Bayesian-multiplicative treatment of count zeros in compositional data sets

Abstract

Compositional count data are discrete vectors representing the numbers of outcomes falling into any of several mutually exclusive categories. Compositional techniques based on the log-ratio methodology are appropriate in those cases where the total sum of the vector elements is not of interest. Such compositional count data sets can contain zero values which are often the result of insufficiently large samples. That is, they refer to unobserved positive values that may have been observed with a larger number of trials or with a different sampling design. Because the log-ratio transformations require data with positive values, any statistical analysis of count compositions must be preceded by a proper replacement of the zeros. A Bayesian-multiplicative treatment has been proposed for addressing this count zero problem in several case studies. This treatment involves the Dirichlet prior distribution as the conjugate distribution of the multinomial distribution and a multiplicative modification of the non-zero values. Different parameterizations of the prior distribution provide different zero replacement results, whose coherence with the vector space structure of the simplex is stated. Their performance is evaluated from both the theoretical and the computational point of viewThis research was supported by the Ministerio de Economia y Competividad under the project 'METRICS' Ref. MTM2012-33236, by the Agencia de Gestio d'Ajuts Universitaris i de Recerca of the Generalitat de Catalunya under the project Ref: 2009SGR424, and by the Scottish Government's Rural and Environment Science and Analytical Services Division (RESAS). The authors also gratefully acknowledge the support by the Operational Program Education for Competitiveness-European Social Fund (project CZ.1.07/2.3.00/20.0170 of the Ministry of Education, Youth and Sports of the Czech Republic

Compositional visualization of intermedia agenda setting by the main media groups and political parties in the Spanish 2015 General Elections

“…If the z vector contains zeros, they must be replaced beforehand (Martín-Fernández et al, 2011). When, like in this case, the data are counts of frequencies, the common framework for replacing the zeros is the Bayesian-multiplicative approach (Martín-Fernández et al, 2015b;Pierotti et al, 2009) In the BayesLaplace's rule, the z cero values are replaced with:…”

Representación composicional de la intermedia agenda-setting de los principales grupos de medios y partidos políticos en las elecciones generales españolas de 2015

“…Si el vector z contiene ceros, deben ser reemplazados antes (Martín-Fernández et al, 2011). Cuando los datos, como en el caso que nos ocupa, son recuentos de frecuencias, el marco común para reemplazar los ceros es el Bayesiano-multiplicativo (Martín-Fernández et al, 2015b, Pierotti et al, 2009. En la regla de Bayes-Laplace, los valores cero z se sustituyen por:…”

“…Como hemos señalado en este mismo artículo, existen métodos para el reemplazo de estos ceros (Martín-Fernández et al, 2015b) cuando su número es reducido. Con todo, cabe apuntar que podemos minimizar esta limitación desarrollando estudios de http://www.revistalatinacs.org/073paper/1255/14es.html Página 287 la AS con un volumen de datos más elevado, cubriendo un mayor número de artículos, programas, tweets, etc.…”

Linear Association in Compositional Data Analysis

“…A common feature of 16S rRNA gene sequencing data sets, is that they are sparse, i.e., zero counts are extremely abundant in the data matrix. Log-ratio analysis of compositional data does not allow null proportions as an argument of a logarithm, thus requiring special treatment of such data (Martín-Fernández, Hron, Templ, Filzmoser, and Palarea-Albaladejo 2015a;Gloor, Macklaim, Pawlowsky-Glahn, and Egozcue 2017). The present contribution is not aimed at discussing procedures and methods for treating these zero count data, although Bayesian estimation methods show some promise (Fernandes et al 2014).…”