Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 86

placement of the mean form into a coordinate space is arbitrary.
Elements of K * calculated from the original data and the landmark
subset are also different. The reason for this difference is that esti-
mates reported by K * are dependent upon the centering that is done
during estimation (see Part 2 of this chapter). Estimation of the mean
form matrix does not require centering, but estimation of the variance-
covariance matrix does. If a different set of landmarks is used, a
different centering occurs, and the variance-covariance matrix
changes. Importantly, values reported in the form matrix remain the
same whether the entire set of landmarks or a subset are used in the
calculation. The mean form matrix is coordinate system-free, but K * is
not. For this reason, specific values of the variance-covariance matrix
should not be directly compared between samples. Relative magni-
tudes of variance estimates within an estimated variance-covariance
matrix can be compared.
The samples presented thus far were chosen as examples of rela-
tively small sample size. Small sample sizes are common in biological
research, and the investigator needs to be aware of the consequences
for the estimation of parameters and for the statistical testing of dif-
ferences in forms. There are often economic, scientific, or biological
reasons for small sample sizes in research situations. In our example,
the aneuploid Ts65Dn sample is small because the transmission of seg-
mental trisomy to offspring is well below 100%, and of the segmentally
trisomic animals produced in any litter, only the females are fertile.
Moreover, the mice are expensive to house and breed. When faced with
a small sample size, our experience suggests that estimates based on
method of moments techniques may be more stable than those based
on maximum likelihood techniques (Lele and McCulloch, 2000). It
should be kept in mind, however, that any estimator might fail given
small sample sizes.
Small sample sizes can impact the estimation of certain parame-
ters. When calculating the mean form matrix, a quantity is calculated
(see Algorithm 1, Step 5, Part 2 of this chapter) that represents the dif-
ference between the sample mean of the squared Euclidean distance
between two landmarks and the sample variance of the squared
Euclidean distance. When the variance of the squared distance is larg-
er than the square of the mean distance between two landmarks, this
quantity is negative, which causes obvious problems. This situation is
more likely to occur in small samples among landmarks that are close-
ly spaced and is less likely to occur as sample size increases.
The most effective way to avoid this problem during estimation is