Bayesian methods for expression-based integration of various types of genomics data

Abstract

: We propose methods to integrate data across several genomic platforms using a hierarchical Bayesian analysis framework that incorporates the biological relationships among the platforms to identify genes whose expression is related to clinical outcomes in cancer. This integrated approach combines information across all platforms, leading to increased statistical power in finding these predictive genes, and further provides mechanistic information about the manner in which the gene affects the outcome. We demonstrate the advantages of the shrinkage estimation used by this approach through a simulation, and finally, we apply our method to a Glioblastoma Multiforme dataset and identify several genes potentially associated with the patients' survival. We find 12 positive prognostic markers associated with nine genes and 13 negative prognostic markers associated with nine genes.

Figure 1

Platform relationships. Schematic representation of the multiple molecular platforms and their biological relationships.

Figure 2

Simulation results. Least squares estimates and posterior means from our method are plotted against the true β values. The vertical lines denote the difference between the estimates from each method thus indicating the shrinkage properties of the NG prior.

Figure 3

GBM data results. The posterior probabilities (based on MCMC samples) that β_j,i > δ + ∗ is plotted, where β_j,i is the clinical model regression coefficient for the marker associated with platform j of gene i, and ${\delta }_{+}^{\ast }=log(1+\delta )$ is the transformed upper practical cutoff. For our analysis, we use δ = 0.05, which corresponds to a 5% change in survival time, so the posterior probability shown here indicates the probability that a one unit increase in the marker results in at least a 5% increase in survival time. We consider the marker j,i to be significant if this probability is greater than 0.5.

Figure 4

GBM data results. The posterior probabilities (based on MCMC samples) that ${\beta }_{j,i}<{\delta }_{-}^{\ast }$ is plotted, where β_j,i is the clinical model regression coefficient for the marker associated with platform j of gene i, and ${\delta }_{-}^{\ast }=log(1-\delta )$ is the transformed lower practical cutoff. For our analysis, we use δ = 0.05, which corresponds to a 5% change in survival time, so the posterior probability shown here indicates the probability that a one unit increase in the marker results in at least a 5% decrease in survival time. We consider the marker j,i to be significant if this probability is greater than 0.5.

Figure 5

Regression coefficient posterior means. The estimates of the regression coefficients in the clinical model (β_j,i’s) are shown, where β_j,i is the coefficient for the marker associated with platform j of gene i; the estimates are computed as the posterior means from our MCMC samples. The multiple platforms for each gene are labeled by color, and solid plot markers indicate that the effect was found to be significant, meaning that the posterior probability that a one unit increase in the marker results in at least a 5% change in survival time is at least 0.5.