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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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