Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 211

STEP 3: Calculate the form difference matrix based on these
Monte Carlo samples.
STEP 4: Repeat Steps 2 and 3 B number of times.

The Bootstrap sample based form difference matrices obtained in
this fashion can be collected in a matrix with K ( K

1)/2 rows and B
columns. In this matrix each column is a form difference matrix
obtained at the end of Step 3 and each row represents B form differ-
ence ratios for a linear distance between a specified pair of landmarks.
To obtain a confidence interval for each linear distance, the ratios
in each row are sorted in an increasing order. If a 90% confidence inter-
val is sought, then the first 5% and the last 5% of the total entries in
this sorted row are deleted. The minimum and the maximum entries
remaining in that row constitute the lower and upper confidence lim-
its for that particular linear distance. This is done separately for each
row to obtain a confidence interval for each linear distance ratio.

Monte Carlo-based EDMA-I testing procedure

STEP 1: Calculate the form difference matrix using the esti-
mated mean forms for the two populations. Calculate

ˆ ˆ

max FDM ( M , M )

ij ij 1 2

T .

ˆ ˆ

obs

min FDM ( M , M )

ij ij 1 2

STEP 2: Assuming that the first population is the base sam-
ple, generate a new set of samples X 1 * , X 2 * ,…, X n * from

ˆ

, ˆ

ˆ

, ˆ

NM ( ,) I

and Y 1 * , Y 2 * ,…, Y m * from NM ( ,) I .

1 K , 1

1 K , 1

STEP 3: Calculate

STEP 4: Repeat Steps 2 and 3 B times to obtain T 1 * , T 2 * ,…, T B * .
STEP 5: Arrange T 1 * , T 2 * ,…, T B * in an increasing order. Reject
the null hypothesis of equality of shapes if T obs is in the top α -th
percentile where α is the size of the test.