An integrative Bayesian Dirichletmultinomial regression model for the analysis of taxonomic abundances in microbiome data
 W. Duncan Wadsworth^{1},
 Raffaele Argiento^{2},
 Michele Guindani^{3},
 Jessica GallowayPena^{4},
 Samuel A. Shelburne^{5} and
 Marina Vannucci^{1}Email authorView ORCID ID profile
DOI: 10.1186/s1285901715160
© The Author(s) 2017
Received: 11 June 2016
Accepted: 31 January 2017
Published: 8 February 2017
Abstract
Background
The Human Microbiome has been variously associated with the immuneregulatory mechanisms involved in the prevention or development of many noninfectious human diseases such as autoimmunity, allergy and cancer. Integrative approaches which aim at associating the composition of the human microbiome with other available information, such as clinical covariates and environmental predictors, are paramount to develop a more complete understanding of the role of microbiome in disease development.
Results
In this manuscript, we propose a Bayesian DirichletMultinomial regression model which uses spikeandslab priors for the selection of significant associations between a set of available covariates and taxa from a microbiome abundance table. The approach allows straightforward incorporation of the covariates through a loglinear regression parametrization of the parameters of the DirichletMultinomial likelihood. Inference is conducted through a Markov Chain Monte Carlo algorithm, and selection of the significant covariates is based upon the assessment of posterior probabilities of inclusions and the thresholding of the Bayesian false discovery rate. We design a simulation study to evaluate the performance of the proposed method, and then apply our model on a publicly available dataset obtained from the Human Microbiome Project which associates taxa abundances with KEGG orthology pathways. The method is implemented in specifically developed R code, which has been made publicly available.
Conclusions
Our method compares favorably in simulations to several recently proposed approaches for similarly structured data, in terms of increased accuracy and reduced false positive as well as false negative rates. In the application to the data from the Human Microbiome Project, a close evaluation of the biological significance of our findings confirms existing associations in the literature.
Keywords
Bayesian hierarchical model Data integration Dirichletmultinomial Microbiome data Variable selectionBackground
The human microbiome is defined as the collection of microorganisms, including bacteria, viruses, and some unicellular eukaryotes, that live in and on our bodies [1]. Research on the microbiome has grown exponentially in the past few years and it has been argued that the microbiota can be regarded as a “second genome”[2, 3]. Indeed, just the human gut microbiome is estimated to be composed of approximately 10^{14} bacterial cells, i.e. ten times more than the total number of human cells in the body [4]. The contribution of the human microbiome on several health outcomes has been frequently reported in the literature. For example, microbial dysbiosis in the gut has been linked to irritable bowel syndrome and Crohn’s disease [5], type 2 diabetes [6], cardiovascular disease [7], and psychological conditions via the socalled “gutbrain axis” [8]. The composition of microbiota at other body sites have also been associated with conditions such as eczema [9] and preterm labor [10]. This stream of research holds great potential for a better understanding of many mechanistic processes in the development of human diseases, especially with respect to immune regulation and barrier defense [11, 12].
Microbiome data is most commonly obtained by sequencing variable regions of the 16S rRNA gene, then grouping the transcripts into Operational Taxonomic Units (OTUs), based on their similarity to one another. The OTUs are then defined as a cluster of reads based on a similarity threshold (typically, 97%) set by the researcher. The membership count of each cluster is then used as a proxy for taxa abundances in the sample [13]. See [14] for a discussion of how the selection of the cutoff might impact the resulting OTUs, in particular for rare species. Many studies summarize the taxa abundances by constructing several indicators of community composition (e.g. alpha and beta diversity indexes, see, [15]). Alternatively, the full OTUs abundance table can be used to obtain more detailed information about existing associations between environment or phenotypes and microbes. Wellestablished statistical models for the analysis of count data (e.g., Poisson or Negative Binomial distribution) can be efficaciously employed for the analysis of taxonomic count data [16]. Although less common, other distributions (e.g., a twoparameter Weibull distribution) have also been shown to provide a good fit to the data for some communities (see, e.g. [17]). One distinctive characteristic of the microbiome data is their overdispersion: while some taxa (e.g., Bacteroides and Lactobacillus species) are common among samples, many other taxa are present at much lower abundances, and often never recorded in a sample, leading to zeroinflated distributions. Many of the existing tools for microbial community analysis (e.g., the QIIME platform, [18]) bypass those characteristics and rely on nonparametric tests to compare species across different conditions [19, 20]. Other approaches use ordination, e.g. multidimensional scaling, to summarize abundances, and are sometimes employed to link the microbiome data with available clinical covariates and phylogenetic information [21, 22]. In those approaches, the choice of the distance metric is often crucial. The interpretation of biological phenomena can also be challenging in low dimensional projections. Most importantly, distancebased methods do not explicitly quantify the relative importance of significant associations between taxa and covariates, and therefore are of limited use for clinical decisions.
In this manuscript, we consider an integrative Bayesian approach based on the use of DirichletMultinomial (DM) distributions [23] for studying the association between taxa abundance data and available measurements on clinical, genetic and environmental covariates. Recently, La Rosa et al. [24] proposed the use of a DM model for hypothesis testing and power calculations in microbiome experiments. Holmes et. al [25] used a finite mixture of DM distributions to directly model the taxa counts. Neither method incorporate predictors to study the influence of external factors on the microbiome’s abundance. A penalized likelihood approach based on a DM regression model has been proposed instead by [26] to determine significant associations between the microbiome composition and a set of covariates which describe the individual dietary nutrients’ intakes. Similarly, [27] develop a structure constrained version of sparse canonical correlation analysis that integrates compositionalized microbiome data, phylogenetic information, and nutrient information. Furthermore, [28] propose penalized regression models to associate the multivariate compositionalized microbiome data with some univariate phenotype of interest, e.g. body mass index, as a response. However, the use of a constrained optimization approach does not allow to fully characterize the uncertainty in the selection of the significant associations, which is of particular importance, especially when dealing with highdimensional and highlycorrelated data.
Here, we propose a probabilistic modeling approach which both flexibly takes into account the typical features of microbiome count data and also allows for straightforward incorporation of available covariate information within a DM loglinear regression framework. With respect to modeling approaches as in [28], our framework allows the study of associations between multivariate microbiome data and multivariable predictors. By imposing sparsity inducing spikeandslab priors on the regression coefficients, our model obtains a parsimonious summary of the effects of the associations and also allows an assessment of the uncertainty of the selection process. We evaluate the performance of our model first on simulated data, where we provide comparisons with methods developed for microbiome or similar type of data. We also illustrate our method on data obtained from the Human Microbiome Project [29], to investigate the association between taxonomic abundances and metabolic pathways inferred from whole genome shotgun sequencing reads. It is known that the combination of environmental and host genetic factors shape the composition of the gut microbiota, and these interactions appear to have a significant effect on several biological mechanisms, which may be related, for example, to the individual immunity and barrier defense, as well as metabolism and diet [30, 31]. The approach has been implemented in a userfriendly R code, which has been made publicly available (see the Licensing Section).
Methods
We describe our Bayesian variable selection approach for the analysis of microbiome data and their association with a set of available covariates in the context of DM loglinear regression models.
Dirichletmultinomial regression with variable selection
where \(\gamma _{+} = \sum _{j}^{J} \gamma _{j}\). First described in [23] as the compound multinomial, the DM(γ) allows more flexibility than the Multinomial when encountering overdispersion in multivariate count data, as it induces an increase in the variance by a factor of (y _{+}+γ _{+})/(1+γ _{+}).
i=1,…,n; j=1,…,J. In this formulation, the intercept term α _{ j } corresponds to the log baseline parameter for the taxon j, whereas the regression parameter β _{ pj } captures the effect of the pth covariate on the abundance for that taxon.
MCMC algorithm
 1.
Update of α : This is a MetropolisHastings step with a symmetric random walk proposal \(\alpha _{j}^{'} \sim \mathcal {N}(\alpha _{j}, t_{\alpha }^{2})\), for j=1,…,J.
 2.Joint update of ( ξ , β ): We sample these parameters jointly via a Gibbs scan that employs a Metropolis acceptance step. For each j=1,…,J and p=1,…,P:

if ξ _{ pj }=1: propose \(\xi _{pj}^{'} = 0\) and \(\beta _{pj}^{'} = 0\).

if ξ _{ pj }=0: propose \(\xi _{pj}^{'} = 1\) and then propose \(\beta _{pj}^{'}\) following an adaptive MetropolisHasting scheme$$\begin{array}{*{20}l} \beta_{pj}^{'} \sim&\, 0.95 \, \mathcal{N}\left(\beta_{pj}, 2.38^{2} \times \hat{\sigma}^{2}_{\beta_{pj}} / J\times P\right)\\ &+ 0.05 \, \mathcal{N}\left(\beta_{pj}, 0.01 / J\times P\right), \end{array} $$
where \(\hat {\sigma }^{2}_{\beta _{pj}}\) is the current estimate of the variance of the target distribution. The value of \(\hat {\sigma }^{2}_{\beta _{pj}}\) is updated using a recursive formula as in [39] on all the previous draws for β _{ pj }.

Accept (\(\xi _{pj}^{'}, \beta _{pj}^{'}\)) with probabilitywhere \(\boldsymbol {\xi }_{pj}^{'} = (\xi _{1,j}^{'}, \dots, \xi _{p1,j}^{'}, \xi _{p + 1,j}, \dots, \xi _{pj})\), and \(\boldsymbol {\beta }_{pj}^{'} = (\beta _{1,j}^{'}, \dots, \beta _{p1,j}^{'}, \beta _{p + 1,j}, \dots, \beta _{pj})\).$$a = \min \left\{ 1, \frac{\pi\left(\xi_{pj}^{'}, \beta_{pj}^{'} \mid \boldsymbol{\xi}_{pj}^{'}, \boldsymbol{\beta}_{pj}^{'}, \text{else}\right) }{ \pi\left(\xi_{pj}, \beta_{pj} \mid \boldsymbol{\xi}_{pj}^{'}, \boldsymbol{\beta}_{pj}^{'}, \text{else}\right)} \right\}, $$

with N=TN+TP+FN+FP, \(\mathrm {P} = \frac {TP + FP} { N }\) and \(\mathrm {S} = \frac {TP + FN} { N }\) [42]. Since the MCC balances TP and FP counts, and can be used even if the classes are of very different sizes, it is generally regarded as one of the most appropriate measures of classification accuracy. We further computed receiving operating curves (ROC) to compare the performance of the selection procedure across the different methods.
Comparison study on simulated data
We carry out a simulation study to assess the performance of our model and compare results to alternative methods. More specifically, we consider two methods which have been specifically employed for the integrative analysis of microbiome data: the penalized approach of Chen and Li [26], and the false discovery ratecorrected pairwise correlation tests considered in [19]. In addition, we consider the factorized maximum a posteriori (MAP) Bayesian lasso of [43], a recently proposed general statistical method for conducting variable selection in multivariate countresponse regression. When fitting the Bayesian Gamma Lasso method of [43], model selection was done using the minimum AIC, while for Chen and Li’s approach the minimum BIC was calculated with the group penalty set to 20%. We also fit the method of Chen and Li to the untransformed data. The false discovery rate threshold for the Spearman’s correlation tests was set to 0.05.
In simulating data, we set n=100, P=50 and J=50, and chose P _{ r }=9 and J _{ r }=5 to obtain a total number of relevant taxa/covariate associations equal to 25. We simulated the covariate matrix X according to a MultivariateNormal (0,Σ) with Σ _{ i,j }=ρ ^{i−j} and ρ=0.4. We drew each vector y _{ i } of counts from a DirichletMultinomial distribution as follows. For i=1,…,n, \(\mathbf {y}_{i}\sim \text {Multinomial}(N_{i},\boldsymbol {\pi }_{i}^{*})\), with the row sum N _{ i }∼DiscreteUnif[1,0000;2,000], and \(\boldsymbol {\pi }^{*}_{i}=(\pi ^{*}_{i1}, \ldots, \pi ^{*}_{iJ})\sim \text {Dirichlet}(\boldsymbol {\gamma }^{*})\). For \(\boldsymbol {\gamma }^{*}=(\gamma ^{*}_{1},\dots,\gamma ^{*}_{J})\), we set \(\gamma _{j}^{*}=\frac {\gamma _{j}}{\gamma _{+}}\,\frac {1\psi }{\psi }\), j=1,…,J, with γ _{ j }= exp{α _{ j }+X β _{ j }}, \(\gamma _{+} = \sum _{j = 1}^{J} \gamma _{j}\) and ψ∈ [ 0,1] an overdispersion parameter. When ψ→0, the simulated values approximate a Multinomial (π) distribution, while for large ψ, the sampled values are more disperse. Here, we set ψ=0.01. We sampled the nonzero β _{ pj }’s from the intervals ±[ 0.5,1.0] and the intercept parameters from a Uniform(–2.3, 2.3). Below we report performance results as averages over 30 replicated simulated datasets.
When running the MCMC, we used a vague prior for the intercept by setting the variance parameter to \(s_{pj}^{2} = 10\). Similarly, we set \(r_{pj}^{2} = 10\), to provide sufficiently vague prior information on the nonzero loglinear regression coefficients. Finally, we set m=0.01 (or a=0.02 and b=1.98), resulting in a sparse prior mean on selected associations of 1% of the total. We provide comments on the sensitivity of the selection results to the choice of these hyperparameters in the Section below. We ran the MCMC algorithm for 10,000 iterations and thinned to every fifth iteration. On a single dataset, the C code took approximately 31.5 min to run on an Intel Xeon E52630 2.30 GHz processor. We assessed convergence visually and via the Geweke diagnostic [44] as implemented in the R package coda. Convergence was checked for a) the number of active variables in each iteration and b) the samples from each of the selected β _{ pj }. The five number summary of the 25 Geweke zscores was (–3.43, –1.06, –0.63, 0.71, 1.98).
Inferring associations between taxonomic abundances and metabolic pathways
We demonstrate our approach on publicly available data obtained from the Human Microbiome Project (HMP) website [29] from which we use 79 samples from healthy individuals. The Y matrix in our model contains 16S rRNA microbial counts from stool samples at the genus taxonomic level. As common in microbiome studies, the genera abundances (Bacteroides, Prevotella, etc.) were filtered by requiring each genus to be present in at least 5% of the samples. This procedure removes extremely lowabundance genera leaving 80 genera for the analysis. From the same 79 individuals, we obtained KEGG orthology group abundances which are used as the matrix of covariates X of our model. The KEGG orthology groups were reconstructed from metagenomic shotgun sequencing (WGS) using the HMP Unified Metabolic Analysis Network (HUMAnN) pipeline [45] and were also provided on the HMP website. These values represent inferred abundances of biochemical functional groups and metabolic pathways present due to the shotgun sequenced reads of bacterial and nonbacterial genes in the sample. To reduce correlation among the covariates we used average linkage clustering on the correlation matrix of the KEGG groups and chose one representative from each cluster, according to its relevance to microbiome research, leaving 76 columns in X. Finally, the columns in X were mean centered and scaled to unit variance. Though the HMP sampled 300 individuals for several timepoints and over many sites, there were relatively few samples that included the WGS used to obtain the KEGG orthology data. Thus, when joining the samples from the 16S rRNA data and the KEGG orthology data, a total of 79 matched samples remained.
We used the same hyperparameter settings as in the simulation study, that is \(s_{pj}^{2} = 10\) and \(r_{pj}^{2} = 10\) and set m=0.01, resulting in a sparse mean selection prior of 1% of the total 6,080 possible associations. The MCMC algorithm described in “MCMC algorithm” section above was run for 500,000 iterations and thinned to every 100th draw. We assessed convergence visually and via the Geweke diagnostic [44] as implemented in the R package coda. The five number summary of the Geweke zscores for the 26 β _{ pj }’s was (–3.83, –1.19, 0.15, 1.46, 3.38).
Results
Simulation study
Simulated data: performance assessment for two different scenarios, characterized by different values of the dispersion parameter ψ
Overall  Taxa  

DMBVS  MAPBL  C&L  CORTEST  DMBVS  MAPBL  C&L  CORTEST  
ψ=0.01  
MCC  0.93  0.64  0.67  0.73  0.89  0.66  0.50  0.85 
FNR  0.05  0.10  0.12  0.31  0.00  0.46  0.43  0.02 
FPR  0.00  0.01  0.01  0.00  0.05  0.01  0.09  0.06 
Accuracy  1.00  0.99  0.99  1.00  0.96  0.91  0.85  0.95 
ψ=0.1  
MCC  0.72  0.42  0.54  0.56  0.73  0.40  0.38  0.70 
FNR  0.39  0.58  0.28  0.63  0.24  0.73  0.52  0.37 
FPR  0.00  0.01  0.01  0.00  0.05  0.01  0.12  0.02 
Accuracy  1.00  0.99  0.99  1.00  0.92  0.86  0.81  0.92 
Sensitivity analysis
Simulated data: sensitivity analysis for varying values of the prior expected value of p _{ pj }, m, and the slab variance \(r^{2}_{pj}\), and for two different scenarios, characterized by different values of the dispersion parameter ψ
m=0.005  m=0.01  m=0.05  

\(r_{pj}^{2} = 1\)  \(r_{pj}^{2} = 10\)  \(r_{pj}^{2} = 100\)  \(r_{pj}^{2} = 1\)  \(r_{pj}^{2} = 10\)  \(r_{pj}^{2} = 100\)  \(r_{pj}^{2} = 1\)  \(r_{pj}^{2} = 10\)  \(r_{pj}^{2} = 100\)  
ψ=0.01  
MCC  0.69  0.93  0.96  0.69  0.93  0.95  0.69  0.93  0.95 
FPR  0.01  0.00  0.00  0.01  0.00  0.00  0.01  0.00  0.00 
FNR  0.02  0.04  0.08  0.02  0.05  0.09  0.02  0.05  0.08 
Accuracy  0.99  1.00  1.00  0.99  1.00  1.00  0.99  1.00  1.00 
AUC  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00 
ψ=0.1  
MCC  0.53  0.73  0.72  0.53  0.73  0.71  0.52  0.73  0.72 
FPR  0.01  0.00  0.00  0.01  0.00  0.00  0.01  0.00  0.00 
FNR  0.26  0.37  0.46  0.26  0.37  0.47  0.26  0.37  0.47 
Accuracy  0.99  1.00  1.00  0.99  1.00  1.00  0.99  1.00  1.00 
AUC  0.96  0.96  0.93  0.96  0.96  0.93  0.95  0.95  0.93 
Data analysis
Real data: selection results using a BFDR of 0.1
KEGG ID  Pathway  Taxa  MPPI  β _{ pj } 

ko04660  T cell receptor signaling pathway  g.Subdoligranulum  1.00  0.40 
ko04512  ECMreceptor interaction  g.Subdoligranulum  1.00  0.44 
ko00680  Methane metabolism  g.Sutterella  1.00  1.47 
ko05200  Pathways in cancer  o.Bacteroidales  1.00  1.04 
ko04540  Gap junction  o.Bacteroidales  1.00  0.92 
ko00534  Glycosaminoglycan biosynthesis  g.Ruminococcus  1.00  0.56 
ko00053  Ascorbate and aldarate metabolism  g.Ruminococcus  1.00  0.46 
ko00650  Butanoate metabolism  g.Ruminococcus  1.00  0.55 
ko00513  Various types of Nglycan biosynthesis  g.Parabacteroides  1.00  0.54 
ko00500  Starch and sucrose metabolism  g.Parabacteroides  1.00  0.61 
ko00904  Diterpenoid biosynthesis  g.Dialister  1.00  1.21 
ko00360  Phenylalanine metabolism  g.Dialister  1.00  2.18 
ko00626  Naphthalene degradation  g.Bacteroides  1.00  0.39 
ko00010  Glycolysis / Gluconeogenesis  g.Bacteroides  1.00  0.57 
ko00513  Various types of Nglycan biosynthesis  g.Prevotella  1.00  0.77 
ko00512  Mucin type OGlycan biosynthesis  g.Prevotella  1.00  1.06 
ko00620  Pyruvate metabolism  g.Prevotella  1.00  1.76 
ko04340  Hedgehog signaling pathway  g.Ruminococcus  1.00  0.45 
ko04660  T cell receptor signaling pathway  g.Roseburia  1.00  0.46 
ko00983  Drug metabolism  g.Sutterella  1.00  1.20 
ko00260  Glycine, serine and threonine metabolism  o.Bacteroidales  1.00  0.88 
ko00330  Arginine and proline metabolism  g.Ruminococcus  0.99  0.39 
ko01057  Biosynthesis of type II polyketide products  o.Bacteroidales  0.99  0.76 
ko00620  Pyruvate metabolism  g.Ruminococcus  0.99  0.56 
ko00920  Sulfur metabolism  f.Prevotellaceae  0.99  1.36 
ko00010  Glycolysis/Gluconeogenesis  o.Clostridiales  0.98  0.54 
Discussion
A close investigation of the biological significance of the associations identified by our model reveals several interesting characteristics and affirms the relevance of these associations. Commensal microbiota that inhabit the human gut are proficient at scavenging glycans and polysaccharides, including those in plants, such as starches or cellulose, animalderived tissues (glycosaminoglycans and Nlinked glycans), and glycans from host mucus (Olinked glycans) [46]. Ruminococcus spp. are known to participate in both resistant starch and glycosaminoglycan degradation [46, 47]. It has been reported that longterm consumption of diets rich in protein and animal fat were associated with an enterotype primarily containing increased Bacteroides and Ruminococcus species [19]. Additionally, Ruminococcus torques and Ruminococcus gnavus have been shown to degrade mucins [48]. Thus, it is logical that Ruminococcus, which is one of the noteworthy genera involved in glycosaminoglycan degradation, would be negatively associated to glycosaminoglycan biosynthesis (ko00534) (Table 3). Similarly, Parabacteroides which is also negatively associated with Nglycan biosynthesis (ko00513), is involved in deglycosylation and utilization of Nglycans [49]. Also, among the associations identified for the glycan pathways, Prevotella was negatively associated with mucin type Oglycan biosynthesis (ko00512). In the literature, Prevotella has implications for mucosal homeostasis, as some Prevotella spp. express a unique mucindesulfating glycosidase that can hydrolyze GlcNAc residues on mucintype Oglycans, and thus is important for mucin degradation [50]. Other associations affirmed through the literature included that of Bacteroides with naphthalene degradation (ko00626). It has been reported that Bacteroidetes possess the capability to degrade polycyclic aromatic hydrocarbons such as naphthalene [51]. Associations of Ruminococcus with pyruvate metabolism (ko00620) are also supported, as phosphoenolpyruvate carboxykinase was previously reported to be associated with Ruminococcus flavefaciens in the rumen [52]. Another supported association is that of Prevotellaceae with sulfur metabolism. Lcysteine desulfhydrase enzymes have been characterized in Prevotella intermedia [53]. Additionally, glycosulfatase enzymes have been described in Prevotella [54]. Equally interesting is the selection of pathways that are expected to be omnipresent among many bacteria, such as glycolysis/gluconeogenesis (ko00010), as glycolysis occurs, with variations, in nearly all organisms, both aerobic and anaerobic. Thus, it is not surprising that taxa like Clostridiales are positively associated with glycolysis/gluconeogenesis as they are abundant taxa within the gut microbiome.
Given the complexity of metabolic pathways and the process of mapping specific genes to pathways, some of the selected associations are unexpected, and might be due to the 16S abundances that were made available at the HMP site and the mapping of metagenomic sequences to specific KEGG orthology groups by HUMAnN. For example, several species of Ruminococcus are known to participate in butanoate (butyrate) metabolism [55], Dialister spp. have phenylalanine arylamidase activities [56], and Prevotella spp. are known to participate in pyruvate metabolism [57, 58]. Since those associations should be driven exclusively by bacterial genes, it is interesting that we find significant associations between the abundance of certain bacterial taxa and KEGG pathways that are primarily reported among eukaryotic species (i.e., Tcell receptor signaling, hedgehog signaling, pathways in cancer, etc.). Indeed, although precautionary steps are performed, the HMP consortium reported that human contaminants are found in 50–90% of the sequences [15]. This might also explain the negative association exhibited by Bacteriodes and glycolysis/gluconeogenesis. These unexpected findings suggest the need for further investigations and validation.
Conclusion
Herein, we have developed a Bayesian approach to the DirichletMultinomial regression models that allows for the selection of significant associations between covariates and taxa from a microbiome abundance table by imposing spikeandslab priors on the loglinear regression coefficients of the model. We have applied our model to simulated data and compared performances with methods developed for similar applications. We have illustrated the performance of our method using publicly available data on taxonomic abundances and metabolic pathways inferred from whole genome shotgun sequencing reads, which we obtained from the Human Microbiome Project website. Our results have revealed interesting links between specific taxa (i.e. genera) and particular metabolic pathways, which we have validated via existing literature.
Several extensions of our model are possible. Because some habitats, e.g. the gut, are thought to have highly variable dynamics, longitudinal sampling may be preferred to crosssectional sampling since it may give a better sense of longterm trends [59]. Thus, incorporating repeated samples with specified correlation structures in the linear predictor could produce additional insights. Another important aspect of microbiome data, which is receiving attention from researchers, is the heterogeneity in community structure across samples, as this can be an indication of the existence of “enterotypes” [60, 61]. This can be addressed within our modeling framework by employing Bayesian nonparametric models that would allow to cluster selected associations across partitions of the samples. These extensions are currently under investigation.
Notes
Abbreviations
 AIC:

Akaike information criterion
 ACC:

True positive
 AUC:

Area under the curve
 BIC:

Bayesian information criterion
 DM:

DirichletMultinomial
 FDR:

False discovery rate
 FN:

False negative
 FNR:

False negative rate
 FP:

False positive
 FPR:

False positive rate
 N:

Negative
 HMP:

Human Microbiome Project
 KEGG:

Kyoto encyclopedia of genes and genomes
 MAP:

Maximum a posteriori
 MCC:

Matthews correlation coefficient
 MCMC:

Markov Chain Monte Carlo
 OTU:

Operational taxonomic unit
 P:

Positive
 PPI:

Posterior probability of inclusion
 ROC:

Receiving operating curve
 TN:

True negative
 WGS:

Whole genome shotgun sequencing
Declarations
Acknowledgements
RA acknowledges the support received by CNRIMATI (Center of National Research  The Institute for Applied Mathematics and Information Technologies “Enrico Magenes”, Milano, Italy) to conduct this research.
Funding
WDW is supported by NIH Grant NCI T32 CA096520 at Rice University. Jessica GallowayPeña is supported by the Odyssey Program and CFP Foundation at The University of Texas MD Anderson Cancer Center. MG, JPG and SAS are been partially supported by the NIH/NCI award P30CA016672, MG used the Biostatistics Shared Resource.
Availability of data and materials
The datasets used in the study are included as Additional files 2 and 3, respectively. The code developed for this manuscript is publicly available on GitHub at https://github.com/duncanwadsworth/dmbvs, at http://www.micheleguindani.info and on http://www.stat.rice.edu/~marina/. An R package is under development and will be announced on those sites.
Authors’ contributions
WDW and RA have collaborated in the development of the algorithm and performed the statistical analyses, JGP and SAS have contributed to the interpretation of the biological findings, MG and MV have conceived the study and supervised the development of the algorithm and the statistical analyses. All authors have contributed to the writing of the manuscript. All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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