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from these data. Typically a sample of observations is available from
which we estimate the parameters. Let us denote the estimated form
matrix corresponding to the mean of the first age group by FM (Â 1 ) .
Estimation of the form matrix is discussed in Chapter 4 . The estimat-
ed form matrix corresponding to the mean of the second group is
represented by FM (Â 2 ) . The estimated growth matrix is given by
( ˆ
( ˆ
( ˆ
Although it is expected that scientists present a valid statistical
test of their analytical work, in the study of growth, it is likely that an
older form will be statistically different from a younger form. Both
hypothesis testing and the confidence interval approach to the statis-
tical comparison of forms presented in Chapter 4 can be used for
growth data. However, we advocate the use of confidence intervals for
the study of growth because simple statistical testing of similarity in
form for different age groups doesn't provide us with much new infor-
mation. The important information that can be obtained from the
study of growth concerns the discovery of local similarities or differ-
ences in form between age groups. Localization of those measures that
change the most, and those that change the least over time can be
accomplished by using confidence intervals.
Information pertaining to magnitude of change local to landmarks
can be obtained by simple inspection of the GM . Magnitude is mea-
sured by the ratio reported in a GM as the relative change in a given
linear distance. Consequently, we can speak of a linear distance dou-
bling (ratio of 2.0), increasing by 25% (ratio of 1.25), or decreasing by
5% (ratio of .95) during growth. The direction of change for any specif-
ic linear distance occurs along the given distance, but by looking at
groups of landmarks and their associated distances or the association
of a single landmark with all others, overall directions of change of
anatomical structures in relation to others can be inferred. The “delete
one” landmark approach ( Chapter 4 ) can be used to determine the rel-
ative importance of the various loci and directions of change. These
procedures were presented in detail in Chapter 4 . More information for
interpreting directions of change from given data sets using these pro-
cedures will be included in the examples at the end of this chapter.
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