Proportionality: A Valid Alternative to Correlation for Relative Data

Abstract

In the life sciences, many measurement methods yield only the relative abundances of different components in a sample. With such relative—or compositional—data, differential expression needs careful interpretation, and correlation—a statistical workhorse for analyzing pairwise relationships—is an inappropriate measure of association. Using yeast gene expression data we show how correlation can be misleading and present proportionality as a valid alternative for relative data. We show how the strength of proportionality between two variables can be meaningfully and interpretably described by a new statistic ϕ which can be used instead of correlation as the basis of familiar analyses and visualisation methods, including co-expression networks and clustered heatmaps. While the main aim of this study is to present proportionality as a means to analyse relative data, it also raises intriguing questions about the molecular mechanisms underlying the proportional regulation of a range of yeast genes.

Linear Association in Compositional Data Analysis

“…For instance the φ-statistic (Lovell et al 2015) and a modification of it (Erb and Notredame 2015). These measures of association are related to the approximate proportionality of single parts, but they can be generalised to association between groups of parts.…”

“…In such circumstances, ordinary least squares (OLS) regression is not an appropriate method. Following Warton, Wright, Falster, and Westoby (2006), and references therein, regression on the major axis (MA), also known as total least squares, and standardized major axis (SMA) (adopted in Lovell et al (2015)) have been chosen for fitting the model (8). These approaches differ on the residual scores used to minimize their sum of squares.…”

“…The first one is the φ-statistic (Lovell et al 2015) applied to balances appearing in the model given by Equation (8). That is…”

“…Linear association of parts in a composition has been examined from a theoretical point of view, beyond the early proposals (Egozcue et al 2013;Lovell et al 2015). Proportionality of two parts in a composition was the idea for introducing b-association which leads to an approximately constant log-ratio.…”

Untitled

“…For DESeq/DESeq2, poor 580 performance may be due to the model's assumption that differentially abundant OTUs are not a 581 large portion of the population (Dillies et al 2013), or the model's overdispersion estimates 582 ). Thus, compositionality is still a large unsolved problem in differential 583 abundance testing (Lovell et al 2015), and we would urge caution in data sets where 584 compositionality may play a large role, e.g. when the alpha diversity of the samples is low 585 (Friedman & Alm 2012).…”