Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 72

above example that as the number of observations tends to infinity, the
maximum likelihood estimate of

2 converges to a quantity that is dif-
ferent from the true value of the patient-to-patient variance. They also
showed that, in this case, is non-estimable. In this example the max-
imum likelihood estimator of patient-to-patient variability,

2 ,is

biased and is also a “statistically inconsistent” estimator.
In Part 2 of this chapter, we prove that certain estimators such as
those based on shape coordinates suggested by Bookstein (1991) and
Procrustean superimposition estimators suggested by Goodall (1991)
are statistically inconsistent. These estimators are inconsistent due to
the effects of nuisance parameters (see Lele and Richtsmeier, 1990;
Lele, 1991; Lele, 1993).

3.4 Invariance and elimination of nuisance parameters

The existence of nuisance parameters, though not appreciated in many
statistical analyses of laboratory or field data, poses significant problems
for the estimation of the mean and variance using landmark coordinate
data. An inconsistent estimator is unsatisfactory for use in scientific
research. Fortunately, there are ways to circumvent these problems.
One approach is to transform the data such that the distribution of
the transformed data is independent of the nuisance parameters.
Statistical inference is then based only on the transformed data.
Consider the blood pressure example introduced earlier. Using the
variables introduced in that example we can define a new variable,
W i Y i 2 . The new observation, W i , represents the difference

between observations Y i 1 and Y i 2 , and is the difference in the reduction
of blood pressure for the two patients at the i -th clinic. This transfor-
mation introduces a new model for the distribution of the transformed
data. After this transformation, the data W i 's are distributed as
W i ~N (0,2

Y i 1

2 ) . Notice that the distribution of W i does not depend upon
the nuisance parameter α i , the clinic effect. We have eliminated the
nuisance parameter α i by transforming the data ( Y i 1 , Y i 2 ) to W i .
Removal of the nuisance parameter enables estimation of the variance
parameter,

2 , using the maximum likelihood method (Casella and
Berger, 1990) where the likelihood is based on W i ~N (0,2

2 ) . This esti-
mator is also statistically consistent (Neyman and Scott, 1948).
In the study of landmark coordinate data, nuisance parameters can
be eliminated using a similar approach. We first transform the land-
mark coordinate matrix to what is known as the Euclidean distance
matrix that consists of all possible linear distances among the land