Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 184

Since this confidence interval contains zero, these results do not allow
us to reject the null hypothesis of similarity in shapes between the two
samples. This result is in contrast to the result obtained using EDMA-
I. Can this be explained? Recall that at the end of the description of
EDMA-II, we remarked that if the magnitudes of the extreme negative
element and the extreme positive element are similar, careful atten-
tion must be paid while applying and interpreting the results of
EDMA-II. Observe that the bootstrap distribution of the test statistics
is bimodal with a dip near zero. This suggests that the two shapes are
different from each other, but because the two ends of the shape dif-
ference matrix are so similar, the test cannot determine if it is the
positive end or the negative end that is more important. Instead of pro-
viding a summary measure that conveys this information, the
summary measure tells the investigator that the forms are not differ-
ent in shape, when they are clearly different. When the distribution of
bootstrapped Z statistics is bimodal, the magnitudes of the extreme
negative element and the extreme positive element should be checked
for similarity in absolute value, and additional data exploration tech-
niques should be used. This is an additional reason why we stress the
use of the confidence interval procedure and advise against the isolat-
ed testing of a simple null hypothesis of equality of shapes.
In a case like this one where evidence points to a difference in the
form of the two populations that are being compared, we may want to
determine the role of scale in the difference. To examine this aspect of
difference in form, confidence intervals for difference in scale (size) of
the two populations can be estimated. To do this, we follow the same
methods for generating bootstrapped samples as described above. For
each pair of bootstrapped samples, the absolute difference in scaling
variable is calculated. A distribution of the scaling variables is obtained
in this way. A confidence interval can be obtained by sorting these boot-
strapped scaling variables in ascending order. If this confidence interval
contains zero, it indicates that there is no significant difference in the
size of the forms representing the two populations (according to the cho-
sen measure of size). For the example above, we calculate the confidence
interval for the difference in the geometric means.