The analysis of interactions among a group of genes is fundamental to fur- ther our understanding of their biological interactions in a cell. Several studies suggested that the co-expression relationship of two genes can be modulated by a third controller gene. These controller genes and the corresponding modulated co-expressed gene pairs are the subjects of interests in this study. This described \controller-modulated genes three-way interactions is referred as liquid association in the literature. Analysis of gene expression data has suggested that these interactions are present in many biological systems.
To quantify the magnitude of liquid association for a given gene triplet, we proposed a statistical measure named generalized liquid association (GLA). To estimate the value of GLA given the data, we propose two approaches: the direct and the model-based estimation approach. For the model-based approach, we introduce the conditional normal model (CNM). This is a generalization of the tri-variate normal distribution that allows us to characterize means, variances, as well as liquid association structures. We provide an approach based on generalized estimation equations to estimate the parameters in the CNM. We validate the proposed approaches through simulation studies and illustrate them in experimental data analysis. We also compare them with the three-product-moment measure suggested by Li in various settings and discuss related computational issues
