CARlasso: An R package for the estimation of sparse microbial networks with predictors

Microbiome data analyses require statistical tools that can simultaneously decode microbes reactions to the environment and interactions among microbes. We introduce CARlasso, the first user-friendly open-source and publicly available R package to fit a chain graph model for the inference of sparse microbial networks that represent both interactions among nodes and effects of a set of predictors. Unlike in standard regression approaches, the edges represent the correct conditional structure among responses and predictors that allows the incorporation of prior knowledge from controlled experiments. In addition, CARlasso 1) enforces sparsity in the network via LASSO; 2) allows for an adaptive extension to include different shrinkage to different edges; 3) is computationally inexpensive through an efficient Gibbs sampling algorithm so it can equally handle small and big data; 4) allows for continuous, binary, counting and compositional responses via proper hierarchical structure, and 5) has a similar syntax to lm for ease of use. The package also supports Bayesian graphical LASSO and several of its hierarchical models as well as lower level one-step sampling functions of the CAR-LASSO model for users to extend.

The Effect of the Prior and the Experimental Design on the Inference of the Precision Matrix in Gaussian Chain Graph Models

Here, we investigate whether (and how) experimental design could aid in the estimation of the precision matrix in a Gaussian chain graph model, especially the interplay between the design, the effect of the experiment and prior knowledge about the effect. We approximate the marginal posterior precision of the precision matrix via Laplace approximation under different priors: a flat prior, the conjugate prior Normal-Wishart, the unconfounded prior Normal-Matrix Generalized Inverse Gaussian (MGIG) and a general independent prior. We show that the approximated posterior precision is not a function of the design matrix for the cases of the Normal-Wishart and flat prior, but it is for the cases of the Normal-MGIG and the general independent prior. However, for the Normal-MGIG and the general independent prior, we find a sharp upper bound on the approximated posterior precision that does not involve the design matrix which translates into a bound on the information that could be extracted from a given experiment. We confirm the theoretical findings via a simulation study comparing the Steins loss difference between random versus no experiment (design matrix equal to zero). Our findings provide practical advice for domain scientists conducting experiments to decode the relationships between a multidimensional response and a set of predictors.

Bayesian Conditional Auto-Regressive LASSO Models to Learn Sparse Microbial Networks with Predictors

Microbiome data analyses require statistical models that can simultaneously decode microbes reactions to the environment and interactions among microbes. While a multiresponse linear regression model seems like a straightforward solution, we argue that treating it as a graphical model is flawed given that the regression coefficient matrix does not encode the conditional dependence structure between response and predictor nodes because it does not represent the adjacency matrix. This observation is especially important in biological settings when we have prior knowledge on the edges from specific experimental interventions that can only be properly encoded under a conditional dependence model. Here, we propose a chain graph model with two sets of nodes (predictors and responses) whose solution yields a graph with edges that indeed represent conditional dependence and thus, agrees with the experimenters intuition on the average behavior of nodes under treatment. The solution to our model is sparse via Bayesian LASSO and is also guaranteed to be the sparse solution to a Conditional Auto-Regressive (CAR) model. In addition, we propose an adaptive extension so that different shrinkage can be applied to different edges to incorporate edge-specific prior knowledge. Our model is computationally inexpensive through an efficient Gibbs sampling algorithm and can account for binary, counting, and compositional responses via appropriate hierarchical structure. We apply our model to a human gut and a soil microbial compositional datasets and we highlight that CAR-LASSO can estimate biologically meaningful network structures in the data. The CAR-LASSO software is available as an R package at https://github.com/YunyiShen/CAR-LASSO.