Three approaches to supervised learning for compositional data with pairwise logratios

The common approach to compositional data analysis is to transform the data by means of logratios. Logratios between pairs of compositional parts (pairwise logratios) are the easiest to interpret in many research problems. When the number of parts is large, some form of logratio selection is a must, for instance by means of an unsupervised learning method based on a stepwise selection of the pairwise logratios that explain the largest percentage of the logratio variance in the compositional dataset. In this article we present three alternative stepwise supervised learning methods to select the pairwise logratios that best explain a dependent variable in a generalized linear model, each geared for a specific problem. The first method features unrestricted search, where any pairwise logratio can be selected. This method has a complex interpretation if some pairs of parts in the logratios overlap, but it leads to the most accurate predictions. The second method restricts parts to occur only once, which makes the corresponding logratios intuitively interpretable. The third method uses additive logratios, so that $K-1$ selected logratios involve exactly $K$ parts. This method in fact searches for the subcomposition with the highest explanatory power. Once the subcomposition is identified, the researcher's favourite logratio representation may be used in subsequent analyses, not only pairwise logratios. Our methodology allows logratios or non-compositional covariates to be forced into the models based on theoretical knowledge, and various stopping criteria are available based on information measures or statistical significance with the Bonferroni correction. We present an illustration of the three approaches on a dataset from a study predicting Crohn's disease. The first method excels in terms of predictive power, and the other two in interpretability.