Assessing statistical significance in causal graphs

Abstract

BackgroundCausal graphs are an increasingly popular tool for the analysis of biological datasets. In particular, signed causal graphs--directed graphs whose edges additionally have a sign denoting upregulation or downregulation--can be used to model regulatory networks within a cell. Such models allow prediction of downstream effects of regulation of biological entities; conversely, they also enable inference of causative agents behind observed expression changes. However, due to their complex nature, signed causal graph models present special challenges with respect to assessing statistical significance. In this paper we frame and solve two fundamental computational problems that arise in practice when computing appropriate null distributions for hypothesis testing.ResultsFirst, we show how to compute a p-value for agreement between observed and model-predicted classifications of gene transcripts as upregulated, downregulated, or neither. Specifically, how likely are the classifications to agree to the same extent under the null distribution of the observed classification being randomized? This problem, which we call "Ternary Dot Product Distribution" owing to its mathematical form, can be viewed as a generalization of Fisher's exact test to ternary variables. We present two computationally efficient algorithms for computing the Ternary Dot Product Distribution and investigate its combinatorial structure analytically and numerically to establish computational complexity bounds.Second, we develop an algorithm for efficiently performing random sampling of causal graphs. This enables p-value computation under a different, equally important null distribution obtained by randomizing the graph topology but keeping fixed its basic structure: connectedness and the positive and negative in- and out-degrees of each vertex. We provide an algorithm for sampling a graph from this distribution uniformly at random. We also highlight theoretical challenges unique to signed causal graphs; previous work on graph randomization has studied undirected graphs and directed but unsigned graphs.ConclusionWe present algorithmic solutions to two statistical significance questions necessary to apply the causal graph methodology, a powerful tool for biological network analysis. The algorithms we present are both fast and provably correct. Our work may be of independent interest in non-biological contexts as well, as it generalizes mathematical results that have been studied extensively in other fields.

Genes Contributing to Pain Sensitivity in the Normal Population: An Exome Sequencing Study

“…None reached significance after multiple testing correction or offered obvious insights into mechanisms of altered pain sensitivity (results not shown). We applied causal reasoning to our data [11], which uses a large curated database of directed regulatory molecular interactions to identify the most plausible upstream regulators of a gene set. Of the 138 genes 86 were present in our database of causal interactions, from which we identified 4 nominally significant regulatory networks (Table 4).…”

“…Causal reasoning [11] uses a large curated database of directed regulatory molecular interactions to identify the most plausible upstream regulators of a gene set with a proposed directionality (eg. down-regulated).…”

“…We looked first for enriched Gene Ontology categories within these genes using DAVID [10] with an EASE p-value<0.05. Then we undertook causal reasoning [11] which uses a large curated database of directed regulatory molecular interactions to identify the most plausible upstream regulators of a gene set. Consequently it allows the recapitulation of regulatory networks/pathways associated with genes of interest.…”

An algorithm for score aggregation over causal biological networks based on random walk sampling

“…In recent years, (reverse) causal reasoning has become increasingly popular in systems biology [ 1 , 9 , 17 , 20 , 21 ]. It is based on the idea that the differential expressions measured in transcriptomics are consequences of the changes in the activities of upstream controlling entities (such as, but not limited to, transcription factor proteins).…”

“…A "causal network model augmented with downstream measurables" is a "causal network model" that additionally contains all the causal edges ending in nodes containing transcript differential expressions (termed "RNA abundance" in BEL). In the literature, the two cases are not always distinguished [ 9 , 20 , 21 ].…”