Introduction

In many cases, the effects of several explanatory variables need to be separated, even when the explanatory variables are correlated. The example below comes from a field fertilization experiment (Pyšek and Lepš 1991). A barley field was fertilized with three nitrogen fertilizers (ammonium sulphate, calcium-ammonium nitrate and liquid urea) and two different total nitrogen doses. For practical reasons, the experiment was not established in a correct experimental design; i.e. plots are pseudo-replications. The experiment was designed by hydrologists to assess nutrient runoff and, consequently, smaller plots were not practical. In 122 plots, the species composition of weeds was characterized by classical Braun–Blanquet relevés (for the calculations, the ordinal transformation was used, i.e. numbers 1–7 were used for grades of the Braun–Blanquet scale: r,+, 1,. …, 5). The percentage cover of barley was estimated in all relevés.

The authors expected the weed community to be influenced both directly by fertilizers and indirectly through the effect of crop competition. Based on the experimental manipulations, the overall fertilizer effect can be assessed. However, barley cover is highly correlated with fertilizer dose. As the cover of barley was not manipulated, there is no direct evidence of the effect of barley cover on the weed assemblages. However, the data enable us to partially separate the direct effects of fertilization from indirect effects of barley competition. This is done in a similar way to the separation of the effects of correlated predictors on the univariate response in multiple regression. The separation can be done using the variable of interest as an explanatory (environmental) variable and the other ones as covariables.