Partialling out the spatial component of ecological variation: questions and propositions in the linear modelling framework

First, we formulate some questions posed by the procedure recently proposed by Borcard et al. (1992) and Borcard and Legendre (1994) to partition the ecological variation of a community into different portions related to spatial and environmental descriptors. Working separately on the two steps of this procedure - linear modelling and ordinations on modelled tables - allows us to propose different solutions to these questions. These solutions, which use little-known properties of a linear regression model with two additive factors and no interaction, are also adapted to the case of mixed factors (qualitative and quantitative). These properties are presented in the framework of canonical correlation analysis. In particular, they allow us to propose an alternative to partial regression, which avoids confounding. A detailed illustration is presented.