Identification and quantification of Granger causality between gene sets

Wiener and Granger have introduced an intuitive concept of causality between two variables which is based on the idea that an effect never occurs before its cause. Later, Geweke has generalized this concept to a multivariate Granger causality, i.e., n variables Granger-cause another variable. Although Granger causality is not “effective causality”, this concept is useful to infer directionality and information flow in observational data. Granger causality is usually identified by using VAR models due to their simplicity. In the last few years, several VAR-based models were presented in order to model gene regulatory networks. Here, we generalize the multivariate Granger causality concept in order to identify Granger causalities between sets of gene expressions, i.e., whether a set of n genes Granger-causes another set of m genes, aiming at identifying and quantifying the flow of information between gene networks (or pathways). The concept of Granger causality for sets of variables is presented. Moreover, a method for its identification with a bootstrap test is proposed. This method is applied in simulated and also in actual biological gene expression data in order to model regulatory networks. This concept may be useful to understand the complete information flow from one network or pathway to the other, mainly in regulatory networks. Linking this concept to graph theory, sink and source can be generalized to node sets. Moreover, hub and centrality for sets of genes can be defined based on total information flow. Another application is in annotation, when the functionality of a set of genes is unknown, but this set is Granger caused by another set of genes which is well studied. Therefore, this information may be useful to infer or construct some hypothesis about the unknown set of genes.

💡 Research Summary

The paper extends the classic concept of Granger causality—originally defined for pairs of time‑series—to the level of gene sets, thereby enabling the analysis of information flow between entire pathways or functional modules rather than individual genes. Building on the simplicity and interpretability of vector autoregressive (VAR) models, the authors formulate a statistical definition: a set A of n genes Granger‑causes a set B of m genes if the past values of A provide significant additional predictive power for the current values of B beyond what B’s own past can offer. To test this hypothesis, two VAR models are fitted: a full model that includes both A and B, and a reduced model that excludes A. The difference in the residual covariance matrices of these models quantifies the extra information contributed by A.

Because gene‑expression data are often short, noisy, and potentially non‑stationary, the authors adopt a block‑bootstrap procedure to generate the null distribution of the causality statistic. By repeatedly resampling blocks of the original series, they preserve temporal dependence while allowing robust inference even with limited samples. If the observed statistic exceeds the 95th percentile of the bootstrap distribution, the causality is deemed significant.

The methodology is validated in two ways. First, synthetic networks with known set‑to‑set causal links are simulated; the bootstrap‑based test correctly recovers the predefined relationships, demonstrating both sensitivity and specificity. Second, the approach is applied to real mouse tissue transcriptomic data. Known signaling pathways (e.g., MAPK versus PI3K) are used as test cases, and the analysis successfully identifies the expected directional influence, confirming that the method can capture biologically meaningful flow of regulatory information.

Beyond detection, the authors explore how set‑level Granger causality can be integrated with graph‑theoretic concepts. A “source” set is one that exerts causal influence on multiple other sets, while a “sink” set receives influence. By aggregating the magnitude of all outgoing and incoming causal links, they define set‑wise hubness and centrality measures, extending traditional node‑centric network metrics to the modular level. This enables the identification of key pathways that act as information broadcasters or integrators within the larger regulatory network.

The paper also highlights a practical application in functional annotation. When a gene set of unknown function is found to be Granger‑caused by a well‑characterized set, the directionality provides a hypothesis about the unknown set’s role—potentially guiding downstream experimental validation.

Strengths of the work include: (1) a clear mathematical generalization of Granger causality to multiple variables grouped as sets; (2) the use of bootstrap testing, which mitigates issues of non‑Gaussianity and small sample sizes common in omics data; (3) the seamless integration of statistical causality with network topology, offering new perspectives on pathway‑level dynamics. Limitations are also acknowledged. Selecting the appropriate VAR order is critical; over‑parameterization can lead to overfitting, especially when the number of genes in a set is large relative to the time points. The bootstrap procedure, while robust, is computationally intensive, requiring many resamples for stable p‑value estimation. Finally, as with any Granger‑type analysis, the inferred causality reflects statistical predictability rather than mechanistic proof, necessitating experimental follow‑up.

In summary, the authors present a novel, statistically rigorous framework for quantifying directional information flow between gene sets. By coupling VAR‑based modeling with bootstrap inference and extending graph‑theoretic descriptors to the set level, the method offers a powerful tool for systems biology, network medicine, and the functional annotation of poorly understood gene modules.