In this chapter, we discuss testing hypotheses in a constrained ordination using the Monte Carlo permutation test and two related issues: stepwise selection of a set of explanatory variables and the variation partitioning procedure.

Permutation tests: the philosophy

Canoco has the ability to test the significance of constrained ordination models described in the preceding section, using Monte Carlo permutation tests. These statistical tests relate to a general null hypothesis, stating the independence of the response data (such as community composition) on the values of the explanatory variables (for example environmental factors or human impact). The principles of the permutation test were introduced in Sections 4.11 and 4.12, in an example of testing a regression model and using the simplest possible type of permutation, i.e. completely random permutation.

The null model of the independence between the corresponding rows of the response data table and of the explanatory data table (the rows referring to the same set of cases in both tables) is the basis for the permutation test in Canoco. While the actual algorithm used in Canoco does not employ exactly the approach described here, we use it to better illustrate the meaning of the permutation test.