A modified EM alr-algorithm for replacing rounded zeros in compositional data sets

The presence of rounded zeros results in an important drawback for the statistical analysis of compositional data. Data analysis methodology based on log-ratios cannot be applied under these conditions. In this paper rounded zeros are considered as a special kind of missing data. Thus, an EM-type computational algorithm for replacing them is provided. The procedure is based on the additive logistic transformation and assumes an additive logistic normal model for the data. First, the alr transformation moves data from the constrained simplex space to the unconstrained real space. Next, missing transformed data are imputed by using modified EM steps. Last, imputed data are transformed back into the simplex space to obtain a compositional data set free of rounded zeros. Additionally, a sequential strategy is proposed for the case of rounded zeros in all the components of a composition. This work focuses on the algorithm's properties and on computational implementation details. Also, its effectiveness on simulated data sets with a range of detection limits is analyzed. Special attention is paid on the effects on the covariance structure of a compositional data set. Results confirm the good behavior of our proposal. Finally, MATLAB routines implementing the algorithm are made available to the reader.