Biology Reference
In-Depth Information
ber of observations), or on biological grounds (e.g., the sample hypoth-
esized as more primitive in evolutionary studies or the unaffected
sample in studies of pathological specimens). Although all four data
sets are used to calculate
G
obs
, only the base population is used to pro-
duce the bootstrapped samples.
In testing
G
obs
, we start with four samples
(A
1
, A
2
, B
1
, B
2
)
of size
m
,
n
,
o
, and
p
, respectively. A landmark coordinate matrix that is rewrit-
ten as a form matrix for analysis represents each individual in these
samples. If population
A
is chosen as the base population, using simple
random sampling with replacement, we obtain a sample of size
m
from
A
1
and of size
n
from
A
2
. These bootstrapped samples are designated
A
1
*
and
A
2
*
. Similarly, we obtain simple random samples with replacement
of size
o
from
A
1
and of size
p
from
A
2
. These samples are designated
B
1
*
and
B
2
*
. Next, we calculate a growth difference matrix for the boot-
strapped samples:
and write the
GDM
as a vector whose elements are sorted from mini-
mum value to maximum value. These steps are repeated
B
times where
B
is sufficiently large (100
≤
≤
1000). In matrix form, this collection
of vectors formed by the analysis of
B
bootstrapped samples has
K
(
K
-
1)/2 rows and
B
columns, each column being a sorted (in ascending
order) growth difference matrix for a set of four bootstrapped samples.
The statistic
G
for each of these bootstrapped samples is defined as the
maximum ratio in the
GDM
divided by the minimum ratio in that
GDM
.
G
is calculated for each column and represents a summary mea-
sure of similarity in shape change due to growth for each bootstrapped
set of samples. The placement of
G
obs
calculated from the original data
in relation to the distribution of the bootstrapped
G
values provides a
means for determining if growth is similar in the two groups being
compared. One may also calculate the probability of getting the
observed (or more extreme) value of the test statistic,
G
obs
, and report
it as a
p
-value. Our goal is to determine the probability of obtaining the
observed (or a larger) maximum-to-minimum ratio value due to under-
lying biological variability when the growth of the two groups is, in
fact, similar. If this probability, the
p-
value, is small, we claim that the
observed value is unlikely under the null hypothesis of similarity in
growth and reject the null hypothesis. By the same results, we enter-
tain the alternative hypothesis.
B
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