Components of statistical methods

It is not possible to prove something conclusively, instead, we can only disprove hypotheses (Popper 1959). Statistical tests begin with a null hypothesis of no effect (no boundary contiguity or no association between boundaries). Then, the pattern of the data is used to evaluate this null hypothesis.

Essential features of these methods (adapted from Waller and Jacquez 1995):

• The null spatial model describes the spatial distribution of the boundaries/boundary elements in the absence of boundary-generating processes.

• The null hypothesis A statement about the boundaries used for testing described in terms of the null spatial model. It describes the pattern of data in the absence of strong boundaries (for subboundary analysis) or boundary overlap (for overlap analysis).

• The alternative hypothesis may be an omnibus alternative to the null hypothesis, such as "not the null hypothesis" or a specific prediction about patterns in the data. For example, an alternative hypothesis can define what the data would look like when a boundary-generating process is at work.

• The test statistic summarizes an aspect of the data, such as boundary branchiness or minimum length between boundaries. It is used to evaluate the null hypothesis.

• The null distribution of the test statistic can be derived empirically through repeated Monte Carlo randomizations of the original data set and recalculation of the test statistic. The randomization procedure is defined by the null spatial model.

Probability values (p-values) for the observed test statistics can be obtained by comparing them to their null distributions. This comparison gives a quantitative estimate of how unlikely the observed value is compared to the expected null distribution. If the patterns in the data are different enough from the prediction of the null hypothesis, then the null hypothesis can be rejected. "Enough" is a difficult concept, see p values for more explanation.