Redundancy analysis includes analysis of variance-simultaneous component analysis (ASCA) and outperforms its extensions

Chemometrics and statistical ecology share interest in the analysis of multivariate data. In ecology, unconstrained and constrained ordination are popular methods to analyze and visualize multivariate data, with principal component analysis (PCA) and redundancy analysis (RDA) as prototype methods. Constraints give more insight and power by focusing on the response of the variables to particular external predictors or experimental factors, after optional adjustment for covariates. In chemometrics, analysis of variance - simultaneous component analysis (ASCA) was proposed decades later, with particular emphasis on the multivariate main and interaction effects in factorial experiments. This paper shows the similarities and differences between ASCA, its extensions, and (partial) RDA, alias reduced-rank regression. ASCA and RDA (understood as a sequence of partial RDAs, just as ASCA uses a sequence of PCAs) are shown to be mathematically identical for equireplicated designed experiments. Differences appear with unequal replication. As a corollary we show that, with equal replication, a particularly attractive form of ASCA, which displays a main effect together with an interaction, is a special case of principal response curve analysis. RDA is a least-squares method and uses the optimal weights in the dimension reduction of the treatment effects, whereas ASCA extensions for unbalanced data use alternative, sub-optimal weights.