Biology Reference
In-Depth Information
Bending energy (1) A measure of the amount of non-uniform shape difference based on the thin-plate spline met-
aphor. In this metaphor, bending energy is the amount of energy required to bend an ideal, infinite and infinitely
thin steel plate by a given amplitude between chosen points. Applying this concept to the deformation of a two-
dimensional configuration of landmarks involves modeling the displacements of landmarks in the
X
,
Y
plane as
if they were displacements above or below the plane (
Z
). (2) Eigenvalues of the bending-energy matrix, repre-
senting the amount of bending energy per unit deformation along a single principal warp (eigenvector of the
bending-energy matrix). This concept of bending energy is useful because it provides a measure of spatial scale; it
takes more energy to bend the plate by a given amount between closely spaced landmarks than between more
distantly spaced landmarks. Thus, principal warps with large eigenvalues represent more localized components
of deformation than principal warps with smaller eigenvalues. The total bending energy (definition 1) of an
observed deformation is a sum of multiples of the eigenvalues, and accounts for the non-uniform deformation of
the reference shape into the target shape. See also Thin-plate spline, Principal warps, Partial warps (Chapter 5).
Bending-energy matrix The matrix used to compute principal warps and their bending energies (eigenvectors
and eigenvalues, respectively). This matrix is a function of the distances between landmarks in the reference
shape. See also Principal warps, Partial warps (Chapter 5).
Between Groups Principal Components Analysis A method for reducing the dimensionality of multivariate
data, performed by extracting the eigenvectors of the variance
6
covariance matrix of the group means. All indivi-
duals in the samples are then scored on these axes; as in ordinary PCA, the scores of individuals are calculated
by taking the dot product between that principal component and the data for that specimen (Chapter 6).
Biorthogonal directions Principal axes of a deformation; the term was used in Bookstein et al., 1985; more
recently, workers refer to principal axes (Chapter 3).
Black Book Marcus,L.F.,Bello,E.andGarcia-Valdcasas,A.(eds)(1993).
Contributions to Morphometrics
.Madrid,Mono-
grafias del Museo Nacional de Ciencias Naturales 8. See also Blue Book, Orange Book, Red Book and White Book.
Blue Book Rohlf, F.J. and Bookstein, F.L. (eds) (1990).
Proceedings of the Michigan Morphometrics Workshop
.
University of Michigan Museum of Zoology, Special Publication No. 2. See also Black Book, Orange Book, Red
Book and White Book.
Bonferroni correction, Bonferroni adjustment An adjustment of the
-value to protect against inflating Type I
error rate when testing multiple
a posteriori
hypotheses. The adjustment is done by dividing the acceptable Type I
error rate (
α
-value for each of the
a posteriori
hypotheses.
For example, if the desired Type I error rate is 5%, and there are 10
a posteriori
hypotheses to test, 0.05/10
α
) by the number of tests. That quotient is the adjusted
α
0.005
5
is the
-value for each of those 10 tests. A less conservative approach is a sequential Bonferroni adjustment in
which the desired
α
for the first test
would be 0.05/10; for the second it would be 0.05/9; for the third it would be 0.05/8, etc. To apply this sequential
adjustment, hypotheses are ordered from lowest to highest p-value; the null hypothesis is rejected for each in
turn until reaching one that cannot be rejected (the analysis stops at that point).
Bookstein coordinates (BC) The shape variables produced by the two-point registration, in which the configura-
tion is translated to fix one end of the baseline at (0, 0), and then rescaled and rigidly rotated to fix the other end
of the baseline at (1, 0). See also Baseline registration (Chapter 3).
Bookstein two-point registration (BTR) See Two-point registration, Bookstein coordinates.
Bootstrap test A statistical test based on random resampling (with replacement) of the data. Usually, the method
is used to simulate the null model that one wishes to test. For example, if using a bootstrap test of the difference
between means, the null hypothesis of no difference is simulated. Bootstrap tests are used when the data are
expected to violate distributional assumptions of conventional analytic statistical tests. Rather than assuming that
the data meet the distributional assumptions, bootstrapping produces an empirical distribution that can be used
either for hypothesis testing or for generating confidence intervals. See also Jackknife test, Permutation test
(Chapters 8, 9).
Canonical variates analysis (CVA) A method for finding the axes along which groups are best discriminated.
These axes (canonical variates) maximize the between-group variance relative to the within-group variance.
Scores for individuals along these axes can be used to assign specimens (including unknowns) to the groups, and
can be plotted to depict the distribution of specimens along the axes. CVA is an ordination rather than statistical
method. See also Ordination methods, Principal components analysis (Chapter 6).
α
-value is divided by the number of remaining tests. Thus, the adjusted
α
