Biology Reference
In-Depth Information
STEP 4. Use the parametric samples generated in Steps 2
and 3 to calculate a new form difference matrix for these sam-
ples,
FDM
(
B*, A*
)
.
STEP 5. Repeat Steps 2 through 4 for
C
times, where
C
is suf-
ficiently large (i.e., from 200 to 1000).
Each
FDM
(
B*, A*
)
calculated from the Monte Carlo samples is writ-
ten as a vector with
K(K-1)
/2
entries. A collection of Monte Carlo
samples compared in this way and written as a matrix has
K(K-1)
/2
rows and
C
columns, each column being sorted according to the land-
marks that define the linear distance. Each column is a form difference
matrix in vector format obtained at the end of Step 4 and each row rep-
resents
C
form difference ratios for a linear distance between a
specified pair of landmarks. Sort the values in each row in ascending
order. The 100(1-
α
)% confidence interval is delimited by removing the
first
α
/2 % and the last
α
/2 % of the sorted entries. If the interval of
entries spanning the lower and upper confidence limits contains the
value 1, then it is likely that the particular distance is not different in
the two populations. As with the Bootstrap approach, this interval also
provides a suggestion of the range of values that a particular ratio
might take. Thus, if the confidence interval is say
(0.85,0.95)
we can
conclude that the linear distance is smaller in sample
B
relative to
sample
A
, and that it is likely that the linear distance is 5 to 15%
smaller in sample
B
. Once a confidence interval is obtained for each
linear distance, similar statements can be made separately using data
from each row.
Notice that the only difference between the Bootstrap procedure
and the Monte Carlo procedure is the way in which the samples in Step
2 are generated. The Bootstrap approach accomplishes this by using
simple random sampling directly from the data, while the Monte Carlo
procedure uses the Gaussian model to generate data from the estimat-
ed parameters.
An additional nuance of both the Bootstrap and Monte Carlo pro-
cedures is that the exact output of running either procedure will vary
even if the input data remain the same. Suppose you run the Bootstrap
program for obtaining confidence intervals at home on Monday eve-
ning. You leave the printout at home and have to re-run the Bootstrap
confidence interval routine again the following morning at work. If you
were to compare the two outputs that have used the exact same input
data, they will be slightly different. The reason for this is that the sam-
ples obtained in Step 2 of both the Bootstrap and Monte Carlo
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