Compositional data analysis tutorial.

Abstract

This article presents techniques for dealing with a form of dependency in data arising when numerical data sum to a constant for individual cases, that is, "compositional" or "ipsative" data. Examples are percentages that sum to 100, and hours in a day that sum to 24. Ipsative scales fell out of fashion in psychology during the 1960s and 1970s due to a lack of methods for analyzing them. However, ipsative scales have merits, and compositional data commonly occur in psychological research. Moreover, as we demonstrate, sometimes converting data to a compositional form yields insights not otherwise accessible. Fortunately, there are sound methods for analyzing compositional data. We seek to enable researchers to analyze compositional data by presenting appropriate techniques and illustrating their application to real data. First, we elaborate the technical details of compositional data and discuss both established and new approaches to their analysis. We then present applications of these methods to real social science datasets (data and code using R are available in a supplementary document). We conclude with a discussion of the state of the art in compositional data analysis and remaining unsolved problems. A brief guide to available software resources is provided in the first section of the supplementary document.

Alpha-cut based compositional representation of fuzzy sets and exploration of associated fuzzy set regression

“…ensuring the estimated left endpoint of intervals is smaller than the right endpoints. While the latter poses one of the key challenges for interval regression models, recent studies have started exploring the adoption of a compositional representation to address it and unlock further potential advantages such as ensuring estimations remain within a given variable's domain [13], [15].…”

“…Recently, an increasing body of work has started exploring the adoption of a compositional representation for modelling interval-valued data reflecting uncertainty or vagueness-for example, interval-valued questionnaire responses [13], [14], [14]. Due to the nature of the mutual dependency of interval endpoint parameters, the compositional transformation is proposed in [15].…”

“…In this paper, we discuss an α-cut based compositional representation of FSs, building on prior interval work [13], [15]. We proceed to explore the adoption of compositional regression approaches to conduct linear regression on the compositional representations of FS valued data.…”

Regression with reduced rank predictor matrices: A model of trade-offs.

“…The various alternatives involve one or more of the following steps: transforming predictors (log ratios, Box Cox transformation, Box & Cox, 1964), reducing the number of predictors to resolve the linear dependency (e.g., isometric log ratios), and/or placing constraints on regression weights to resolve the linear dependency (centered log ratios). For discussions of alternatives, see Aitchison (1982), Aitchison and Bacon-Shone (1984), Aitchison et al (2000), Chen et al (2017), Egozcue et al (2003), Greenacre and Grunsky (2019), Hron et al (2012), and Smithson and Broomell (2022). All of these alternatives suffer from a common problem: the magnitudes of the regression coefficients are difficult, if not impossible, to interpret for one or more of the following reasons.…”

“…(2003), Greenacre and Grunsky (2019), Hron et al. (2012), and Smithson and Broomell (2022). All of these alternatives suffer from a common problem: the magnitudes of the regression coefficients are difficult, if not impossible, to interpret for one or more of the following reasons.…”