On the variation explained by ordination and constrained ordination axes

Total inertia (TI), the sum of eigenvalues for all ordination axes, is often used as a measure of total variation in a data set. By use of simulated data sets, I demonstrate that lack-of-fit of data to the response model implicit in any eigenvector ordination method results in polynomial distortion ordination axes, with eigenvalues that normally contribute 30 - 70 % to TI (depending on data set properties). The amount of compositional variation extracted on ecologically interpretable ordination axes (structure axes) is thus underestimated by the eigenvalue-to-total-inertia ratio. I recommend that the current use of total inertia as a measure of compositional variation is discontinued. Eigenvalues of structure axes can, however, be used with some caution to indicate their relative importance. I also demonstrate that when the total inertia is partitioned on different sets of explanatory variables and unexplained variation by use of (partial) constrained ordination, (35) 50 - 85 % of the variation 'unexplained' by the supplied explanatory variables represents lack-of-fit of data to model. Thus, the common interpretation of 'unexplained variation' as random variation ('noise') or coenoclinal variation caused by unmeasured explanatory variables, is generally inappropriate. I recommend a change of focus from the variation-explained-to-total inertia ratio and 'unexplained' variation to relative amounts of variation explained by different sets of explanatory variables.