The Annals of Applied Statistics

A phylogenetic scan test on a Dirichlet-tree multinomial model for microbiome data

Yunfan Tang, Li Ma, and Dan L. Nicolae

Full-text: Open access


In this paper, we introduce the phylogenetic scan test (PhyloScan) for investigating cross-group differences in microbiome compositions using the Dirichlet-tree multinomial (DTM) model. DTM models the microbiome data through a cascade of independent local DMs on the internal nodes of the phylogenetic tree. Each of the local DMs captures the count distributions of a certain number of operational taxonomic units at a given resolution. Since distributional differences tend to occur in clusters along evolutionary lineages, we design a scan statistic over the phylogenetic tree to allow nodes to borrow signal strength from their parents and children. We also derive a formula to bound the tail probability of the scan statistic, and verify its accuracy through simulations. The PhyloScan procedure is applied to the American Gut dataset to identify taxa associated with diet habits. Empirical studies performed on this dataset show that PhyloScan achieves higher testing power in most cases.

Article information

Ann. Appl. Stat. Volume 12, Number 1 (2018), 1-26.

Received: October 2016
Revised: March 2017
First available in Project Euclid: 9 March 2018

Permanent link to this document

Digital Object Identifier

Dirichlet-multinomial microbiome Dirichlet-tree multinomial phylogenetic tree PhyloScan scan statistics union probability


Tang, Yunfan; Ma, Li; Nicolae, Dan L. A phylogenetic scan test on a Dirichlet-tree multinomial model for microbiome data. Ann. Appl. Stat. 12 (2018), no. 1, 1--26. doi:10.1214/17-AOAS1086.

Export citation


  • Caporaso, J. G., Kuczynski, J., Stombaugh, J., Bittinger, K., Bushman, F. D., Costello, E. K., Fierer, N., Pena, A. G., Goodrich, J. K., Gordon, J. I. et al. (2010). QIIME allows analysis of high-throughput community sequencing data. Nat. Methods 7 335–336.
  • Chen, Y. and Hanson, T. E. (2014). Bayesian nonparametric $k$-sample tests for censored and uncensored data. Comput. Statist. Data Anal. 71 335–346.
  • Chen, J. and Li, H. (2013). Variable selection for sparse Dirichlet-multinomial regression with an application to microbiome data analysis. Ann. Appl. Stat. 7 418–442.
  • David, L. A., Maurice, C. F., Carmody, R. C., Gootenberg, D. B., Button, J. E., Wolfe, B. E., Ling, A. V., Devlin, A. S., Varma, Y., Fischbach, M. A. et al. (2014). Diet rapidly and reproducibly alters the human gut microbiome. Nature 505 559–563.
  • Dennis, S. Y. III (1991). On the hyper-Dirichlet type $1$ and hyper-Liouville distributions. Comm. Statist. Theory Methods 20 4069–4081.
  • Dohmen, K. (2000). Improved Bonferroni inequalities via union-closed set systems. J. Combin. Theory Ser. A 92 61–67.
  • Dohmen, K. (2002). Improved inclusion-exclusion identities and Bonferroni inequalities with reliability applications. SIAM J. Discrete Math. 16 156–171.
  • Dohmen, K. and Tittmann, P. (2004). Bonferroni–Galambos inequalities for partition lattices. Electron. J. Combin. 11 Article ID 85.
  • Efron, B. (1997). The length heuristic for simultaneous hypothesis tests. Biometrika 84 143–157.
  • Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics. Springer, New York.
  • Hahn, T. (2005). Cuba—A library for multidimensional numerical integration. Comput. Phys. Commun. 168 78–95.
  • Holmes, I., Harris, K. and Quince, C. (2012). Dirichlet multinomial mixtures: Generative models for microbial metagenomics. PLoS ONE 7 Article ID e30126.
  • Holmes, C. C., Caron, F., Griffin, J. E. and Stephens, D. A. (2015). Two-sample Bayesian nonparametric hypothesis testing. Bayesian Anal. 10 297–320.
  • Human Microbiome Project Consortium (2012). A framework for human microbiome research. Nature 486 215–221.
  • Hunter, D. (1976). An upper bound for the probability of a union. J. Appl. Probab. 13 597–603.
  • Lavine, M. (1992). Some aspects of Pólya tree distributions for statistical modelling. Ann. Statist. 20 1222–1235.
  • La Rosa, P. S., Brooks, J. P., Deych, E., Boone, E. L., Edwards, D. J., Wang, Q., Sodergren, E., Weinstock, G. and Shannon, W. D. (2012). Hypothesis testing and power calculations for taxonomic-based human microbiome data. PLoS ONE 7 Article ID e52078. DOI:10.1371/journal.pone.0052078.
  • Ma, L. and Wong, W. H. (2011). Coupling optional Pólya trees and the two sample problem. J. Amer. Statist. Assoc. 106 1553–1565.
  • McDonald, D., Birmingham, A. and Knight, R. (2015a). Context and the human microbiome. Microbiome 3 1–8.
  • McDonald, D., Hornig, M., Lozupone, C., Debelius, J., Gilbert, J. and Knight, R. (2015b). Towards large-cohort comparative studies to define the factors influencing the gut microbial community structure of ASD patients. Microb. Ecol. Health Dis. 26 26555.
  • Mosimann, J. E. (1962). On the compound multinomial distribution, the multivariate $\beta$-distribution, and correlations among proportions. Biometrika 49 65–82.
  • Naiman, D. Q. and Wynn, H. P. (1992). Inclusion-exclusion-Bonferroni identities and inequalities for discrete tube-like problems via Euler characteristics. Ann. Statist. 20 43–76.
  • Naiman, D. Q. and Wynn, H. P. (1997). Abstract tubes, improved inclusion-exclusion identities and inequalities and importance sampling. Ann. Statist. 25 1954–1983.
  • Neuman, H., Debelius, J. W., Knight, R. and Koren, O. (2015). Microbial endocrinology: The interplay between the microbiota and the endocrine system. FEMS Microbiol. Rev. 39 509–521.
  • Silverman, J. D., Washburne, A., Mukherjee, S. and David, L. A. (2017). A phylogenetic transform enhances analysis of compositional microbiota data. ELife 6 Article ID e21887.
  • Soriano, J. and Ma, L. (2017). Probabilistic multi-resolution scanning for two-sample differences. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 547–572.
  • Tang, Y., Ma, L. and Nicolae, D. L. (2018). Supplement to “A phylogenetic scan test on a Dirichlet-tree multinomial model for microbiome data.” DOI:10.1214/17-AOAS1086SUPP.
  • Tang, Z., Chen, G., Alekseyenko, A. V. and Li, H. (2017). A general framework for association analysis of microbial communities on a taxonomic tree. Bioinformatics 33 1278–1285. DOI:10.1093/bioinformatics/btw804.
  • Taylor, J. E., Worsley, K. J. and Gosselin, F. (2007). Maxima of discretely sampled random fields, with an application to ‘bubbles’. Biometrika 94 1–18.
  • Turnbaugh, P. J., Ridaura, V. K., Faith, J. J., Rey, F. E., Knight, R. and Gordon, J. I. (2014). The effect of diet on the human gut microbiome: A metagenomic analysis in humanized gnotobiotic mice. Sci. Transl. Med. 1 Article ID 6ra14. DOI:10.1126/scitranslmed.3000322.
  • Wang, T. and Zhao, H. (2017). A Dirichlet-tree multinomial regression model for associating dietary nutrients with gut microorganisms. Biometrics 73 792–801.
  • Weir, B. S. and Hill, W. G. (2002). Estimating F-statistics. Annu. Rev. Genet. 36 721–750.
  • Worsley, K. J. (1982). An improved Bonferroni inequality and applications. Biometrika 69 297–302.

Supplemental materials

  • Theorem proofs. This supplementary file contains proofs of independence of DTM p-value under the global null (Theorem 1) and error bound of PhyloScan statistic approximation (Theorem 2).