Biology Reference
In-Depth Information
From the paired samples, we calculate the angles between the vectors, reiterating
this procedure to generate a distribution of within-group angles. Because sample sizes
can differ for different groups, the two bootstrap sets formed from the group with the
larger sample size match the sample sizes of the two groups (i.e. one of the bootstrap
sets will have a sample size equal to that of the group with more observations and one
will have the sample size equal to that of the group with fewer observations). Both boot-
strap sets formed from the data of the group with the smaller sample size have that
group's smaller sample size because we ought not form bootstrap sets larger than the
original data set. We then determine the statistical significance of the inter-group angle
by comparing it to the 95th percentile of the range of both within-group angles. Should
it be larger, the inter-group difference is judged to be statistically significant at a 5%
level.
We can also use a permutation test to determine whether the difference between the
trajectories is greater than expected by chance ( Adams and Collyer, 2009 ). One approach
is to use the residuals from the reduced model, which includes two factors “species” and
“size” (or “age”) but not the interaction term “species
age”). The rea-
son for using the residuals from the reduced model is to hold constant the relationship
between shape and size (age) within each species. The residuals are then randomly
assigned to the species and the randomized residuals are added to the predicted values.
The full model is then used to calculate the predicted values from the random data.
Repeating that procedure numerous times yields the empirical null distribution. The
p-value is determined by the proportion of values that are as extreme or more extreme
than the observed value.
We might also want to know if two species are no more similar than expected
by chance. To test this null hypothesis, we can randomly permute the coefficients of the
allometric vectors and compute the mean correlation between random vectors and
the 95% and 97.5% confidence intervals. If the observed correlation is higher than 97.5% of
the correlations obtained by randomly permuting the coefficients, then the observed corre-
lation is higher than expected by chance.
size” (or “species
3
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A Traditional Approach to Estimating the Contribution of Scaling Makes to
Morphological Variation
If the null hypothesis of no difference between trajectories is rejected, we might still
want to know whether heterochrony or scaling is the dominant cause of disparity the
two trajectories might not be identical but the difference between them might be slight
and have very little impact on the evolution of morphology. We would also like a method
for assessing the degree to which scaling accounts for the variation in morphology. One
widely used method is to conduct a PCA of the pooled ontogenies series, i.e. a multigroup
PCA ( Shea, 1985 ). PC1 is expected to show ontogenetic scaling, or the shape variation
resulting from extension or truncation of shared allometry and PC2 and subsequent com-
ponents show differences between groups in allometric coefficients as well as differences
due to transpositions. The contribution that ontogenetic scaling (and the differential trun-
cation/extension of it) makes to the overall variation is then quantified by the eigenvalue
of the first component relative to subsequent ones ( Shea, 1985 ).
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