Changes in human microbiome are associated with many human diseases. Next generation sequencing technologies make it possible to quantify the microbial composition without the need for laboratory cultivation. One important problem of microbiome data analysis is to identify the environmental/biological covariates that are associated with different bacterial taxa. Taxa count data in microbiome studies are often over-dispersed and include many zeros. To account for such an over-dispersion, we propose to use an additive logistic normal multinomial regression model to associate the covariates to bacterial composition. The model can naturally account for sampling variabilities and zero observations and also allow for a flexible covariance structure among the bacterial taxa. In order to select the relevant covariates and to estimate the corresponding regression coefficients, we propose a group penalized likelihood estimation method for variable selection and estimation. We develop a Monte Carlo expectation-maximization algorithm to implement the penalized likelihood estimation. Our simulation results show that the proposed method outperforms the group penalized multinomial logistic regression and the Dirichlet multinomial regression models in variable selection. We demonstrate the methods using a data set that links human gut microbiome to micro-nutrients in order to identify the nutrients that are associated with the human gut microbiome enterotype.