Statistical Models for Landmark Coordinate Data - An Invariant Approach to Statistical Analysis of Shapes - page 94

Lele (Lele, 1993) provided the first evidence of errors associated with
this intuitively appealing approach, which were subsequently support-
ed and extended by Kent and Mardia (1997). The problems associated
with the Procrustes method when applied under the Gaussian pertur-
bation model used in this chapter (and used by Goodall, 1991) have
consequences for data analysis. Choosing a single coordinate system
for Procrustes superimposition does not eliminate the nuisance param-
eters. Instead, it constrains the nuisance parameters to take a certain
form. Moreover, because the nuisance parameters are not eliminated
properly, the Procrustes mean form and mean shape estimators are
statistically inconsistent. This means that as the sample size increas-
es, the estimator converges to a quantity that is different from the true
mean form, or the true mean shape. Even more important, however, is
the observation that the variance-covariance estimator is statistically
inconsistent. We have found that even when the amount of error in the
estimation of the mean form or mean shape is small, the error in the
covariance estimator can be substantial.
The implications of inconsistency of the variance-covariance esti-
mator are serious. Any statistical inference procedure that uses the
Procrustes estimator of variance will yield incorrect results.
Confidence intervals for form or shape difference cannot have correct
coverage probabilities if the variance estimators are wrong. For exam-
ple, when the variance estimators are wrong and it is claimed that we
have a 95% confidence interval, the true value may not actually be cov-
ered 95% of the time. Similarly, Principal Components Analysis based
on the Procrustes residuals (Kent, 1994) can be patently misleading
when variance is estimated incorrectly. In biology, perhaps more than
in any other science, variability is one of the most important parame-
ters than can be estimated using statistics. The Procrustes method
fails in its estimation of this parameter.
The following example, which our readers can carry out using trans-
parencies, illustrates how the Procrustes approach fails to correctly
measure variability (a rigorous mathematical discussion is provided in
Part 2 of this chapter). Let us begin by examining how the Procrustes
superimposition is implemented. Consider the two quadrangles in
Figure 3.6 . On one transparency, use a red pen to draw the solid quad-
rangle, and on another transparency use a green pen to draw the
dashed quadrangle. The centroid of each quadrangle is defined as the
point at which the two diagonals cross. The first step in any superim-
position method is to fix one of the figures (say the red quadrangle) and
translate the other figure (the green quadrangle) so that it matches the