Log-Ratio Analysis of Finite Precision Data: Caveats, and Connections to Digital Lines and Number Theory

Log-ratio analysis (LRA) is a popular and theoretically coherent framework for investigating and modelling compositional data. Most empirical compositional data will be measured and recorded with finite precision; count data is a special instance of this in which the fundamental quantity of interest is discrete, but it is also common and practical to round continuous variables to the nearest convenient multiple of the unit of measurement. LRA is often applied to such finite precision measurements without considering the underlying discrete nature of the data (with the exception of the special case of zero values). Here we examine how the characteristics of finite precision data can manifest in LRA so that theoreticians and practitioners can be mindful of situations in which finite precision might affect their conclusions. We focus in particular on log-ratio variance—a fundamental measure of pairwise association between components—and demonstrate situations in which finite precision can have a profound effect on this statistic and related measures of proportionality. We also make connections to computer science concepts about digital lines and to mathematical concepts in number theory, including Farey sequences, to understand how finite precision approximations can affect the value of log-ratio variance even when the underlying continuous variables are perfectly proportional.