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With the development of next generation sequencing technology, researchershave now been able to study the microbiome composition using direct sequencing,whose output are bacterial taxa counts for each microbiome sample. One goal ofmicrobiome study is to associate the microbiome composition with environmentalcovariates. We propose to model the taxa counts using a Dirichlet-multinomial(DM) regression model in order to account for overdispersion of observedcounts. The DM regression model can be used for testing the association betweentaxa composition and covariates using the likelihood ratio test. However, whenthe number of covariates is large, multiple testing can lead to loss of power.To address the high dimensionality of the problem, we develop a penalizedlikelihood approach to estimate the regression parameters and to select thevariables by imposing a sparse group $\ell_1$ penalty to encourage bothgroup-level and within-group sparsity. Such a variable selection procedure canlead to selection of the relevant covariates and their associated bacterialtaxa. An efficient block-coordinate descent algorithm is developed to solve theoptimization problem. We present extensive simulations to demonstrate that thesparse DM regression can result in better identification of themicrobiome-associated covariates than models that ignore overdispersion or onlyconsider the proportions. We demonstrate the power of our method in an analysisof a data set evaluating the effects of nutrient intake on human gut microbiomecomposition. Our results have clearly shown that the nutrient intake isstrongly associated with the human gut microbiome.

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