Why Integrate Diverse Data Sources?

• Complex traits arise from interacting molecular layers — genetic, transcriptomic, proteomic, and metabolic.
  
• Single data types provide only partial insights complex biological systems (GWAS: DNA → disease phenotype)
  
• Integration connects variants → genes → pathways → phenotypes may help reveal molecular mechanisms that drive traits and diseases.
• Enables functional interpretation, better prediction, and new discoveries across molecular systems.

The gact R Package

gact provides an infrastructure for efficient processing of large-scale genomic association data, with core functions for:

• Establishing and populating a database of genomic associations
  
• Downloading and processing biological databases
  
• Handling and processing GWAS summary statistics
  
• Linking genetic markers to genes, proteins, metabolites, and biological pathways
  
• Integrates with statistical machine learning tools in the qgg R package

gact is intended to serve as a practical implementation of integrative genomics, bridging statistical modeling and biological interpretation, and supporting reproducible and extensible workflows.

Integrating Data with gact

The gact() function is a single R command that creates and populates the Genomic Association of Complex Traits (GACT) database.

It automates three main tasks:

• Infrastructure creation – sets up a structured folder-based database
  (glist, gstat, gsets, marker, gtex, download, etc.)
• Data acquisition – downloads and organizes multiple biological data sources
  (e.g., GWAS Catalog, Ensembl, GTEx, Reactome, STRING, STITCH, DGIdb)
• Marker and feature set generation - integrates data across sources to create curated genomic feature sets that form the basis for the integrative genomic analyses.

Biological Databases Used by gact

gact constructs gene and marker sets from a wide range of curated biological databases:

From Database to Model Inputs

The gact R package includes utility functions to extract and structure data from the GACT database into analysis-ready inputs — \(\mathbf{Y}\) (e.g., summary statistic outcomes) and \(\mathbf{X}\) (genomic or biological features).

• getMarkerStat() — retrieve GWAS summary statistics (Y’s)
  
• getFeatureStat() — extract gene-, protein-, or pathway-level results (Y’s)
  
• getMarkerSets() — define biological groupings (basis for X’s)
  
• designMatrix() — build feature matrices (X) linking variants or genes to biological feature sets

Together, these functions form a reproducible workflow for generating standardized input data for Bayesian Hierarchical Models and other machine learning approaches.

Bayesian Hierarchical Models

Bayesian Hierarchical Models provide a flexible statistical framework for modeling complex biological and healthcare data.

They support key applications such as:

• Genome-wide association studies (GWAS) and fine-mapping of causal variants
  
• Polygenic risk scoring (PRS) for predicting complex traits and disease risk
  
• Gene and pathway enrichment analyses to test biological hypotheses
  
• Integrative multi-omics modeling across the genome, transcriptome, epigenome, and proteome
  

These models build directly on the Y’s and X’s generated by gact, providing a unified framework for integrative genomic analysis.

The Bayesian Linear Regression Model

The Bayesian Linear Regression (BLR) model builds:

\[ \mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}, \qquad \boldsymbol{\varepsilon} \sim \mathcal{N}(0, \sigma^2 \mathbf{I}) \]

• \(\mathbf{Y}\) represents the observed outcomes or association measures corresponding to the features in \(\mathbf{X}\).
• \(\mathbf{X}\) represents molecular or genomic predictors (e.g., genotypes, gene scores, annotations, pathway indicators).
  
• \(\boldsymbol{\beta}\) — effect sizes quantifying how features in \(\mathbf{X}\) explain variation in \(\mathbf{Y}\)
  
• \(\boldsymbol{\varepsilon}\) — residual noise capturing unexplained variation

In the Bayesian formulation, each \(\beta_j\) is assigned a prior distribution reflecting beliefs about effect size magnitude or sparsity and determine how information is shared across features or biological layers.

Why Bayesian Hierarchical Models?

Regression effects can be estimated in many ways, but we focus on a Bayesian hierarchical framework because it:

• Combines data and prior knowledge to improve inference
  
• Provides a natural way to regularize and handle noisy or high-dimensional data
  
• Enables flexible modeling of diverse effect patterns:
  • Many small vs. few large effects
    
  • Structured effects (e.g., by pathway, gene set, or omic layer)
    
• Returns uncertainty estimates for all parameters → improving interpretability and model comparison

Through their hierarchical structure, BLR models naturally integrate multiple biological layers — linking genomic, transcriptomic, and other molecular data

Hierarchical Structure

The three levels in the model:

This hierarchical structure allows the model to learn how strongly to shrink effect estimates from the data, while accounting for uncertainty in prior parameters and automatically regularizing effect sizes.

Simple and robust, but may not capture diverse effect-size distributions.

Adapting to Complex Biological Architectures

Complex traits arise from heterogeneous effect-size distributions — some features have large effects, many have small, and others are likely null. To capture this diversity, the BLR framework can be extended in two ways:

• Data-driven grouping of molecular features:
  The model learns effect-size classes from the data using a mixture of variances \(\{\tau_k^2\}\) with probabilities \(\{\pi_k\}\).

• Biologically informed grouping of molecular features:
  The model uses prior biological knowledge to assign features to groups a priori, each with its own variance \(\tau_g^2\) capturing within-group variability.

Both approaches enable the model to adapt to complex genetic and molecular architectures and share information across related features or omic layers

Indicator Variables and Posterior Inclusion Probabilities

In Bayesian variable selection, each feature \(j\) is assigned an indicator variable:

\[ \delta_j = \begin{cases} 1, & \text{if feature $j$ has a non-zero effect} \\ 0, & \text{if feature $j$ has no effect.} \end{cases} \]

After inference, we estimate \(\text{PIP}_j = P(\delta_j = 1 \mid \text{data})\) — the posterior inclusion probability for feature j.

Summary of BLR Model Structures

These models form a hierarchy of increasing flexibility and biological realism —
from global shrinkage → to data-driven heterogeneity → to biologically structured mixtures that model variation within and between feature sets.

Motivation for Multivariate BLR

Many traits and molecular layers are correlated — they share genetic architecture and biological pathways.

To model these dependencies, we extend BLR to the multivariate setting:

• Jointly models multiple traits or omic layers
  → captures shared genetic or molecular effects
  
• Borrows strength across correlated traits
  → improves fine-mapping resolution and prediction accuracy
  
• Estimates cross-trait effect patterns
  → helps identify pleiotropic genes and shared biological pathways

The Multivariate Bayesian Linear Regression (MV-BLR) Model

In the multivariate BLR model, we model multiple correlated outcomes jointly:

\[ \mathbf{Y} = \mathbf{X}\mathbf{B} + \mathbf{E} \]

• \(\mathbf{Y}\): \((n \times T)\) matrix of outcomes
  (e.g., association measures for \(T\) traits or omic layers)
  
• \(\mathbf{X}\): \((n \times p)\) feature matrix
  
• \(\mathbf{B}\): \((p \times T)\) matrix of effect sizes
  
• \(\mathbf{E}\): \((n \times T)\) residual matrix

Each row of \(\mathbf{Y}\) corresponds to an observation or gene, and each column to a trait, phenotype, or molecular layer.

Multivariate Error and Effect Priors

We extend the univariate priors to the multivariate setting:

\[ \mathbf{e}_{i\cdot} \sim \mathcal{N}_T(\mathbf{0}, \boldsymbol{\Sigma}_e) \] \[ \mathbf{b}_j \sim \mathcal{N}_T(\mathbf{0}, \boldsymbol{\Sigma}_b) \]

• \(\boldsymbol{\Sigma}_e\): residual covariance among traits
  
• \(\boldsymbol{\Sigma}_b\): covariance of effect sizes across traits
  
• When \(\boldsymbol{\Sigma}_e\) and \(\boldsymbol{\Sigma}_b\) are diagonal, the model reduces to \(T\) independent univariate BLR models.

Allows information sharing across correlated traits or omic layers and can be used to identify pleiotropic effects and cross-trait genetic architectures.

Multivariate BLR (Structured MV-BLR)

The hierarchical structure can be extended to model multiple traits
while preserving biological grouping of features:

\[ \mathbf{b}_j \sim \mathcal{N}_T\!\big(\mathbf{0}, \boldsymbol{\Sigma}_{b, g(j)}\big), \qquad \boldsymbol{\Sigma}_{b, g} \sim p(\boldsymbol{\Sigma}_{b, g}) \]

• Each biological group \(g\) has its own trait-level covariance matrix \(\boldsymbol{\Sigma}_{b,g}\)
  
• \(\boldsymbol{\Sigma}_{b,g}\) captures correlations and scale of effects across traits within that group

Enables information sharing both within biological sets and across correlated traits.

Indicator Variables and PIPs (Multivariate BLR)

In the multivariate BLR, each feature \(j\) may affect multiple outcomes (traits).
We extend the indicator variable to capture cross-trait activity patterns:

\[ \boldsymbol{\delta}_j = \begin{bmatrix} \delta_{j1} \\ \delta_{j2} \\ \vdots \\ \delta_{jT} \end{bmatrix}, \qquad \delta_{jt} = \begin{cases} 1, & \text{if feature $j$ affects trait $t$} \\ 0, & \text{otherwise.} \end{cases} \]

After inference, we estimate \(\text{PIP}_{jt} = P(\delta_{jt} = 1 \mid \text{data})\) — the posterior inclusion probability that feature j affects trait t.

Interpreting Posterior Inclusion Probabilities (PIPs)

• PIPs quantify the probability that a feature is truly associated with one or more outcomes.
  
• High PIPs (e.g., > 0.9) indicate strong evidence for inclusion.
• A probabilistic view of how molecular variation drives outcomes
• Patterns in \(\boldsymbol{\delta}_j\) reveal shared vs. trait-specific effects — all zeros (no effect), few ones (trait-specific), many ones (shared or pleiotropic).

BLR Models in Integrative Genomics

Learning at Different Levels

Bayesian Linear Regression for Polygenic Risk Scoring

Bayesian Linear Regression (BLR) provides a probabilistic framework for estimating genome-wide effects used in polygenic risk scores (PRS).

\[ \mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}, \qquad \boldsymbol{\varepsilon} \sim \mathcal{N}(0, \sigma^2 \mathbf{I}) \]

After fitting the BLR model:

• Posterior means \(\hat{\boldsymbol{\beta}} = E(\boldsymbol{\beta} \mid \text{data})\) represent regularized SNP effect estimates.
  
• Individual-level PRS values are computed as:

\[ \text{PRS}_i = \sum_j X_{ij} \, \hat{\beta}_j \]

The qgg R Package

qgg provides tools for statistical modeling and analysis of large-scale genomic data, including:

• Fine-mapping of genomic regions using Bayesian Linear Regression (BLR) models
  
• Polygenic scoring using Bayesian Linear Regression (BLR) models
  
• Gene set enrichment analysis using Bayesian Linear Regression (BLR) models

qgg handles large-scale genomic data through efficient algorithms and sparse matrix techniques, combined with multi-core processing using OpenMP, multithreaded matrix operations via BLAS libraries (e.g., OpenBLAS, ATLAS, or MKL), and fast, memory-efficient batch processing of genotype data stored in
binary formats such as PLINK .bed files.

Tutorials using the qgg and gact R packages

Summary

From Data Integration to Modeling

• Bridges data integration, statistical modeling and biological interpretation, enabling reproducible and extensible workflows.

• Integrates biological information across molecular layers — from genome to pathways, complexes, and drug–gene interactions
  

• Uses structured priors and hierarchical modeling to share information, regularize effect estimates, and quantify uncertainty
  

• Enables data-driven discovery and prediction

Next Steps

• Apply the framework to new and relevant use cases
  
• Expand integration with more biological data resources
  
• Incorporate additional statistical and machine learning methods

Further Reading and Collaboration

💡 We are open to collaboration!

If you’re interested in applying BLR methods or contributing to the gact framework, please reach out.

Further Reading

• Bayesian Linear Regression – teaching materials
  
• gact – R package documentation
  
• qgg – R package documentation

References

Gene Set Analyses

Gene and biological pathway prioritization can provide valuable insights into the underlying biology of diseases and potential drug targets.

MAGMA: Multi-marker Analysis of GenoMic Annotation (Leuww et al 2015) generalized gene set analysis of GWAS data

• Compute gene-level association statistics (\(y\))
• Create a design matrix based on annotation (\(X\))
• Fits single or multiple regression models (\(y=Xb\))
  
• Identifies associated features using standard procedures (e.g., t-tests)

Gene-level Association Statistics

Compute gene-level (or other feature-level) association statistics:

• Account for correlation among marker statistics (i.e., linkage disequilibrium, LD)
• Different LD-adjustment methods (e.g., SVD, clumping and thresholding, BLR)
• The choice of method depends on the quality of the available GWAS summary statistics and LD reference panel

Bayesian MAGMA

• Fits a Bayesian regression model that allows regularization and variable selection
  
• Supports single- or multi-trait analyses
  
• Identifies associated features based on posterior inclusion probabilities (PIPs) for the regression effects

Bayesian MAGMA – KEGG Pathway

Bayesian MAGMA – DGIdb

Bayesian MAGMA – Across Ancestries