Nonmetric ANOVA: a generic framework for analysis of variance on dissimilarity measures

Abstract

Classic Analysis of Variance (ANOVA; cA) tests the explanatory power of a partitioning on a set of objects. Nonparametric ANOVA (npA) extends to a case where instead of the object values themselves, their mutual distances are available. While considerably widening the applicability of the cA, the npA does not provide a statistical framework for the cases where the mutual dissimilarity measurements between objects are nonmetric. Based on the central limit theorem (CLT), we introduce nonmetric ANOVA (nmA) as an extension of the cA and npA models where metric properties (identity, symmetry, and subadditivity) are relaxed. Our model allows any dissimilarity measures to be defined between objects where a distinctiveness of a specific partitioning imposed on those are of interest. This derivation accommodates an ANOVA-like framework of judgment, indicative of significant dispersion of the partitioned outputs in nonmetric space. We present a statistic which under the null hypothesis of no differences between the mean of the imposed partitioning, follows an exact F-distribution allowing to obtain the consequential p-value. Three biological examples are provided and the performance of our method in relation to the cA and npA is discussed. Significance StatementThe Nonmetric Analysis of Variance (nmANOVA) conveys a framework that allows a compatible type of ANOVA for the cases where the proper metric measurements between objects are either lost, unknown or however inaccessible. While classic ANOVA is based on the measurements of the data from a base datum, the nmANOVA is formulated on the dissimilarity outputs (not necessarily metric) defined between all objects. As the main goal of ANOVA in providing a statistical test for assessing the significance of a considered partitioning on the data, the nmANOVA is yielding a paralleled scheme of inference with 1) accommodating the outcomes dissimilarities into within and between groups statistics, 2) assessing their respective divergence with a parametric distribution, and 3) providing a resultant p-value indicative of evidences fore rejecting the null hypothesis.