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 1 error in all cases. The modification consists of adjusting the significance level from √&agr; to &agr; 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. [ABSTRACT FROM AUTHOR]