A Logistic Normal Mixture Model for Compositional Data Allowing Essential Zeros

Abstract

The usual candidate distributions for modeling compositions, the Dirichlet and the logistic normal distribution, do not include zero components in their support. Methods have been developed and refined for dealing with zeros that are rounded, or due to a value being below a detection level. Methods have also been developed for zeros in compositions arising from count data. However, essential zeros, cases where a component is truly absent, in continuous compositions are still a problem.The most promising approach is based on extending the logistic normal distribution to model essential zeros using a mixture of additive logistic normal distributions of different dimension, related by common parameters. We continue this approach, and by imposing an additional constraint, develop a likelihood, and show ways of estimating parameters for location and dispersion. The proposed likelihood, conditional on parameters for the probability of zeros, is a mixture of additive logistic normal distributions of different dimensions whose location and dispersion parameters are projections of a common location or dispersion parameter. For some simple special cases, we contrast the relative efficiency of different location estimators.

A phylogenetic transform enhances analysis of compositional microbiota data

“…Here, we employ zero replacement, which implies an assumption that all zero values represent rounded zeros. New mixture models that explicitly allow for both essential and rounded zeros ( Bear and Billheimer, 2016 ), as well as more advanced methods of zero replacement ( Martın-Fernandez et al, 2011 ; Martin-Fernandez et al, 2015 ), may enable us to handle zeros in a more sophisticated manner. Lastly, in regards to informational loss caused by normalization, it is known that the number of counts measured for a given taxon influences the precision with which we may estimate its relative abundance in a sample ( Gloor et al, 2016a ; Good, 1956 ; McMurdie and Holmes, 2014 ).…”

Dirichlet composition distribution for compositional data with zero components: An application to fluorescence in situ hybridization (FISH) detection of chromosome

“…The data amalgamation, which was proposed by Aitchison ( 1990 ), is to eliminate the components with zero elements by combining them with some other nonzero components. The second approach models the zeros separately (e.g., Aitchison, 1986 ; Bear & Billheimer, 2016 ; Zadora et al., 2010 ). For instance, Bear and Billheimer ( 2016 ) projected compositions with zeros onto smaller dimensional subspaces.…”

“…The second approach models the zeros separately (e.g., Aitchison, 1986 ; Bear & Billheimer, 2016 ; Zadora et al., 2010 ). For instance, Bear and Billheimer ( 2016 ) projected compositions with zeros onto smaller dimensional subspaces. As a result, they developed a mixture of logistic normals which successfully addresses the issues of division by zero and the log of zero.…”

Score matching for compositional distributions

“…Maximum likelihood estimation for the Dirichlet, logistic normal distribution and the Tsagris and Stewart (2020) model cannot be applied whenever there are zeros in the data since the loglikelihood is not finite. Stewart and Field (2010) and Bear and Billheimer (2016) both modified the logistic normal model to handle zero's by separating the components each into two parts and modelling the zeros separately. However, these two methods based on maximum likelihood estimation were applied only in the case where the dimension was small and Bear and Billheimer (2016) assumed that one of the components was always non-zero.…”

“…Stewart and Field (2010) and Bear and Billheimer (2016) both modified the logistic normal model to handle zero's by separating the components each into two parts and modelling the zeros separately. However, these two methods based on maximum likelihood estimation were applied only in the case where the dimension was small and Bear and Billheimer (2016) assumed that one of the components was always non-zero. In the case of the Dirichlet distribution with some zero components, moment estimators are still valid, however this model is not fully flexible because it has a restrictive correlation structure, for example all covariances are negative between the components.…”