Biology Reference

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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.

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