  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.
