Variable selection for sparse Dirichlet-multinomial regression with an application to microbiome data analysis

“…are either mapped to an existing phylogenetic tree in a taxonomic dependent way [e.g., Mat-sen, Kodner and Armbrust (2010)] or clustered into operational taxonomic units (OTUs) at a certain similarity level in a taxonomic independent way [e.g., Caporaso et al. (2010), Schloss et al. (2009)].”

“…are either mapped to an existing phylogenetic tree in a taxonomic dependent way [e.g., Mat-sen, Kodner and Armbrust (2010)] or clustered into operational taxonomic units (OTUs) at a certain similarity level in a taxonomic independent way [e.g., Caporaso et al. (2010), Schloss et al. (2009)].”

“Details of the study and the data set can be found in Section 6 and in Wu et al. (2011).”

“The collective genomes of these microbes constitute an extended human genome that provides us with genetic and metabolic capabilities that we do not inherently possess [B¨ackhed et al. (2005)].”

“For example, we found that Naringenin (flavanone) was positively associated with Faecalibacterium, an anti-inflammatory commensal bacterium identified by gut microbiota analysis of Crohn’s disease patients [Sokol et al. (2008)].”

“en calculated and the signifi- cance is then evaluated by permutation test. Alternatively, one can define a distance between the samples and then use a PERMANOVA procedure to test for covariate effects [McArdle (2001)]. It is easy to show that when the distance is Euclidean, these two procedures are equivalent. In this p…”

“· Minimization of (11) with respective to d ¯ Sk can be performed in a similar fashion as in Meier, van de Geer and B¨uhlmann (2008) for the group ℓ 1 penalty.”

“Benson et al. (2010) demonstrated that genetic variants are associated with the mouse gut microbiome.”

“The sparse group ℓ 1 estimates of β can be obtained by minimizing the penalized negative log-likelihood function (8): Using the general block coordinate gradient descent algorithm of Tseng and Yun (2008), we develop in the following an efficient algorithm to solve this optimization problem.”

“The Dirichlet-multinomial (DM) distribution [Mosimann (1962)] results from a compound multinomial distribution with weights from the Dirichlet distribution (parametrization I): f DM ( y 1 , y 2 , . . . , y q ; γ ) = f M ( y 1 , y 2 , . . . , y q ; φ ) f D ( φ ; γ ) d φ Γ( y j + γ j ) Γ( γ j )Γ( y j…”

“To further assess the relevance of the nutrients selected, we used the bootstrap to analyze the stability of the selected nutrients [Bach (2008)].”

“Besides overdispersion, the taxa count data can also exhibit zero-inflation [Barry and Welsh (2002)], where the count data contain more zeros than expected from the DM model.”

“A similar penalty involving both group ℓ 1 and ℓ 1 terms is discussed in Peng et al. ( 2010) and Friedman, Hastie and Tibshirani (2010) for regularized multivariate linear regression.”

“Recent studies have linked the microbiome with human diseases including obesity and inflammatory bowel disease [Virgin and Todd (2011)].”

“The multilevel zero-inflated DM regression model for overdispersed count data with extra zeros [Moghimbeigi et al. (2008), Lee et al. (2006)] can potentially provide a solution to this problem.”

“data set using an R implementation of the algorithm (available at http://statgene.med.upenn.edu/). Besides the sparse l 1 group penalty, other group penalty functions such as the sup-norm penalty in Zhang et al. (2008) and the composite absolute penalties in Zhao, Rocha and Yu (2009) can also be used in the setup of t…”

“[50, 67, 68], we include those distributions in our simulation study for purely comparative purposes.”

“As typically done (13), to avoid sparsely observed OTUs,”

“Examples include a novel statistical regression framework for relating microbiome compositions and covariates in the context of nutrient intake [8], observations that microbiome compositions strongly correlate with disease status in new-onset Crohn’s disease [9], and the connections between helminth infection and the…”

“To evaluate the statistical power of rank sum metaanalysis method, we simulated microbial composition data following Dirichlet-multinomial distribution [46].”

“nd others, 2015; Sankaran and Holmes, 2017). Hierarchical models of absolute abundances are often used to constrain the number of parameters (e.g. La Rosa and others (2012); Holmes and others (2012); Chen and Li (2013); Sankaran and Holmes (2017); Cao and others (2017)). However, modeling the variance structure is cha…”

“MDiNE addresses compositionality by utilizing a Dirichlet-multinomial logistic-normal distribution model [61,62].”

“Specifically, we generate zero-inflated count data based on a Dirichlet-multinomial model with known Chen et al. (2018), PeerJ, DOI 10.7717/peerj.4600 5/20 library sizes (Chen & Li, 2013).”

“A semiparametric simulation framework Traditional simulations are usually based on parametric models such as Dirichlet-multinomial model [50, 51] and logistic normal multinomial model [41].”

“DCA has been applied to the functional analysis of genes (Cho et al., 2016) and drug-target interaction network (Luo et al., 2017).”

“Alternative methods that treat microbiome as dependent variables include sparse Dirichlet-Multinomial (DM) method (Chen and Li, 2013) and multivariate zero-inflated logistic-normal method (Li et al.”

“Several studies have recently performed simulations for microbiome data [25, 29, 30, 49, 50], most of which use negative binomial dis-”

“While both the multinomial Dirichlet model and the multinomial logistic normal model can handle over-dispersed count data 70,71 , the logistic normal model also allows for both positive and negative covariation between microbial features 69 .”

“02 obtained from these data, which is also the median value we observed in an admittedly brief survey of the literature (Chen and Li, 2013; La Rosa et al., 2012; Morgan et al., 2015).”

“For instance, in a sparse Dirichlet-multinomial regression model proposed by Chen and Li (2013), βk j being nonzero indicates association of predictor k with bacterial taxon j.”

“So far, compositional analysis has been an active and ongoing area in metagenomic data (Norouzi-Beirami et al., 2021) and microbiome research (Chen and Li, 2013; Bian et al., 2017; Rivera-Pinto et al., 2018), due to the compositional nature of metagenomic and microbiome data.”

“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.”

“This was quantitatively compared against alternative models (Dirichlet-multinomial, DM [20] and gamma-MGH, namely metaSPARSim [18]) with the PERMANOVA R 2 statistic [25] ( Meth-ods ).”

“Preliminary versions of the group lasso within the multinomial regression framework have been considered by Tutz (2012), Vincent and Hansen (2014), Chen and Li (2013) and Simon et al. (2013).”

“The observed overfitting, especially among lowly abundant orders, is not surprising since LOWESS and ML estimation do not take into account the statistical nature of count data.”

“Simulating OTU Abundance Data We generated the OTU counts using a Dirichlet-multinomial distribution with the parameters (the mean proportion vector and the dispersion parameter φ) estimated based on a real OTU dataset from a study of the microbiome of the human upper respiratory tract (Charlson et al., 2010; Chen and…”

“5 was added to zero-count data before calculating relative abundance and log transformation (Chen and Li, 2013).”

“To make feasible the comparison of ROC curves between our method and that of Chen and Li (2013), we can calculate the true positive rate and false positive rate in terms of how each method identifies the relevant covariates.”

“Multiple testing results in a loss of power in a high-dimensional setting and so regularization would help to increase the power [67].”

“For comparisons between H-FCR and L-FCR groups, we investigated only bacterial and archaeal profiles with a relative abundance > 0.1% prevalent in at least three samples (3 out 6) to avoid sparsely observed counts, which tend to introduce noise in the analysis (Chen and Li, 2013).”

“The second type directly models the compositional taxa counts using Dirichletmultinomial (DM) distribution (Chen and Li, 2013; La Rosa et al., 2012).”

“…example, the DM has proven useful for microbiome analysis in the areas of hypothesis testing and power calculations (La Rosa et al. 2012), sparse variable selection (Chen and Li 2013), inference of microbial community structure (Shafiei et al. 2015), and regression modeling (Wadsworth et al. 2017).”

“Other model-based methods (Chen and Li, 2013; Wadsworth and others , 2017) have been proposed to detect covariate-taxa associations, but the taxon-outcome associations haven been ignored from these models.”

“On the other hand, a joint microbiome selection approach usually models all taxa collectively using penalized regression (Chen and Li, 2013; Lin et al., 2014).”

“However, they are not suitable for microbiome data in which apparent overdispersion has been evidenced (Chen and Li, 2013; McMurdie and Holmes, 2014).”

“Regression models for 16 s sequencing have recently been developed and will allow for analysis of the multiple covariates in future studies [18,19].”

“One approach for joint modeling of multivariate microbial sequence count data is Dirichlet-multinomial regression (Holmes et al., 2012; La Rosa et al., 2012; Chen and Li, 2013; Wadsworth et al., 2017).”

“After combining the multinomial model with the Dirichlet model, the mixture model is a Dirichlet-multinomial model with the probability mass function (PMF) shown below: [23, 25].”

“Another method, MetaSPARSim [18], uses a gamma-multivariate hypergeometric (gamma-68 MHG) model, in which the gamma distribution models the biological variability of taxa counts, accounting for overdispersion, and the MHG distribution models technical variability originating from the sequencing…”

“The uniform distribution is a commonly used distribution to generate total reads [6].”

“The overall profile of the distribution of species and genera at 6 and at 18 months across the four randomised groups was tested with the Dirichlet-multinomial distribution for the normalised multivariate count data without log-transformation [24, 25], using the R package HMP.”

“We illustrate the usefulness of our proposed methodology by applying it to the data set reported by Wu et al. (2011) and previously studied by Chen and Li (2013) and Lin et al. (2014), where a cross-sectional study of 98 healthy volunteers was carried out to investigate the habitual diet effect on…”

“The grouping operation based on taxonomy is a common practice in recent papers including La Rosa et al. (2012) and Chen and Li (2013).”

“In biology, sequence data—for example, SAGE data, RNA-Seq and microbiome data [8, 17, 40]—are represented as a matrix of short sequence tags (eg taxonomic units in a microbiome experiment) and corresponding observed reads for several samples.”

“We model the OTU counts using Dirichlet-multinomial (DM) model to account for the overdispersion seen in real microbiome data [11].”

“To account for the excess variation, the Dirichlet-multinomial (DM) distribution is commonly used in practice [27, 28].”

“crobial count data, researchers have previously used Dirichlet-multinomial models, since these models can handle overdispersed data that arise from within- and between-subject variability in microbial data (La Rosa et al., 2012; Chen and Li, 2013; Wadsworth et al., 2017).”

“For these reasons, Dirichlet-Multinomial distributions are particularly appropriate to model microbiome data, as demonstrated by several authors (Chen and Li, 2013; Tang et al., 2018; Wadsworth et al., 2017).”

“When treating compositional data as response, a sparse Dirichlet-multinomial regression model was employed in Chen and Li (2013) to associate microbiome composition with environmental covariates.”

“Here, the exponentiation of a regression coefficient is interpreted as the multiplicative factor of change in the proportion of a compositional element with a one unit change in the corresponding standardized covariate while holding all else constant (Chen and Li, 2013).”

“…amplicons are compositional, sparse, and have complex distributions such as negative binomial (Robinson et al. , 2010; Love et al. , 2014), zero-inflated gaussian (Paulson et al. , 2013), and Dirichlet (Holmes et al. , 2012; Chen and Li, 2013; Wadsworth et al. , 2017; Harrison et al. , 2019).”

“[17] proposed an approach which assumes that multiple OTUs follow Dirichlet multinomial (DM) distribution.”

“An alternative test is based on the Dirichlet multinomial (DM) distribution to model the counts of sequence reads from microbiome samples (La Rosa et al., 2012; Chen and Li, 2013).”

“Instead of using a parametrical model such as the Dirichlet-multinomial model (Chen and Li, 2013), the microbiome data were directly resampled from a large gut microbiome study by Hale et al. (2017).”

“A limitation of our approach, which is shared by other analytical approaches such as [4,16,17], is the assumption that taxonomic assignment for all organisms at all phylogenetic levels is equally accurate.”

“It is shown in Appendix A that, in the case where the total count does not depend on the random effect ui, the likelihoods (10) and (12) are equivalent.”

“Indeed, in some contexts, linking metabolome to microbiome using models such as the Dirichlet regression could be achieved 36,37 .”

“Extensions such as the Sparse Dirichlet-multinomial regression technique [25] and finite mixtures of the Dirichlet-multinomial model [26,27] have been proposed to improve on different statistical aspects.”

“el (Zhang et al., 2017), zero-in ated negative binomial model (Zhang and Yi, 2020), zero-in ated Poisson model(Joseph et al., 2013; Xu et al., 2020), Dirichlet-multinomial model (Holmes et al., 2012; Chen and Li, 2013; Subedi et al., 2020), and the logistic normal multinomial model(Xia et al., 2013). While modelling s…”

“One simple attempt to alleviate this problem is to aggregate OTUs from the same genus and study their composition at the genus level as suggested by Chen and Li (2013).”

“The outlined limitation has prompted the introduction of joint models of microbial abundance (Chen and Li, 2013; Xia et al., 2013; Wadsworth et al., 2017; Grantham et al., 2019).”

“Microbiome datasets were simulated according to Chen and Li’s approach (Chen and Li, 2013).”

“In particular, the Dirichlet-multinomial model (Mosimann, 1962; Chen and Li, 2013), which imposes a Dirichlet distribution on the taxa composition, is computationally simple but entails correlation structures among the taxa that are too restrictive to be satisfied in practice.”

“By treating Zij as the true compositions, it is also overlooking the heteroskedasticity or overdispersion of Zij caused by enormous uncontrollable factors of variation in sequencing, e.g., time, sampling location, or technical variability (Chen and Li, 2013).”

“To allow for overdispersion, Dirichlet multinomial (DM) regression (Chen & Li, 2013) and logistic normal multinomial regression (Xia, Chen, Fung, & Li, 2013) were proposed for modeling microbiome data.”

“Few group penalized approaches have been developed for MLM except for a few work by [8], [36], and [43].”

“We select a log-linear regression model of the Dirichlet multinomial (DMN) count profiles on PEG MW as continuous, nonzero predictor.(32,33) The DMN distribution has been proposed as an extension to the multinomial for estimating proportions arising from count data in multiple categories, providing better handling of…”