Granger causality (GC) (Granger, 1969) is ideally suited to infer causal relationships from bivariate time series such as first-order vector-autoregressive processes VAR (1) (Hamilton, 1994). Recent studies have used GC to infer causal relationships and network structure from temporal gene expression profiles (Mukhopadhyay and Chatterjee, 2007; Fujita et al., 2007). Mukhopadhyay and Chatterjee (2007) proposed a method to infer causality from temporal gene expression data. Their approach determine the p-values from F-test (Hamilton, 1994) across each pair of genes (gi, gj) addressing the null hypotheses (gi fails to Granger cause gj) and (gj fails to Granger cause gi). Subsequently, they chose the desired uni-directional causal relationship as the direction with the lowest p-value. Mukhopadhyay and Chatterjee (2007) also caution the audience that their approach does not guarantee true causality but may prove to be a strong starting point in uncovering “….functional, metabolic relationship and biological causation models for temporal gene expressions……”. They demonstrate their approach on stationary and non-stationary networks modeled as vector auto-regressive processes as well as microarray gene expression profiles.#R##N##R##N##R##N#In this brief letter, we investigate the usefulness of such unidirectional/acyclic approximations within the context of Granger causality and vector auto-regressive processes (VAR). Subsequently, biological significance of the various parameters in a VAR (1) model that can give rise to acyclic approximations is elucidated. In this respect, synthetic two-gene networks modeled as VAR (1) with and without auto-regulatory feedback are investigated. The choice of VAR (1) is especially encouraged by (i) its close connection with the formulation of Granger causality (Granger, 1969; Hamilton, 1994); (ii) recent simulation studies presented by the authors Mukhopadhyay and Chatterjee (2007) and (iii) its prevalence in modeling genetic networks [Fujita et al., 2007; Opgen-Rhein and Strimmer, 2007). We show that such uni-directional/acyclic approximations reflected by discrepancy in p-values as proposed by (Mukhopadhyay and Chatterjee, 2007) may provide biologically meaningful results under the assumptions of #R##N##R##N##R##N#equal transcriptional noise variance, reflected by the variance of the noise terms in VAR (1).#R##N##R##N#and#R##N##R##N##R##N#equal strength of auto-regulatory feedback, reflected by the magnitude of the diagonal elements in VAR (1).#R##N##R##N##R##N##R##N#The results presented encourage cautious interpretation of the results of Granger causality (Granger, 1969; Hamilton, 1994) and its variants (Mukhopadhyay and Chatterjee, 2007) in acyclic representation of transcriptional networks inferred from their expression profiles.
['Radhakrishnan Nagarajan', 'Meenakshi Upreti']
Comment on causality and pathway search in microarray time series experiment