bayesianmice Data Methods Results Code

Ensemble Bayesian modeling

Preliminary data processing

After confirming that each of the two independent experimental replicates showed similar trends, sequence count data and the cytokine expression data from these two independent replications were combined for all the downstream analytical steps. The analyses that included the bacterial microbiome was performed using 30 mice; 10 in each treatment group while those with the fungal microbiome was performed on 36 mice; 12 in each treatment group. The GAPDH normalized cytokine mRNA expression levels and genera level relative abundances of sequences were log transformed to bring the numerical attributes of the dataset into the same dynamic range.

Bayesian model averaging

All multivariable statistical models were estimated using Bayesian model averaging (BMA)1. with a spike-and-slab prior distribution2 implemented in the BoomSpikeSlab R package3,4. The spike prior is based on an assumption that a sparse set of variables can explain the response. It consists of a Bernoulli distribution that specifies whether or not a variable is selected as influential. The slab prior is a Gaussian distribution that models the effect sizes of the variables, conditional on their being chosen as influential.


BMA combines information from these two priors and uses a Markov Chain Monte Carlo procedure to compute a space of likely variable configurations that explain the response5. The 10,000 most likely models from this space are retained. The median effect size and its associated 95% Bayesian CI for each variable was computed using the distribution of its regression coefficient across the 10,000 models. The Bayesian 95% CI was free of distributional assumptions. The initial 1,000 models were excluded as burn-in6. The posterior inclusion probability (PIP) for each variable is the proportion of the models that selected the variable as influential. In our findings, the median effect size and its 95% Bayesian CI and the PIP are presented as the two formal measures of statistical significance and consistency for each variable.

BMA model specifications

Three separate sets of BMA models were estimated: a) logistic regression with antibiotic treatment (PSG or vancomycin) as response and controls as reference; linear regression with b) log(mRNA cytokine expression) as response, and c) colonization levels measured in log(CFU) as response. These models were used to evaluate a) impact of antibiotics on specific microbiota and the immune response, b) influence of specific microbiota on cytokine mRNA expression c) the role of specific microbiota and cytokine mRNA expression on the level of C. albicans colonization. In each model, antibiotic treatment, exposure to C. albicans, and time-points of sampling were included as indicator variables. Mice in control groups served as the reference baseline. Each bacterial model~ was estimated from 30 mice, 10 in each treatment group (controls, vancomycin and PSG). Each fungal model was estimated from 36 mice, 12 in each treatment group. All data analyses were performed in the R language for statistical computing7.

References

  1. Hoeting, J. A., Madigan, D., Raftery, A. E. & Volinsky, C. T. Bayesian Model Averaging: A Tutorial. Stat. Sci. 14, 382–417 (1999).↩

  2. George, E. I. & Mcculloch, R. E. Approaches for Bayesian variable selection. in Statistica Sinica, 7, 339–373 (1997).↩

  3. Scott, S. L. & Varian, H. R. Predicting the Present with Bayesian Structural Time Series. IJMNO 5, 4+ (2013).↩

  4. Scott, S. L. BoomSpikeSlab: MCMC for spike-and-slab regression. (CRAN, 2014).↩

  5. George, E. I. & Mcculloch, R. E. Approaches for Bayesian variable selection. in Statistica Sinica, 7, 339–373 (1997).↩

  6. Scott, S. L. & Varian, H. R. Predicting the Present with Bayesian Structural Time Series. IJMNO 5, 4+ (2013).↩

  7. R Development Core Team. R: A Language and Environment for Statistical Computing.(R Foundation for Statistical Computing, 2014).↩