Biology Reference

In-Depth Information

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

FM

A

)

(
ˆ

ˆ

ij

2

GM A

,

A

)

(
ˆ

21

FM

A

)

ij

1

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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