The Study of Growth - An Invariant Approach to Statistical Analysis of Shapes - page 218

from more than one individual such that a different individual repre-
sents each time point. Cross-sectional data are often collected from a
large population, and each individual is assigned to an age category. In
this case, data for a developmentally staged or chronologically aged
class consists of data from many individuals, and data from any one
individual is only used in a single age class. Mixed longitudinal data
results when longitudinal data are included within a cross-sectional
data set.
Longitudinal data are rare, but extremely valuable when available.
Longitudinal data must be analyzed as dependent data since the con-
dition of a data point at time t+1 is dependent upon the condition of the
data point at time t . Individual modulations in growth patterns can
only be identified with closely spaced longitudinal data points (Wilson,
1999). Longitudinal data are of particular use in the study of secular
trends, and are a requirement for predictive models of development.
On the other hand, closely spaced longitudinal data may obscure more
general patterns and reveal seemingly erratic, idiosyncratic patterns
of individual growth. To study general population patterns, cross-sec-
tional data may be more useful. The research question determines
which types of data are most useful and appropriate.
Growth is a phenomenon of the individual, but samples are needed
to speak to the statistical aspects of growth patterns. When longitudi-
nal data for many individuals are plotted over the same age interval,
it is clear that growth is highly individualized and quite variable (see
Figure 5.2 ). An average might be offered as representative of the pat-
tern, but these averages can obscure the irregularities observed in
individual patterns. A study of averages provides useful information,
but because the variability noted in individual growth patterns is
responsible in part for inter-individual morphological variability, the
variability of individual growth patterns warrants investigation.
5.3 Assigning age and forming age-related groups
Statistical analysis of growth calls for analysis of samples of individuals.
To do this, individuals need to be placed into age-graded groups or class-
es according to some criteria. There are no rigid rules for the formulation
of age-specific groups for analysis of growth patterns. If chronological
ages are known, it may help to survey the available data and look for
natural breaks in the age distribution for forming age groups for analy-
sis. If chronological ages are not known, features of the organisms that