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In-Depth Information

5.7 Statistical analysis of form and shape difference due

to growth

As discussed in previous sections, although statistical testing is a stan-

dard part of any quantitative analysis of morphology, it should not be

considered the definitive answer to questions concerning similarity of

growth patterns. If
GDM (A
2
,A
1
: B
2
,B
1
)
does not equal 1, patterns of dif-

ferences from 1 should be sought by close examination of the
GDM
.

This process is labor intensive but graphic aids are available (some of

which are discussed in
Chapter 4
; also see Cole and Richtsmeier, 1998).

To test for the statistical significance of differences in shape change

due to growth we use the statistic
G
obs
:

where max (= maximum) is the ratio with the largest value within the

GDM
, and min (= minimum) is the ratio with the smallest value. This

test statistic is scale invariant thereby eliminating the effects of scal-

ing. Growth patterns are interpreted as being more similar in shape

change as
G
obs
approaches 1. A null hypothesis of similarity in shape

change due to growth is stated as:

H
o
:GM
ij
(
A
2
,A
1
) = c
GM
ij
(
B
2
, B
1
)

where
c
is some scaling factor and
c >
0. This is a one-way hypothesis

that states that the growth of sample
B
is similar to the growth of sam-

ple A. If accepted, it does not imply that the growth of sample
A
is

similar to the growth of sample
B
. This would have to be tested by

restating the null hypothesis as:

H
o
:GM
ij
(
B
2
,B
1
) = c
GM
ij
(
A
2
, A
1
)

and rerunning the analysis using the other sample as the base population.

The one-way nature of this hypothesis demands a specific design

for the statistical testing of
G
obs
. Our design uses an extension of the

bootstrap approach presented in
Chapter 4
. The statistical comparison

of growth patterns from two populations using a one-way hypothesis

requires that one of the samples be chosen to represent the base pop-

ulation. The choice of which sample to use as the base population can

be made for statistical reasons (e.g., the sample with the largest num-

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