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A Note on Inferring Acyclic Network Structures Using Granger Causality Tests

Granger causality (GC) and its extension have been used widely to infer causal relationships from multivariate time series generated from biological systems. GC is ideally suited for causal inference in bivariate vector autoregressive process (VAR). A zero magnitude of the upper or lower off-diagonal element(s) in a bivariate VAR is indicative of lack of causal relationship in that direction resulting in true acyclic structures. However, in experimental settings, statistical tests, such as F-test that rely on the ratio of the mean-squared forecast errors, are used to infer significant GC relationships. The present study investigates acyclic approximations within the context of bi-directional two-gene network motifs modeled as bivariate VAR. The fine interplay between the model parameters in the bivariate VAR, namely: (i) transcriptional noise variance, (ii) autoregulatory feedback, and (iii) transcriptional coupling strength that can give rise to discrepancies in the ratio of the mean-squared forecast errors is investigated. Subsequently, their impact on statistical power is investigated using Monte Carlo simulations. More importantly, it is shown that one can arrive at acyclic approximations even for bi-directional networks for suitable choice of process parameters, significance level and sample size. While the results are discussed within the framework of transcriptional network, the analytical treatment provided is generic and likely to have significant impact across distinct paradigms.