range
is from
0.1950, an interval that again excludes zero. Both results thus argue
against the hypothesis of a random distribution and also against over-dispersion. Instead,
they suggest clustering, the hypothesis we will explicitly test after we have tested the
hypothesis of over-dispersion for adults.
2
0.3450 to
2
