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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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