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 (e.g., no clustering). 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 modelis a mechanism for generating the reference distribution. This may be based on distribution theory, or it may use randomization (e.g. Monte Carlo) techniques.

• The null hypothesis (Ho)describes the spatial pattern expected when the alternative hypothesis is false (e.g. CSR).

• The alternative hypothesis (Ha)describes the spatial pattern that the test is designed to detect.  This may be a specific alternative, such as direct overlap of objects or clustering, or it may be the omnibus ”r;not the null hypothesis.”

• The test statisticquantifies a relevant aspect of spatial pattern (e.g. spatial clustering, overlap).

• The reference distributionis the distribution of the test statistic when the null hypothesis is true.

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, by convention people often choose "enough" to mean "expected by chance less than 5% of the time", or an alpha level of 0.05.