Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 156

procedures will not, in general, be the same from one analysis to the
next. The full statistical explanation for this is beyond the scope of this
monograph but a colloquial explanation should help here.
Both the Bootstrap approach and the Monte Carlo approach involve
a randomized procedure. In the case of the Bootstrap approach, if we
randomly pick individuals from the known sample, it is highly unlikely
that the same random sample will be chosen upon repetition. This will
result in slight differences in output when the whole procedure is run
different times. The Monte Carlo approach is also a randomized proce-
dure, but it is model-based. Even though the estimate of the mean and
variance-covariance matrix is constant from analysis to analysis, the
generation of data under the Gaussian model using these estimates
will result in slightly different Monte Carlo samples each time the
analysis is done. Consequently, the exact values of the limits of the con-
fidence intervals will change as you repeat a confidence interval
estimation analysis. The differences between analyses, however, are
not so large as to affect your conclusions.
Let us try to understand the interpretation of the confidence inter-
vals in an intuitive fashion. Suppose that there are 100 researchers in
the world who are studying the difference in craniofacial form between
unaffected children and children with Apert syndrome. Each scientist
has a sample of patients and controls from his own clinic and collects
the coordinates for the same landmarks from these individuals. When
the affected and unaffected individuals from each clinic are compared
using the above procedures, 90% confidence intervals are obtained.
Each of the researchers uses the confidence intervals derived from
their clinic data to make a decision regarding whether or not a partic-
ular linear distance is larger in Apert syndrome children as compared
to unaffected children. Following the confidence interval interpreta-
tion, we say that approximately 90 of the 100 researchers are correct
in this decision. Knowing this statistic does not guarantee that any
particular researcher is correct about their decision, but on average,
the community of researchers is right 90% of the time.
We provide percentile intervals because they are simple to interpret
and are sufficient in most biological applications. We stress that the
intervals provided are element-wise confidence intervals. This means
that each confidence interval is computed for a specific linear distance
and that they should be interpreted in that way. Finally, although the
confidence intervals are reported separately for each linear distance
(Lele and Richtsmeier, 1995), our computational procedures do not
assume that the distances in the form matrix are independently dis-