On the complementary nature of ANOVA simultaneous component analysis (ASCA+) and Tucker3 tensor decompositions on designed multi-way datasets

Abstract

The complementary nature of analysis of variance (ANOVA) Simultaneous Component Analysis (ASCA+) and Tucker3 tensor decompositions is demonstrated on designed datasets. We show how ASCA+ can be used to (a) identify statistically sufficient Tucker3 models; (b) identify statistically important triads making their interpretation easier; and (c) eliminate non-significant triads making visualization and interpretation simpler. For multivariate datasets with an experimental design of at least two factors, the data matrix can be folded into a multi-way tensor. ASCA+ can be used on the unfolded matrix, and Tucker3 modeling can be used on the folded matrix (tensor). Two novel strategies are reported to determine the statistical significance of Tucker3 models using a previously published dataset. A statistically sufficient model was created by adding factors to the Tucker3 model in a stepwise manner until no ASCA+ detectable structure was observed in the residuals. Bootstrap analysis of the Tucker3 model residuals was used to determine confidence intervals for the loadings and the individual elements of the core matrix and showed that 21 out of 63 core values of the 3 7 3 model were not significant at the 95% confidence level. Exploiting the mutual orthogonality of the 63 triads of the Tucker3 model, these 21 factors (triads) were removed from the model. An ASCA+ backward elimination strategy is reported to further simplify the Tucker3 3 7 3 model to 36 core values and associated triads. ASCA+ was also used to identify individual factors (triads) with selective responses on experimental factors A, B, or interactions, A B, for improved model visualization and interpretation.