Microbiome studies often report within-sample diversity and between-sample distances without clearly defining how local communities relate to a larger regional ensemble. This work develops a common framework for making those relationships explicit. Hellinger–Riemann intrinsic coordinates (HRIC) represent compositional data on the closed simplex while preserving zero-valued taxa. Within this representation, local and regional diversity, additive between-community diversity, individual taxon contributions, distances between samples, and variation explained by statistical models can be analyzed together. Simulations connect HRIC-based dispersion to the between-condition component of PERMANOVA. Applications to Arctic and North Atlantic microbial communities reveal comparable relationships between relative local diversity and vertical environmental gradients. In an autologous fecal microbiota transplantation trial, recipients recovered toward their own baseline compositions earlier, while alpha-diversity differences were less clear. Together, the results connect diversity partitioning with compositional analysis across environmental, host-associated, and longitudinal microbiome studies.

Beta-diversity analysis is central to comparing microbiome communities, yet widely used dissimilarity measures do not always satisfy the metric, Euclidean, or conditionally negative definite properties required by common downstream methods. We systematically evaluate popular measures, organize them into four mathematical classes, and show how property violations can distort PCoA, PERMANOVA, and kernel-based inference. We also introduce diagnostic tools and correction strategies that restore valid geometry while preserving ordination structure. Analyses across real microbiome datasets translate the theory into practical guidance, supported by an R package, interactive Shiny application, and reproducible tutorials for selecting and refining beta-diversity metrics.

Microbiome abundance data combine unequal sequencing depth, compositional constraints, and frequent zeros, which can make standard scaling and transformation methods yield inconsistent conclusions. This work reviews the connections and distinctions among existing approaches and introduces a unified framework that pairs proportion conversion with contrast transformation. The framework recovers familiar methods such as ALR and CLR while enabling new transformations, including Centered Arcsine Contrast and Additive Arcsine Contrast. Simulation and data analyses indicate that the arcsine-based methods are particularly useful under heavy zero inflation, whereas log-ratio methods perform better when zeros are less prevalent, offering practical guidance for robust microbiome analysis.