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Although this analysis was developed specifically for the comparison
of growth patterns, it can be used to study related problems in biology.
For example, Lague and Jungers (1999) used GDMA to study differences
in patterns of sexual dimorphism in joint surface morphology among
hominoid species. Other problems that require a quantitative method for
comparing comparisons may also be studied using the method.
5.8 Growth difference matrix analysis: comparing
patterns of growth using growth matrices
“Great as are the differences between the rates of growth in different parts
of a complex organism, the marvel is that the ratios between them are so
nicely balanced as they are, and so capable of keeping the form of the
growing organism all but unchanged for long periods of time, or of slow-
ly changing it in its own harmonious way. There is the nicest possible
balance of forces and resistances in every part of the complex body; and
when this normal equilibrium is disturbed, then we get abnormal growth,
in the shape of tumours and exostoses, and other malformations and
deformities of every kind.”
D'Arcy Thompson (1992)
Our interest may be less in defining a single growth pattern as in
understanding how the growth pattern for one group differs from that
of another. For example, an abnormal growth pattern needs to be com-
pared to a normal growth pattern to be understood. The comparison of
growth patterns can only be done in a coordinate system-free method of
measurement. Our presentation of a coordinate system-free system for
the comparison of growth patterns follows methods previously present-
ed by Richtsmeier and Lele (1993).
To compare growth patterns of two populations, four samples are
used in analysis. Each sample must contain data for the exact same
landmarks. Let us say that we have collected sample data from indi-
viduals of population A from two age groups, 1 and 2. We refer to these
samples as A 1 and A 2 . We have also collected sample data from corre-
sponding age groups in population B . We refer to these samples as B 1
and B 2 . In this example, we assume that the populations have been
aged correctly, according to comparable aging criteria.
The first step is to calculate the growth pattern for the samples that
represent population A . The growth matrix is estimated using two
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