Modeling Liquid Association

Abstract

In 2002, Ker-Chau Li introduced the liquid association measure to characterize three-way interactions between genes, and developed a computationally efficient estimator that can be used to screen gene expression microarray data for such interactions. That study, and others published since then, have established the biological validity of the method, and clearly demonstrated it to be a useful tool for the analysis of genomic data sets. To build on this work, we have sought a parametric family of multivariate distributions with the flexibility to model the full range of trivariate dependencies encompassed by liquid association. Such a model could situate liquid association within a formal inferential theory. In this article, we describe such a family of distributions, a trivariate, conditional normal model having Gaussian univariate marginal distributions, and in fact including the trivariate Gaussian family as a special case. Perhaps the most interesting feature of the distribution is that the parameterization naturally parses the three-way dependence structure into a number of distinct, interpretable components. One of these components is very closely aligned to liquid association, and is developed as a measure we call modified liquid association. We develop two methods for estimating this quantity, and propose statistical tests for the existence of this type of dependence. We evaluate these inferential methods in a set of simulations and illustrate their use in the analysis of publicly available experimental data.

Statistical Analysis Reveals Co-Expression Patterns of Many Pairs of Genes in Yeast Are Jointly Regulated by Interacting Loci

“…In this case, if a genetic marker affects post-transcriptional regulation, its effect may be captured by the change of co-expression patterns of the targets of TFs, so a LA analysis may lead to the identification of such markers, where it may be difficult to detect these signals using single gene expressions as the response. Recently, Ho et al [14] proposed a conditional bi-variate normal model to analyze LA that simultaneously captures means, variances, and correlation between a pair of genes. Under a similar framework, Chen et al [15] proposed a penalized likelihood approach to effectively detecting causal genetic loci using iterative reweighted least squares, and Daye et al [16] further considered the heteroscedastic problem.…”

Identification of markers associated with global changes in DNA methylation regulation in cancers

“…For example, the liquid association model was developed to identify mediator genes that can modulate coexpression of other pairs of genes [8]. A few other similar models have been proposed to describe three-way relationships among genes [9-11]. Cancer type dependent coexpression patterns have been reported previously [12,13].…”

A Penalized Likelihood Approach for Bivariate Conditional Normal Models for Dynamic Co-expression Analysis

“…However, some of these mediator variables can also affect the mean expression levels of these two genes and even their variances. The conditional normal model of Ho et al (2009) allows such a dependency when there is only one mediator variable. When the set of potential mediator variables is large, as in our analysis of the real data sets, we first regressed out the effects of these variables on the mean expressions using regression approaches and then applied our model on the residuals.…”

“…It has also been applied to analysis of expression quantitative trait loci (eQTL) data in order to identify the genetic variants that can mediate the co‐expression patterns between two genes (Sun, Yuan, and Li, 2008). Ho et al (2009) further extended the LA using a conditional normal model that allows one to characterize means, variances, and liquid association structures. These applications clearly demonstrated that the co‐expression patterns between two genes are often affected by the other genes or variables, which we call dynamic co‐expression in this article.…”

“…Under such a model, we can consider more than one mediating gene that may affect the co‐expression patterns between two genes. The model is more general than the original LA definition of Li (2002) and the generalized LA defined in Ho et al (2009) since the liquid association between two genes may be determined by the expression levels of several genes simultaneously or some combination of multiple SNPs. In some situations, conditioning on a set of genes makes more biological sense and can lead to more powerful methods for identifying the mediating genes.…”