ter Braak C., Cormont A., Dray S. (2012). Improved testing of species traits-environment relationships in the fourth corner problem. Ecology, 93:1525--1526.

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 sqrt(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.

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