The Annals of Applied Statistics

Kernel-penalized regression for analysis of microbiome data

Timothy W. Randolph, Sen Zhao, Wade Copeland, Meredith Hullar, and Ali Shojaie

Full-text: Open access


The analysis of human microbiome data is often based on dimension-reduced graphical displays and clusterings derived from vectors of microbial abundances in each sample. Common to these ordination methods is the use of biologically motivated definitions of similarity. Principal coordinate analysis, in particular, is often performed using ecologically defined distances, allowing analyses to incorporate context-dependent, non-Euclidean structure. In this paper, we go beyond dimension-reduced ordination methods and describe a framework of high-dimensional regression models that extends these distance-based methods. In particular, we use kernel-based methods to show how to incorporate a variety of extrinsic information, such as phylogeny, into penalized regression models that estimate taxon-specific associations with a phenotype or clinical outcome. Further, we show how this regression framework can be used to address the compositional nature of multivariate predictors comprised of relative abundances; that is, vectors whose entries sum to a constant. We illustrate this approach with several simulations using data from two recent studies on gut and vaginal microbiomes. We conclude with an application to our own data, where we also incorporate a significance test for the estimated coefficients that represent associations between microbial abundance and a percent fat.

Article information

Ann. Appl. Stat., Volume 12, Number 1 (2018), 540-566.

Received: January 2017
Revised: May 2017
First available in Project Euclid: 9 March 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Compositional data distance-based analysis kernel methods microbial community data penalized regression


Randolph, Timothy W.; Zhao, Sen; Copeland, Wade; Hullar, Meredith; Shojaie, Ali. Kernel-penalized regression for analysis of microbiome data. Ann. Appl. Stat. 12 (2018), no. 1, 540--566. doi:10.1214/17-AOAS1102.

Export citation


  • Aitchison, J. (1982). The statistical analysis of compositional data. J. Roy. Statist. Soc. Ser. B 44 139–177. With discussion.
  • Aitchison, J. (2003a). A concise guide to compositional data analysis. In 2nd Compositional Data Analysis Workshop.
  • Aitchison, J. (2003b). The Statistical Analysis of Compositional Data. The Blackburn Press, Caldwell, NJ.
  • Anderson, M. J. (2006). Distance-based tests for homogeneity of multivariate dispersions. Biometrics 62 245–253, 320.
  • Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Ann. Statist. 29 1165–1188.
  • Bühlmann, P., Kalisch, M. and Meier, L. (2014). High-dimensional statistics with a view toward applications in biology. Annu. Rev. Statist. Appl. 1 255–278.
  • Cai, T. T. and Hall, P. (2006). Prediction in functional linear regression. Ann. Statist. 34 2159–2179.
  • Chen, J., Bittinger, K., Charlson, E. S., Hoffmann, C., Lewis, J., Wu, G. D., Collman, R. G., Bushman, F. D. and Li, H. (2012). Associating microbiome composition with environmental covariates using generalized UniFrac distances. Bioinformatics 28 2106–2113.
  • Claesson, M. J., Jeffery, I. B., Conde, S., Power, S. E., O’Connor, E. M., Cusack, S., Harris, H. M., Coakley, M., Lakshminarayanan, B., O’Sullivan, O., Fitzgerald, G. F., Deane, J., O’Connor, M., Harnedy, N., O’Connor, K., O’Mahony, D., van Sinderen, D., Wallace, M., Brennan, L., Stanton, C., Marchesi, J. R., Fitzgerald, A. P., Shanahan, F., Hill, C., Ross, R. P. and O’Toole, P. W. (2012). Gut microbiota composition correlates with diet and health in the elderly. Nature 488 178–184.
  • Egozcue, J. J. and Pawlowsky-Glahn, V. (2011). Basic concepts and procedures. In Compositional Data Analysis: Theory and Applications (V. Pawlowsky-Glahn and A. Buccianti, eds.) 12–28. Wiley, Chichester.
  • Evans, S. N. and Matsen, F. A. (2012). The phylogenetic Kantorovich–Rubinstein metric for environmental sequence samples. J. R. Stat. Soc. Ser. B. Stat. Methodol. 74 569–592.
  • Franklin, J. N. (1978). Minimum principles for ill-posed problems. SIAM J. Math. Anal. 9 638–650.
  • Freytag, S., Manitz, J., Schlather, M., Kneib, T., Amos, C. I., Risch, A., Chang-Claude, J., Heinrich, J. and Bickeböller, H. (2013). A network-based kernel machine test for the identification of risk pathways in genome-wide association studies. Hum. Hered. 76 64–75.
  • Friedman, J. and Alm, E. J. (2012). Inferring correlation networks from genomic survey data. PLoS Comput. Biol. 8 e1002687.
  • Fukuyama, J., McMurdie, P. J., Dethlefsen, L., Relman, D. A. and Holmes, S. (2012). Comparisons of distance methods for combining covariates and abundances in microbiome studies. In Pacific Symposium on Biocomputing 2012 213–224.
  • Golub, G. H. and van Loan, C. F. (2012). Matrix Computations. Johns Hopkins Univ. Press, Baltimore, MD.
  • Goodrich, J. K., Rienzi, S. C. D., Poole, A. C., Koren, O., Walters, W. A., Caporaso, J. G., Knight, R. and Ley, R. E. (2014). Conducting a microbiome study. Cell 158 250–262.
  • Gower, J. C. (1966). Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53 325–338.
  • Greenacre, M. J. (1984). Theory and Applications of Correspondence Analysis. Academic Press, Cambridge, MA.
  • Gretton, A., Herbrich, R., Smola, A., Bousquet, O. and Schölkopf, B. (2005). Kernel methods for measuring independence. J. Mach. Learn. Res. 6 2075–2129.
  • Hamady, M. and Knight, R. (2009). Microbial community profiling for human microbiome projects: Tools, techniques, and challenges. Genome Res. 19 1141–1152.
  • Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York.
  • Hoerl, A. E. and Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12 55–67.
  • Hullar, M. A., Lancaster, S. M., Li, F., Tseng, E., Beer, K., Atkinson, C., Wähälä, K., Copeland, W. K., Randolph, T. W., Newton, K. M. and Lampe, J. W. (2015). Enterolignan-producing phenotypes are associated with increased gut microbial diversity and altered composition in premenopausal women in the United States. Cancer Epidemiol. Biomark. Prev. 24 546–554.
  • Josse, J. and Holmes, S. (2016). Measuring multivariate association and beyond. Stat. Surv. 10 132–167.
  • Kim, S. and Xing, E. P. (2010). Tree-guided group lasso for multi-task regression with structured sparsity. In Proceedings of the 27th International Conference on Machine Learning (ICML 2010) (J. Fürnkranz and T. Joachims, eds.) 543–550.
  • Koren, O., Knights, D., Gonzalez, A., Waldron, L., Segata, N., Knight, R., Huttenhower, C. and Ley, R. E. (2013). A guide to enterotypes across the human body: Meta-analysis of microbial community structures in human microbiome datasets. PLoS Comput. Biol. 9 e1002863.
  • Kuczynski, J., Liu, Z., Lozupone, C., McDonald, D., Fierer, N. and Knight, R. (2010). Microbial community resemblance methods differ in their ability to detect biologically relevant patterns. Nat. Methods 7 813–819.
  • Kurtz, Z. D., Mueller, C. L., Miraldi, E. R., Littman, D. R., Blaser, M. J. and Bonneau, R. A. (2015). Sparse and compositionally robust inference of microbial ecological networks. PLoS Comput. Biol. 11 e1004226.
  • Li, H. (2015). Microbiome, metagenomics, and high-dimensional compositional data analysis. Annu. Rev. Statist. Appl. 2 73–94.
  • Li, C. and Li, H. (2008). Network-constrained regularization and variable selection for analysis of genomic data. Bioinformatics 24 1175–1182.
  • Lovell, D., Pawlowsky-Glahn, V., Egozcue, J. J., Marguerat, S. and Bähler, J. (2015). Proportionality: A valid alternative to correlation for relative data. PLoS Comput. Biol. 11 e1004075.
  • Lozupone, C. and Knight, R. (2005). UniFrac: A new phylogenetic method for comparing microbial communities. Appl. Environ. Microbiol. 71 8228–8235.
  • Lozupone, C. A., Hamady, M., Kelley, S. T. and Knight, R. (2007). Quantitative and qualitative $\beta$ diversity measures lead to different insights into factors that structure microbial communities. Appl. Environ. Microbiol. 73 1576–1585.
  • Mardia, K. V., Kent, J. T. and Bibby, J. M. (1980). Multivariate Analysis. Academic Press, San Diego, CA.
  • Matsen, F. A. and Evans, S. N. (2013). Edge principal components and squash clustering: Using the special structure of phylogenetic placement data for sample comparison. PLoS ONE 8 e56859.
  • Pan, W. (2011). Relationship between genomic distance-based regression and kernel machine regression for multi-marker association testing. Genet. Epidemiol. 35 211–216.
  • Pavoine, S., Dufour, A.-B. and Chessel, D. (2004). From dissimilarities among species to dissimilarities among communities: A double principal coordinate analysis. J. Theoret. Biol. 228 523–537.
  • Pearson, K. (1896). Mathematical contributions to the theory of evolution—On a form of spurious correlation which may arise when indices are used in the measurement of organs. Proc. R. Soc. Lond. 60 489–498.
  • Pekalska, E., Paclik, P. and Duin, R. P. (2002). A generalized kernel approach to dissimilarity-based classification. J. Mach. Learn. Res. 2 175–211.
  • Purdom, E. (2011). Analysis of a data matrix and a graph: Metagenomic data and the phylogenetic tree. Ann. Appl. Stat. 5 2326–2358.
  • Randolph, T. W., Harezlak, J. and Feng, Z. (2012). Structured penalties for functional linear models—Partially empirical eigenvectors for regression. Electron. J. Stat. 6 323–353.
  • Randolph, W. T., Zhao, S., Copeland, W., Hullar, M. and Shojaie, A. (2018). Supplement to “Kernel-penalized regression for analysis of microbiome data.” DOI:10.1214/17-AOAS1102SUPP.
  • Robert, P. and Escoufier, Y. (1976). A unifying tool for linear multivariate statistical methods: The $RV$-coefficient. J. R. Stat. Soc. Ser. C. Appl. Stat. 25 257–265.
  • Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression. Cambridge Univ. Press, New York.
  • Schaid, D. J. (2010). Genomic similarity and kernel methods I: Advancements by building on mathematical and statistical foundations. Hum. Hered. 70 109–131.
  • Schifano, E. D., Epstein, M. P., Bielak, L. F., Jhun, M. A., Kardia, S. L. R., Peyser, P. A. and Lin, X. (2012). SNP set association analysis for familial data. Genet. Epidemiol. 36 797–810.
  • Schölkopf, B. and Smola, A. J. (2002). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, Cambridge, MA.
  • Srinivasan, S., Hoffman, N. G., Morgan, M. T., Matsen, F. A., Fiedler, T. L., Hall, R. W., Ross, F. J., McCoy, C. O., Bumgarner, R., Marrazzo, J. M. et al. (2012). Bacterial communities in women with bacterial vaginosis: High resolution phylogenetic analyses reveal relationships of microbiota to clinical criteria. PLoS ONE 7 e37818.
  • Sun, T. and Zhang, C.-H. (2012). Scaled sparse linear regression. Biometrika 99 879–898.
  • Székely, G. J. and Rizzo, M. L. (2009). Brownian distance covariance. Ann. Appl. Stat. 3 1236–1265.
  • Tanaseichuk, O., Borneman, J. and Jiang, T. (2014). Phylogeny-based classification of microbial communities. Bioinformatics 30 449–456.
  • The Human Microbiome Project Consortium (2012). Structure, function and diversity of the healthy human microbiome. Nature 486 207–214.
  • Tibshirani, R. J. and Taylor, J. (2011). The solution path of the generalized lasso. Ann. Statist. 39 1335–1371.
  • Tolosana-Delgado, V. and Van Den Boogart, K. G. (2011). Linear models with compositions in R. In Compositional Data Analysis: Theory and Applications (V. Pawlowsky-Glahn and A. Buccianti, eds.) 12–28. Wiley, Chichester.
  • Van de Geer, S., Bühlmann, P., Ritov, Y. and Dezeure, R. (2014). On asymptotically optimal confidence regions and tests for high-dimensional models. Ann. Statist. 42 1166–1202.
  • Van Loan, C. F. (1976). Generalizing the singular value decomposition. SIAM J. Numer. Anal. 13 76–83.
  • Yatsunenko, T., Rey, F. E., Manary, M. J., Trehan, I., Dominguez-Bello, M. G., Contreras, M., Magris, M., Hidalgo, G., Baldassano, R. N., Anokhin, A. P., Heath, A. C., Warner, B., Reeder, J., Kuczynski, J., Caporaso, J. G., Lozupone, C. A., Lauber, C., Clemente, J. C., Knights, D., Knight, R. and Gordon, J. I. (2012). Human gut microbiome viewed across age and geography. Nature 486 222–227.
  • Zhang, C.-H. and Zhang, S. S. (2014). Confidence intervals for low dimensional parameters in high dimensional linear models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 217–242.
  • Zhao, S. and Shojaie, A. (2016). A significance test for graph-constrained estimation. Biometrics 72 484–493.
  • Zhao, N., Chen, J., Carroll, I. M., Ringel-Kulka, T., Epstein, M. P., Zhou, H., Zhou, J. J., Ringel, Y., Li, H. and Wu, M. C. (2015). Testing in microbiome-profiling studies with MiRKAT, the microbiome regression-based kernel association test. Am. J. Hum. Genet. 96 797–807.

Supplemental materials