Simple Size and Shape Variables: Shape Coordinates - Geometric Morphometrics for Biologists - page 59

coordinates of the free coordinates involves (clockwise) rotations around the X , Y and Z
axes (through angles

φ ω θ

, , and , respectively). The three rotation matrices are:

2
4

3
5

1 0 0

R x φ φ

5 0 cos sin

0 sin φ cos φ

2

2
4

3
5

ω ω

cos 0 sin

R y 0 1 0 (3.2)

5

sin ω 0 cos ω

2

2
4

3
5

θ θ

cos sin 0

R z sin θ cos θ 0

5 2

0 0 1

So, designating the translated and scaled coordinates by A ts ,B ts ,C ts , the three-dimensional
shape coordinates are R x R y R z (A ts ,B ts ,C ts ) T (for a more detailed presentation of the calculation
of three-dimensional Bookstein shape coordinates see Dryden and Mardia, 1998; Claude, 2008 ).

STATISTICS OF SHAPE COORDINATES

Now that we have shape coordinates, we can answer the basic “existential” questions
as defined in Chapter 1, such as “do these samples differ in shape?” All conventional
statistical methods and tests can be applied to shape coordinates and centroid size. For
example, an average value for the shape coordinate at point C is computed by averaging
the X -coordinates for that point across all individuals within a sample, then dividing that
sum by the total number of individuals in that sample and applying that same procedure
to the Y -coordinates. Variances and standard deviations are also calculated by standard
formulae. Because the two endpoints of the baseline are fixed, they have no variance and
should not be included in statistical analyses. If you use conventional statistical packages
to analyze these coordinates, remember to exclude them from the analysis because many
programs will not run if the variables do not vary.
Because every landmark has two dimensions (its X-, and Y -coordinates), statistical anal-
yses are necessarily multivariate. Even if we are asking whether two samples of triangles
differ in average shape, we must use a multivariate test. In particular, we would use the
multivariate form of the familiar Student's t -test, Hotelling's T 2 test (see, for example,
Morrison, 1990 ). When comparing two samples of triangles, the test is applied to the two
coordinates of landmark C. When we are comparing more than two samples, we can use
Wilks'

( Rao, 1973 ) or one of the related statistics obtained by a multivariate analysis of
variance (MANOVA). In studies of allometry, we use multivariate regression. However,
an important consideration that needs to be taken into account when applying statistical

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