Biology Reference
In-Depth Information
Lele and Richtsmeier (1995) introduced the procedure for obtain-
ing confidence intervals for elements of the form difference matrix.
There are two different approaches available for the computation of
confidence intervals for the form difference matrix. The first approach,
called the Monte Carlo approach, is model-based and assumes that the
underlying perturbation model is Gaussian. In addition to assuming
Gaussian perturbation, the Monte Carlo approach makes direct use of
the parameters estimated from the samples under study (i.e., mean,
variance-covariance matrix). The second approach, called the
Bootstrap approach, is less model-based as dependence on the assump-
tion of the underlying Gaussian perturbation model is relaxed (Efron
and Tibshirani, 1991). The Monte Carlo approach is preferable if one
has reasonable confidence that the data being analyzed follow the
underlying model as Monte Carlo confidence intervals can be sensitive
to deviations from the underlying model (Huber, 1972). For this reason,
if evidence suggests that the data do not follow the model, the
Bootstrap approach might be more suitable. However, if only a small
number of observations are available, the Bootstrap approach may
prove problematic and it may be necessary to adopt the Monte Carlo
approach. Both the Monte Carlo and Bootstrap methods for obtaining
confidence intervals are described below. The computational algo-
rithms for these approaches are presented in
Part 2
of this chapter.
4.8.2 The Bootstrap method for obtaining confidence intervals
The following steps briefly describe the Bootstrap procedure. A proper
computational algorithm is provided in
Part 2
for any reader who wish-
es to program this procedure.
Let
A
1
,
A
2
,
A
3
,...,
A
n
and
B
1
,
B
2
,
B
3
,...,
B
m
be the landmark coordinate
matrices for individuals from samples representing populations
A
and
B
. Suppose we have obtained a sample of size
n
from population
A
and
a sample of size
m
from population
B
.
STEP 1. Obtain a simple random sample with replacement of
size
n
from sample
A
1
,
A
2
,...,
A
n
and of size
m
from sample
B
1
,
B
2
,...,
B
m
. Let us refer to the samples from the first population
as
A*
1
,
A*
2
,...
A*
n
and those from the second population as
B*
1
,
B*
2
,...
B*
m
.
STEP 2. Calculate the mean form difference matrix,
FDM
(
B*
,
A*
)
for the samples obtained in Step 1 using the
equations given previously.
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