Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 63

mark data is the specification of a model for variability within a group.
This step is important because the model that is adopted and used in
analysis influences the analytical results. An inappropriate model can
produce illusory or incorrect results. As discussed above, we use the
term model to describe the mathematical construct that attempts to
characterize (not calculate) how individuals within a population vary.
Unfortunately, there is no such thing as the correct model. All models,
even highly complex ones, are approximations.
The choice of a suitable model is dependent on several factors
including previous experience of the investigator, knowledge of the
specimens, familiarity with the nature of the data, and mathematical
convenience. In choosing a model, we need to be concerned that it is
realistic enough to model observable facts reasonably well, but at the
same time simple enough mathematically to allow statistical infer-
ence. The parameters of the chosen statistical model (e.g., the mean
and the variance) should help elucidate the biological phenomena, and
should be estimable using a sample of observations. For example, if
technical problems do not allow the collection of three-dimensional
data from biological organisms that are truly three-dimensional, the
observer needs to make an informed decision regarding whether the
organisms can be adequately represented by two-dimensional data.
Additionally, any model of the variability of landmark coordinate data
should be such that all the observations generated under such a model
belong to the same dimensional space as the original data (i.e., either
two-dimensional or three-dimensional Euclidean space).
In short, three criteria should be met in designing the model and
the parameters: 1) the model should be a reasonable approximation of
the variability within a population; 2) the parameters should convey a
direct, tangible meaning to the phenomena under study (i.e., a biolog-
ical meaning for biological data sets); and 3) the parameters should be
estimable using available data. The first two criteria require knowl-
edge of the biology of the specimens. The third criterion requires
statistical knowledge. The aim of this chapter is to suggest a reason-
able statistical model for landmark coordinate data and to discuss the
estimation of the relevant parameters.
3.2 Model for intra-group variability
Our model of intra-group variability can be illustrated by conducting
the following experiment. Imagine that you have eleven transparen-