Improved testing of species traits--environment relationships in the fourth-corner problem

The fourth-corner problem entails estimation and statistical testing of the relationship between species traits and environmental variables from the analysis of three data tables. In a 2008 paper, S. Dray and P. Legendre proposed and evaluated five permutation methods for statistical significance testing, including a new two-step testing procedure. However, none of these attained the correct type I error in all cases of interest. We solve this problem by showing that a small modification of their two-step procedure controls the type I error in all cases. The modification consists of adjusting the significance level from mean square root of alpha to alpha or, equivalently, of reporting the maximum of the individual P. values as the final one. The test is also applicable to the three-table ordination method RLQ.