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G.K. van~den~Boogaart, R. Tolosano-Delgado, M. Templ:
"Regression with Compositional Response Having Unobserved Components or Below Detection Limit Values";
Statistical Modelling, 15 (2015), 2; 191 - 213.

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 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 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 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. Within this framework, the regression parameters and all missing
values (including BDLs) 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 BDLs are not actually observed, something imputation methods cannot achieve.

bayesian regression; compositional regression; missing values; nondetects; MCMC

http://dx.doi.org/10.1177/1471082X14535527