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tributed. Our procedures fully account for the dependence between the
distances.
The confidence intervals described above can localize form differ-
ence to particular linear distances, and in some cases to particular
landmarks. For that reason, they are more useful in revealing biological
information and relationships than they are for testing equality of
form or shape. The testing of global null hypotheses has a long history
and is desired by certain scientists. For this reason, we present two dif-
ferent statistical procedures for testing the hypothesis of equality of
shapes using point estimates.
4.9 Statistical hypothesis testing for shape difference
The testing of statistical hypotheses has become a standard and near-
ly required part of biological analyses. We deemphasize the importance
of this procedure because it can distract the researcher from looking
carefully at the data and direct the researcher away from data based
inferences that are not directly revealed by the results of hypothesis
testing. We include statistical hypothesis testing so as to satisfy the
more traditionally minded biologists and to offer a complete suite of
methods.
The rules for conducting a valid and rigorous statistical test of a null
hypothesis are fairly standard in statistics (see Casella and Berger,
1990). Any statistical testing procedure has the following components:
1) a statement of the null and the alternative hypotheses; 2) a test
statistic which takes a specified value if the null hypothesis is true and
a different set of values under the alternative; and 3) the distribution of
the test statistic when the null hypothesis is true. If the observed value
of the test statistic lies in the extreme tails of the null distribution, the
null hypothesis is rejected. One may also calculate the probability of
getting the observed (or more extreme) value of the test statistic and
report it as a p-value. Various approaches to hypothesis testing in the
study of difference in form using EDMA are detailed below.
When comparing shapes using EDMA, the null hypothesis is that
the mean form of one population is a scaled version of the other popu-
lation (i.e., the ratios of all linear distances are equal to a constant).
The inability to reject this null hypothesis would mean that the two
samples of forms are similar in shape and differ only in size. The alter-
native hypothesis is that the two populations differ in form. In this
case, the two populations are not merely scaled versions of one anoth-
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