Compositional regression with unobserved components or below detection limit values

The typical way to deal with zeroes 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 a 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 ane linear combination of some fully-observed explanatory variables. Both the coecients of this linear combination and the missing values can be estimated with standard Gibbs sampling techniques. In a second step, a normally-distributed additive error is considered superimposed on the compositional response, and values are taken as \below the detection limit" (BDLs) if they are \too small" in comparison with the additive standard deviation of each variable (usually, a 3 rule is applied here). Within this framework, the regression parameters and all missing values (including BDLs) can be estimated, albeit this time with a less ecient Metropolis-Hastings algorithm. Both methods estimate the regression coecients without need of any preliminary imputation step, and adequately propagate the uncertainty derived from the fact that the missing values and BDLs are not actually observed, something imputation methods cannot achieve.