On the variation explained by ordination and constrained ordination axes

Abstract. 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.