Biology Reference
In-Depth Information
STEP 2: Generate a new set of samples X 1 * , X 2 * ,…, X n *
with
replacement from the first sample X 1 , X 2 ,…, X n
and
Y 1 * , Y 2 * ,…, Y m *
with replacement from the second sample
Y 1 , Y 2 ,…, Y m .
STEP 3: Calculate the form difference matrix based on these
bootstrap 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 procedures
The difference between the Bootstrap procedures described above and
the Monte Carlo procedures described below is that in the Monte Carlo
procedure, instead of obtaining Bootstrap samples using simple ran-
dom sampling with replacement from the original sample, we generate
Bootstrap samples using the Gaussian model and the estimated mean
form and the variance-covariance matrix. This procedure is valid pro-
vided the Gaussian perturbation model is a reasonable model. This
procedure may be used when the sample sizes are small, say 10-15
individuals in each group.
Monte Carlo confidence intervals for the form difference
matrix
STEP 1: Calculate the form difference matrix using the esti-
mated mean forms for the two populations.
STEP 2: 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
2
K
,
2
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