Zero problems with compositional data of physical behaviors: a comparison of three zero replacement methods.

Background: Researchers applying compositional data analysis to time-use data (e.g., time spent in physical behaviors) often face the problem of zeros, that is, recordings of zero time spent in any of the studied behaviors. Zeros hinder the application of compositional data analysis because the analysis is based on log-ratios. One way to overcome this challenge is to replace the zeros with sensible small values. The aim of this study was to compare the performance of three existing replacement methods used within physical behavior time-use epidemiology: simple replacement, multiplicative replacement, and log-ratio expectation-maximization (lrEM) algorithm. Moreover, we assessed the consequence of choosing replacement values higher than the lowest observed value for a given behavior.

Method: Using a complete dataset based on accelerometer data from 1310 Danish adults as reference, multiple datasets were simulated across six scenarios of zeros (5-30% zeros in 5% increments). Moreover, four examples were produced based on real data, in which, 10 and 20% zeros were imposed and replaced using a replacement value of 0.5 min, 65% of the observation threshold, or an estimated value below the observation threshold. For the simulation study and the examples, the zeros were replaced using the three replacement methods and the degree of distortion introduced was assessed by comparison with the complete dataset.

Results: The lrEM method outperformed the other replacement methods as it had the smallest influence on the structure of relative variation of the datasets. Both the simple and multiplicative replacements introduced higher distortion, particularly in scenarios with more than 10% zeros; although the latter, like the lrEM, does preserve the ratios between behaviors with no zeros. The examples revealed that replacing zeros with a value higher than the observation threshold severely affected the structure of relative variation.

Conclusions: Given our findings, we encourage the use of replacement methods that preserve the relative structure of physical behavior data, as achieved by the multiplicative and lrEM replacements, and to avoid simple replacement. Moreover, we do not recommend replacing zeros with values higher than the lowest observed value for a behavior.