Karl Gerald van den Boogaart

Regression with compositional response having unobserved components or below detection limit values

The typical way to deal with zeros and missing values in compositional data sets is to impute them with a reasonable value, and then the desired statistical model is estimated with the imputed data set, e.g., a regression model. This contribution aims at presenting alternative approaches to this problem within the framework of Bayesian regression with a compositional response. In the first step, a compositional data set with missing data is considered to follow a normal distribution on the simplex, which mean value is given as an Aitchison affine linear combination of some fully observed explanatory variables. Both the coefficients of this linear combination and the missing values can be estimated with standard Gibbs sampling techniques. In the second step, a normally distributed additive error is considered superimposed on the compositional response, and values are taken as ‘below the detection limit’ (BDL) if they are ‘too small’ in comparison with the additive standard deviation of each variable. Within this framework, the regression parameters and all missing values (including BDL) can be estimated with a Metropolis-Hastings algorithm. Both methods estimate the regression coefficients without need of any preliminary imputation step, and adequately propagate the uncertainty derived from the fact that the missing values and BDL are not actually observed, something imputation methods cannot achieve.

Robust Regression with Compositional Response: Application to Geosciences

Compositional data as multivariate observations carrying relative information frequently occur in geochemistry and they are popularly expressed in relative units, like proportions or percentages or mg/kg. The aim of multivariate regression is to quantify relations between a multivariate response and one or more explanatory variables. The standard theory on linear regression models—the least squares methodology—is appropriate if the data do not include outlying observations, deviating from the main linear trend. Although robust regression tolerates a certain amount of deviating data points, it may lead to distorted results if it is directly applied to compositional data. As a way out, the isometric logratio (ilr) transformation is used to develop robust regression, where a compositional response depends on (non-compositional) explanatory variables. For theoretical and practical reasons, the estimation of the regression parameters is carried out by the multivariate least trimmed squares (MLTS) method that fulfills all required concepts of robustness for regression with compositional data. Consequently, the robust regression model with compositional response can be used for statistical inference like hypotheses testing. Theoretical results are applied to a real-world problem from geosciences.