Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 159

STEP 1. Estimate the mean form matrices for sample A and
for sample B .
STEP 2. Calculate the form difference matrix for these sam-
ples, FDM ( B, A ) and sort the vector in ascending order.
STEP 3. Calculate the test statistic, T , which is the ratio of the
maximum ratio in the matrix FDM ( B, A ) to the minimum
entry in FDM ( B, A ) . Using our observations from samples A

and B , .

The next step constitutes a Bootstrap approach to estimate the null
distribution of the test statistic, T . To do this you must first choose one
of the samples, say B , as your baseline sample.

STEP 4. Select n individuals randomly and with replacement
from sample B , and call this sample A 1 . Then, select m indi-
viduals randomly and with replacement from sample B , and
call this sample B 1 . Follow steps 1, 2, and 3 using A 1 and B 1 to
obtain a T value for the bootstrapped sample.
STEP 5. Repeat Step 4 an adequate number of times (e.g., 200
to 1000 times).
STEP 6. Use the distribution of T values produced by Steps 4
and 5 in the form of a histogram as the null distribution of T
when the null hypothesis is true. If T obs falls in the upper α %
tail of the null distribution, we reject the null hypothesis at
the

% level of significance.

Notice that Step 4 requires the choice of a baseline sample. This
choice is required because the test that we have designed does not sim-
ply test whether or not the two mean forms are similar. Instead, we
test whether or not the average form of the first sample is similar to
the average form of the baseline group. Our test is designed to answer
the specific question, is the mean form of Group A similar to the mean
form of Group B ? Importantly, the answer to this question may not be
the same as the answer to the question, is the mean form of Group B
similar to the mean form of Group A ? Our test is a one-way test and
due to the design of the test, one group (generally the group with the
larger sample size) must be chosen to serve as the baseline group.
The question that this test addresses is similar to the question,
could sample A have arisen from the distribution of the population
from which sample B was obtained? This subtle point that pertains to
this testing procedure means that if one wants to know for certain if