 Methodology
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A distance based multisample test for highdimensional compositional data with applications to the human microbiome
BMC Bioinformatics volume 21, Article number: 205 (2020)
Abstract
Background
Compositional data refer to the data that lie on a simplex, which are common in many scientific domains such as genomics, geology and economics. As the components in a composition must sum to one, traditional tests based on unconstrained data become inappropriate, and new statistical methods are needed to analyze this special type of data.
Results
In this paper, we consider a general problem of testing for the compositional difference between K populations. Motivated by microbiome and metagenomics studies, where the data are often overdispersed and highdimensional, we formulate a wellposed hypothesis from a Bayesian point of view and suggest a nonparametric test based on interpoint distance to evaluate statistical significance. Unlike most existing tests for compositional data, our method does not rely on any data transformation, sparsity assumption or regularity conditions on the covariance matrix, but directly analyzes the compositions. Simulated data and two real data sets on the human microbiome are used to illustrate the promise of our method.
Conclusions
Our simulation studies and real data applications demonstrate that the proposed test is more sensitive to the compositional difference than the meanbased method, especially when the data are overdispersed or zeroinflated. The proposed test is easy to implement and computationally efficient, facilitating its application to largescale datasets.
Background
Data that lie on the simplex \(\mathcal {S}^{d1}=\big \{ (x_{1}, x_{2},..., x_{d}), s.t. \min _{j}x_{j}\geq 0, \sum _{j=1}^{d}x_{j}=1 \big \}\) are often called (d−1)dimensional compositional data, and they arise in many scientific disciplines such as genomics, geology and economics [1–3]. As the components in a composition must sum to one, classic statistical tests including twosample ttest and Wilcoxon ranksum test become inappropriate as they require unconstrained data, and directly applying these standard methods to compositional data could result in misleading inference [4]. To overcome this difficulty, Aitchison (1982) proposed to use a logratio transformation to relax the unitsum constraint, so that some classic tests can be applied to the transformed data. For instance, the generalized likelihood ratio test based on logratios [4] has been widely used to test compositional difference between groups due to its simplicity and good empirical performance.
It is noteworthy that Aitchison’s test only applies to lowdimensional settings where the dimension is less than sample size. In recent microbiome and metagenomic studies, however, the compositional data are often highdimensional. For instance, in the Human Microbiome Project, it is common to have hundreds to thousands of bacterial taxa while only tens of samples are available for analysis. To this end, Cao et al. (2017) developed a powerful twosample test for highdimensional means using a centered logratio transformation [3]. Cao et al.’s test achieves satisfactory statistical power under highdimensional sparse settings, and the consistency of the test has been well established under some regularity conditions. Nevertheless, this test has several shortcomings which has limited its application in practice. For instance, Cao et al.’s test can only deal with twosample comparison, and its validity depends on a list of regularity conditions on the underlying covariance matrices. In addition, this test is a maximumtype test, and its performance relies on the sparsity assumption, i.e., only a small proportion of components in the composition are different across groups.
In this paper, we formulated a new hypothesis from a Bayesian point of view, to handle highdimensionality and overdispersion that are commonly seen in recent microbiome data. Different from those meanbased hypotheses, we assumed that the abundances follow a multinomial model with random composition parameters, and redefined the compositional equivalency using the distribution of random compositions. To directly target the distributional difference in composition, we suggested a distance based nonparametric test for detecting the difference between multiple groups. Unlike most existing tests for compositional data, our method does not rely on any data transformation, sparsity assumption or regularity conditions on the covariance matrix, but directly analyzes the compositions. Simulation studies demonstrated that our test is more sensitive to the compositional difference than the meanbased method, especially when the data are overdispersed or zeroinflated. The proposed test is easy to implement and computationally efficient, facilitating its application to largescale datasets.
The remainder of the paper is structured as follows: “Methods” section formulates the hypothesis testing and introduces our distance based method. “Results” section compares the proposed method with a logratio based test using simulated data from negative binomial models. In addition, we apply the new method to two real datasets on human throat microbiome and intestinal microbiome. “Discussion” section discusses our method with some future perspectives. “Conclusions” section concludes the paper.
Methods
Problem formulation
In this part, we briefly reviewed the test by Cao et al. (2017), and then formulated our new hypothesis. We begin with the notations. Let k∈{1,2,...,K} be the group index and j∈{1,2,...,p} be the index of components in the composition. Denote by \(\pmb {X}^{(k)}=\left (\pmb {X}^{(k)}_{1},..., \pmb {X}^{(k)}_{n_{k}}\right)^{T}\) the observed n_{k}×p data matrix for group k, where \(\pmb {X}^{(k)}_{i}=\left (X_{i1}^{(k)},..., X_{ip}^{(k)}\right)^{T}\) represents the composition for subject i that lie on the (p−1)dimensional simplex. We assume that the observed compositional data X^{(k)} arise from a latent matrix \(\pmb {W}^{(k)}=\left (\pmb {W}^{(k)}_{1},..., \pmb {W}^{(k)}_{n_{k}}\right)^{T} \left (\pmb {W}^{(k)}_{1},..., \pmb {W}^{(k)}_{n_{k}} \text {are iid samples}\right)\) by normalization
where the unobserved W^{(k)} may refer to the true abundance of bacterial taxa for microbiome data.
As the true abundances W^{(k)} are generally unknown, Cao et al. (2017) formulated a testable hypothesis for two groups:
where 1_{p} stands for the vector of p ones. The hypothesis above is meanbased and it can be tested through a centered logratio transformation
It can be shown that the centered logratio variables \(Y_{ij}^{(k)}\)’s are only weakly dependent and satisfy certain desired properties, and Cao et al. (2017) suggested the following test statistics based on these logratio variables
where \(\bar {Y}_{j}^{(k)}=\sum _{i=1}^{n_{k}}Y_{ij}^{(k)}/n_{k}, \hat {\gamma }_{jj}=\sum _{k=1}^{2}\sum _{i=1}^{n_{k}}\left (Y^{(k)}_{ij}\bar {Y}_{j}^{(k)}\right)^{2}/(n_{1}+n_{2})\). The pvalue can be then obtained through Gumbel distribution
Cao et al.’s test targets the difference in highdimensional means, and its validity relies on several assumptions on the underlying covariance matrices, which are impractical to check in reality. Here, we considered a different hypothesis based on the distribution of composition instead of means. Under multinomial model, we have
where \(N^{(k)}_{i}\) stands for the total abundance of bacterial taxa for sample i from group k, and \(\pmb {\pi }^{(k)}_{i}\) represents the true composition. In order to model overdispersion, we assumed random parameters, \(N^{(k)}_{i}\sim f_{N}(\alpha)\) and \(\pmb {\pi }^{(k)}_{i}=\left (\pi ^{(k)}_{i1},..., \pi ^{(k)}_{ip})\sim f_{\pi }(\pmb {\Theta }^{(k)}\right)\), where α and Θ^{(k)} are hyperparameters. We then define the compositional equivalence between two groups based on the distribution of parameter π:
Definition 1
Two groups k and k^{′} are said to be compositionally equivalent if \(\phantom {\dot {i}\!}f_{\pi }\left (\pmb {\Theta }^{(k)}\right)=f_{\pi }\left (\pmb {\Theta }^{(k')}\right)\).
By Definition 1, we formulate the null and alternative hypotheses for K groups
Throughout this paper, we assume that the total abundance or sequencing depth \(N^{(k)}_{i}\) is independent of \(\pmb {\pi }^{(k)}_{i}\), and \(N^{(k)}_{i}\sim f_{N}(\alpha)\) for i∈{1,...,n_{k}} and k∈{1,2,...,K}, therefore testing H_{0} amounts to testing the distributional equality of the compositions between K groups. Let \(f^{(k)}_{\pmb {X}}(\pmb {x})\) be the density function of \(\pmb {X}^{(k)}_{i}\), one can test the following equivalent hypothesis
Here, it is noteworthy that \(H^{*}_{0}\) is equivalent to the independence between the composition X and the grouping variable k∈{1,2,...,K} (i.e., phenotype), which converts the problem to testing the independence between the continuous random vector and a categorical variable.
Distance based test
In this part, we proposed a distance based method to test \(H^{*}_{0}\), i.e., to detect the association between composition and phenotype. To begin with, we briefly introduce the notion of distance covariance. The distance covariance between two random vectors X and Y (can be of different sizes and different types) is defined as the square root of
where ϕ(·) represents a characteristic function, d_{x} and d_{y} are the dimensions of X and Y, \(c_{d_{x}}=\frac {\pi ^{(1+d_{x})/2}}{\Gamma \{(1+d_{x})/2\}}\) and \(c_{d_{y}}=\frac {\pi ^{(1+d_{y})/2}}{\Gamma \{(1+d_{y})/2\}}\). Unless otherwise specified, \(\\pmb {z}\_{d_{z}}\) denotes the Euclidean norm of \(\pmb {z}\in \mathbb {R}^{d_{z}}\), and \(\\phi \^{2} = \phi \bar {\phi }\) for a complexvalued function ϕ and its conjugate \(\bar {\phi }\).
One remarkable property of distance covariance is that dCov(X, Y)=0 if and only if X and Y are statistically independent, indicating that the distance covariance can also capture nonlinear associations. In their seminal work, Szekely et al. (2007) also provided the following alternative definition of distance covariance based on Euclidean distance and established its equivalency to the original definition in Eq. (1) (see Theorem 1, [5]):
where (X_{1},Y_{1}), (X_{2},Y_{2}) and (X_{3},Y_{3}) be three independent copies of (X,Y). Here, we choose to use this alternative definition to derive the explicit formula of distance covariance between composition and phenotype. For ease of notations, let Y be the phenotype, taking values from a discrete set {1,2,...,K} with probabilities {p_{1},...,p_{K}}, and X={X_{1},...,X_{p}} be the composition. For illustration purpose, here we assume Y is nominal (without ordering between categories), however, our test can be easily extended to ordinal Y and the formula is given in the “Discussion” section. Let (X_{1},Y_{1}), (X_{2},Y_{2}) and (X_{3},Y_{3}) be three independent copies of (X,Y), we define ∥Y_{1}−Y_{2}∥=1, if Y_{1}≠Y_{2} and 0 otherwise. In addition, we define expected interpoint distance as
The distance covariance between Y and X can then be derived from the second definition
Summarizing the results above, we have
By CauchySchwarz inequality, it can be shown that dCov(X,Y)≥0 and the equality holds if and only if \(\mathcal {D}_{ii}=\mathcal {D}_{jj}=\mathcal {D}_{ij}\) for all (i,j)’s. When K=2, we have the following special case
The sample version of dCov(X,Y) can be expressed as
Let n_{i} be the sample size in group i, the maximum likelihood estimate of p_{i} is \(\hat {p}_{i}=n_{i}/n\), and the sample interpoint distance can be computed as follows:
where \(\left \{\pmb {X}^{(i)}_{1},...,\pmb {X}^{(i)}_{n_{i}}\right \}\) and \(\left \{\pmb {X}^{(j)}_{1},...,\pmb {X}^{(j)}_{n_{j}}\right \}\) stand for samples of X_{i} and X_{j}, respectively.
As the distribution of sample distance covariance is impractical to evaluate [5], we suggest a simple permutation procedure to obtain pvalues. In practice, one can randomly shuffle the vector of Y for M times, and calculate sample distance covariance between composition and the permuted Y, then the permutation pvalue can be computed as the proportion of distance covariance from permuted data that exceed the observed one.
It is noteworthy that in addition to distance correlation, there are many other dependence measures that could be used in our framework, including the energydivergence metric [6], multiscale graph correlation [7] and projection correlation [8], among others. One may refer to Josse and Holmes (2014) [9] for a general review of existing dependence measures between random vectors, and Szekely and Rizzo (2013) [10] for a review of energy and distancebased measures.
Results
Numerical study
We conduct three simulation studies to compare the distance based test and the logratio based test [3] in detecting the compositional differences between groups. In the first study, we focus on twogroup comparison under various highdimensional and overdispersed models. The dimension is fixed at p=200 and two different sample sizes n_{1}=n_{2}=50 and n_{1}=n_{2}=100 are used. The abundance \(W_{ij}^{(k)}\) are generated from three different settings

Setting 1: \(W_{ij}^{(k)} \sim \text {NegBin}\left (\mu ^{(k)}_{j}, r^{(k)}_{j}\right)\), i=1,...,n_{i}, j=1,...,p, \(r^{(1)}_{j} \sim \text {Unif}(0.1, 1), r^{(2)}_{j}=r^{(1)}_{j}, \mu ^{(1)}_{j} \sim \text {Unif}(10, 15)\). Let I={I_{+},I_{−}} be the set of taxa with different abundances in two conditions, \(\mu ^{(2)}_{j}=\mu ^{(1)}_{j}+\Delta \) for j∈I_{+} and \(\mu ^{(2)}_{j}=\mu ^{(1)}_{j}\Delta \) for j∈I_{−}, \(\mu ^{(2)}_{j}=\mu ^{(1)}_{j}\) for j∉I, I_{+}=I_{−}=dp, where · represents set cardinality, and d is the proportion of differential means. We chose d=5%,20%, representing relatively sparse and dense signals in mean difference, and used Δ={0.5,1.0,1.5,2.0,2.5,3.0}.

Setting 2: Same as Setting 1, but \(\mu ^{(1)}_{j} \sim \text {Unif}(5, 10)\).

Setting 3 (Negative binomial model with excess zeros): \(W_{ij}^{(k)}=0\) with probability π (π=10%,20%), and \(W_{ij}^{(k)} \sim \text {NegBin}\left (\mu ^{(k)}_{j}, r^{(k)}_{j}\right)\) with probability 1−π. Other settings are same as in Setting 1, and we used d=10%, Δ={0.5,1.0,1.5,2.0,2.5} in the simulation.
We compute the composition \(X_{ij}^{(k)}\) by normalizing the abundance \(W_{ij}^{(k)}\), and test the null hypothesis using compositional data at the level of 0.05. For Cao et al.’s test, we calculate the test statistics \(\mathcal {M}_{n}\) and directly compute the pvalue using Gumbel distribution. For our distance correlation test, pvalue is computed based on 5,000 permutations.
For each setting, we simulate 1,000 datasets, and compare the true positive rates (TPRs) by the two tests. Figures 1, 2 and 3 summarize the TPRs under three settings. It can be seen that our distance based test consistently outperforms the logratio based method in all settings. Particularly, in the dense setting (d=20%), our test achieves substantially higher TPR than the logratio test. For instance, in Setting 1, when Δ=2.0, n_{1}=n_{2}=50, our test achieves a high TPR of 0.97 while the TPR by logratio test is only 0.41. However, when Δ is subtle, e.g., Δ=0.50, both tests fail to detect the difference, even for relatively large sample size, e.g., n_{1}=n_{2}=100.
In the second simulation study, we investigate the effect of dimension on the true positive rate. The sample size is fixed at n_{1}=n_{2}=100, and the dimension p is varied from 100 to 500. The abundance \(W_{ij}^{(k)}\) are generated from two different settings (similar to settings 1 and 3)

Setting 4: \(W_{ij}^{(k)} \sim \text {NegBin}\left (\mu ^{(k)}_{j}, r^{(k)}_{j}\right)\), i=1,...,n_{i}, j=1,...,p, \(r^{(1)}_{j} \sim \text {Unif}(0.1, 1), r^{(2)}_{j}=r^{(1)}_{j}, \mu ^{(1)}_{j} \sim \text {Unif}(10, 15)\). Let I={I_{+},I_{−}} be the set of taxa with different abundances in two conditions, \(\mu ^{(2)}_{j}=\mu ^{(1)}_{j}+1.5\) for j∈I_{+} and \(\mu ^{(2)}_{j}=\mu ^{(1)}_{j}1.5\) for j∈I_{−}, \(\mu ^{(2)}_{j}=\mu ^{(1)}_{j}\) for j∉I, I_{+}=I_{−}=10, where · represents set cardinality.

Setting 5 (Negative binomial model with excess zeros): \(W_{ij}^{(k)}=0\) with probability π=10%, and \(W_{ij}^{(k)} \sim \text {NegBin}\left (\mu ^{(k)}_{j}, r^{(k)}_{j}\right)\) with probability 1−π. Other settings are same as in Setting 4.
Figure 4 summarizes the TPRs by the two tests based on 1,000 replicates and significance level α=0.05. It can be seen that the distance based test outperforms the logratio test especially when the dimension is relatively low. When the dimension is high, for instance p=500, the two tests are comparable. More importantly, there is a substantial decrease of TPR as p increases, indicating that a feature screening could improve the test performance when p is large.
In the third study, we consider testing the compositional difference between multiple groups. We set K=4 with sample sizes n_{1}=n_{2}=n_{3}=n_{4}=50. The dimension p is fixed at 200. The abundance \(W_{ij}^{(k)}\) are generated from the negative binomial model with excess zeros. Let \(\pi =P\left (W_{ij}^{(k)}=0\right)\), with probability 1−π, \(W_{ij}^{(k)} \sim \text {NegBin}\left (\mu ^{(k)}_{j}, r^{(k)}_{j}\right)\), i=1,...,n_{i}, j=1,...,p, \(r^{(1)}_{j} \sim \text {Unif}(0.1, 1), r^{(3)}_{j} \sim \text {Unif}(0.1, 1), r^{(2)}_{j}=r^{(1)}_{j}, r^{(4)}_{j}=r^{(3)}_{j}, \mu ^{(1)}_{j} \sim \text {Unif}(10, 15), \mu ^{(3)}_{j} \sim \text {Unif}(10, 15)\). Let I={I_{+},I_{−}} be the set of taxa with different abundances in four conditions, \(\mu ^{(2)}_{j}=\mu ^{(1)}_{j}+\Delta \) and \(\mu ^{(4)}_{j}=\mu ^{(3)}_{j}+\Delta \) for j∈I_{+}, \(\mu ^{(2)}_{j}=\mu ^{(1)}_{j}\Delta \) and \(\mu ^{(4)}_{j}=\mu ^{(3)}_{j}\Delta \) for j∈I_{−}, \(\mu ^{(1)}_{j}=\mu ^{(2)}_{j}=\mu ^{(3)}_{j}=\mu ^{(4)}_{j}\) for j∉I, I_{+}=I_{−}=20, where · represents set cardinality. We use Δ={0.5,1.0,1.5,2.0,2.5} and π={10%,20%} in the simulation.
For the distance correlation test, pvalue is computed based on 5,000 permutations. For Cao et al.’s test, we calculate the pvalues by Gumbel distribution from six pairwise comparisons, and use the smallest pvalue for decisionmaking. Figure 5 summarizes the TPRs by the two tests, where it can be seen that our proposed test performs consistently better than the logratio based test. Notably, in the setting π=20% and Δ=2.0, the distance correlation test achieves a TPR of 0.83, compared to the TPR of 0.46 by the logratio test.
Two microbiome applications
Analysis of throat microbiome data
In this part, we use the proposed hypothesis and distance based test to reanalyze a throat microbiome dataset. Cigarette smokers have an increased risk of infectious diseases involving the respiratory tract, however, the consequences for global airway microbial community composition remains unclear. Charlson et al. (2010) used cultureindependent highdensity sequencing to analyze the microbiota from the right and left nasopharynx and oropharynx of 29 smoking and 33 nonsmoking healthy adults to assess microbial composition and effects of cigarette smoking [11]. Bacterial communities were profiled using 454 pyrosequencing of 16S sequence tags, aligned to 16S rRNA databases.
We are interested in whether there is any significant difference in microbial compositions between smokers and nonsmokers. The processed data (observed abundance) were downloaded from R package GUniFrac [12], which included the read counts of 856 predefined operational taxonomic units (OTUs, also called phylotypes) on 62 samples. We first deleted OTUs with extremely small number of reads (less than 20 reads in total), resulting a final set of 190 OTUs.
Two methods, including the logratio based test and distance based test, are applied to the compositional data. The proposed distance correlation test is implemented in the following steps

Step 1: Compute the composition X for each sample by normalizing the abundance W.

Step 2: Calculate the sample proportions \(\hat {p_{i}}\), and the intergroup distances \(\hat {\mathcal {D}}_{ij}\) and \(\hat {\mathcal {D}}_{ii}\), i,j=1,...,K (e.g., Euclidean distance) using Eqs. (2) and (3).

Step 3: Compute the permutation pvalue based on \(\widehat {\text {dCov}(\pmb {X}, Y)}\).
The proposed test yields a pvalue of 0.0027, indicating a significant difference between smokers and nonsmokers in microbial composition. In contrast, the test based on logratio transformation gives a pvalue of 0.098, thus fails to reject the null hypothesis of equal means at the level of 0.05.
The disagreement between the two tests may indicate the existence of nonlinear effects and overdispersion, because the logratio test only targets the mean difference while our test targets the distributional difference. We illustrate this point by carrying out additional analyses. Figure 6 gives two examples (bacteria 2434 and bacteria 2831), where the centered logratios exhibite substantially different distributions. However, the mean difference is not significant due to the nonlinear effect and heavy tails, which inflates the variance estimates in Cao et al.’s test.
We also compare the distributions of interpoint distance within smokers and nonsmokers. Szekely et al. (2007) illustrated that if two multivariate distributions are identical, the interpoint distances within each group have the same distribution. Figure 7 showed the interpoint distance distributions of two groups, where a substantial discrepancy was observed. Furthermore, we used the 3minimum spanning tree (3MST), a treebased visualization method, to confirm our findings. Figure 8 shows the 3MST based on the compositional data, where a connection in the network represents compositional similarity between two samples. In theory, if the two groups have the same distribution, then each sample has equal chance to connect with any other sample, regardless of which group it is from. However, it can be seen that certain samples from the same group formed clusters in the network. For instance, we identified a set of 12 smokers (circled) that are highly connected each other, but with very few connections with nonsmokers, indicating a distributional difference in composition between the two groups.
Analysis of intestinal microbiome data
The microbial communities living in the human intestine have profound impact on our wellbeing and health. To understand the mechanisms that control this complex ecosystem, Lahti et al. (2014) conducted a deep phylogenetic analysis of the intestinal microbiota in 1,006 western adults from Europe and the United States [13]. The analysis is based on 130 genuslike phylogenetic groups that cover the majority of the known bacterial diversity of the human intestine. Clinical variables include age, nationality, BMI and DNA extraction method etc.
One of the key research questions is whether different age groups have different microbiome compositions. We use the cutoffs suggested by Lahti et al. (2014) to define three age groups: young (18–40), middleaged (41–60) and older (61–77). The distance test yields an overall pvalue of 3.0×10^{−6}. In addition, we calculate the pvalues for the three pairwise comparisons: 8.2×10^{−5} for young vs middle aged, 2.2×10^{−5} for young vs older, and 0.081 for middleaged vs older, indicating a significant difference in microbiome compositions between young and middleaged/older subjects, but a minor difference between middleaged and older subjects. To confirm this finding, we identified a list of microbiome groups with different distributions between age groups. Figures 9 and 10 show two examples of these, including group 25 and group 60. The distribution of interpoint distance within each age group is given in Fig. 11, where a discrepancy can be observed between young and middleaged/older subjects.
Discussion
Microbiome data are often compositional, highdimensional and overdispersed, which poses great challenges to the statistical analysis. To overcome these obstacles, in this work, we formulated a new testable hypothesis from a Bayesian point of view, and suggested a nonparametric test to detect the compositional difference between multiple populations. Compared to the existing tests, our method has several advantages. First, the distance based test is free of parametric assumptions but directly targets the distributional difference, therefore it is capable of detecting nonlinear effects. The application in throat microbiome provided a good example, where the new test successfully captured the difference between two phenotypes, while the mean based test failed to do so. In addition, our method can deal with multiple groups, while most of existing methods are only for twogroup comparison. Third, our test does not require sparsity assumption on the mean differences as in Cao et al.’s test, and in our simulation study, the new test worked quite well against both sparse and relatively dense alternatives.
There are several possible extensions of the proposed test. First, the distance based method can be readily extended to ordinal phenotypes (or conditions), although we have been using nominal phenotypes for illustrative purpose. For ordinal phenotype, Y∈{1,2,...,K}, where there is a natural ordering 1<2...<K, (e.g., {mild, moderate, severe} for severity of a disease, {I, II, III, IV} for cancer stage, or {nonsmoking, light smoking, heavy smoking} for smoking status), we need predefine the distance matrix between categories i and j, for instance, d_{ij}=i−j, or d_{ij}=i−j^{2}. The distance covariance between composition X and ordinal phenotype Y has the following expression
and one may use the same permutation procedure to obtain pvalues. In practice, the distance matrix d_{ij} should be carefully chosen to reflect the true spacings between categories. An inappropriate choice of d_{ij} may result in misleading conclusions. Second, our test might be improved by incorporating more information about bacteria taxa. For instance, one can assign different weights for different bacterial taxa based on their position in the polygenetic tree [14], and use weighted Euclidean distance to construct the test statistic.
In addition to the microbiome application that we illustrated in this paper, the proposed test can be readily applied to several other fields. For instance, the market share data in economics are compositional and often highdimensional [15]. One may apply our test to detect the market share difference between multiple countries. In geology, it is often of interest to study the compositions of species in sediment [16] and it is possible to apply our test to detect the difference in species compositions between multiple locations.
Conclusions
We formulate a Bayesian testing framework to identify the compositional differences between multiple populations. In addition, we propose to use the distance correlation measure to test the null hypothesis. Simulation studies and two real applications in the human microbiome demonstrate that our test is more sensitive to the compositional difference than the meanbased method, especially when the data are overdispersed or zeroinflated. The proposed test is easy to implement and computationally efficient, facilitating its application to largescale datasets.
Availability of data and materials
The throat microbiome data can be downloaded from R package GUniFrac at https://cran.rproject.org/web/packages/GUniFrac/index.html. The intestinal microbiome data can be downloaded at https://datadryad.org/stash/dataset/doi:10.5061/dryad.pk75d.
Abbreviations
 dCov:

Distance covariance
 OTU:

Operational taxonomic unit
 MST:

Minimum spanning tree
 TPR:

True positive rate
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Acknowledgements
QZ’s research is supported in part by the Arkansas Biosciences Institute, the major research component of the Arkansas Tobacco Settlement Proceeds Act of 2000.
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This article has been published as part of BMC Bioinformatics Volume 21 Supplement 9, 2020: Selected Articles from the 20th International Conference on Bioinformatics & Computational Biology (BIOCOMP 2019). The full contents of the supplement are available online at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume21supplement9.
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This work and publication costs are funded by the Arkansas Biosciences Institute (ABIUAF17). The funder had no role in data analysis, decision to publish, or preparation of the manuscript.
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QZ conceived of the presented idea and developed the theory. QZ and TD performed the computations and wrote the manuscript. Both authors read and approved of the final manuscript.
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Zhang, Q., Dao, T. A distance based multisample test for highdimensional compositional data with applications to the human microbiome. BMC Bioinformatics 21 (Suppl 9), 205 (2020). https://doi.org/10.1186/s128590203530x
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DOI: https://doi.org/10.1186/s128590203530x