The Study of Growth - An Invariant Approach to Statistical Analysis of Shapes - page 241

tion, reflection, and translation. Keep in mind that to obtain a graphi-
cal representation that is biologically meaningful, the coordinate
matrix may need to be reflected. This can be done by multiplying one
or more of the axes by -1.
It is important to remember that transformation matrices can be
incompatible with form matrices. When this occurs, the hypothetical
form produced is of a different dimension than the starting form. To
avoid such errors, it is important to check for the dimensionality of the
hypothetical form. Checking the dimensionality of a form is a neces-
sary step in the production of hypothetical forms (Richtsmeier and
Lele, 1993). The first thing that should be done is to check that the
eigenvalues are all positive. There should be no large negative eigen-
values. If there are small, negative eigenvalues, the researcher must
choose whether or not to ignore them. If there are large negative eigen-
values, then the growth matrix and the form matrix are incompatible
and the geometry of the hypothetical form cannot be resolved. If the
majority of the eigenvalues are non-negative, then the dimension of
the hypothetical form must be determined.
To determine the dimensionality of a hypothetical form, we suggest
adding the first D eigenvalues (where D = the number of dimensions of
the starting form) to see if these eigenvalues constitute a given per-
centage of the sum of the K eigenvalues. We suggest the following
criterion be used in determining the dimensionality of a hypothetical
form. Let
Choose that value of D that makes P D larger than 0.95. The 95% cutoff
is arbitrary, but sensible. A more or less restricted dimensionality
check may be required depending upon the purpose of the experiment.
If we expect a two-dimensional hypothetical form, then the sum of the
first two eigenvalues should exceed 95% of the sum of all eigenvalues.
If we expect a three-dimensional hypothetical form, then the sum of
the first three eigenvalues (but not the first two) should exceed 95% of
the sum of all eigenvalues. The first 6 eigenvalues and their associat-
ed percentages for our hypothetical form are given below. The
remaining eigenvalues are 0.0.