Biology Reference

In-Depth Information

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

Search WWH ::

Custom Search