Testing differential networks with applications to the detection of gene-gene interactions

Abstract

Summary Model organisms and human studies have led to increasing empirical evidence that interactions among genes contribute broadly to genetic variation of complex traits. In the presence of gene-by-gene interactions, the dimensionality of the feature space becomes extremely high relative to the sample size. This imposes a significant methodological challenge in identifying gene-by-gene interactions. In the present paper, through a Gaussian graphical model framework, we translate the problem of identifying gene-by-gene interactions associated with a binary trait D into an inference problem on the difference of two high-dimensional precision matrices, which summarize the conditional dependence network structures of the genes. We propose a procedure for testing the differential network globally that is particularly powerful against sparse alternatives. In addition, a multiple testing procedure with false discovery rate control is developed to infer the specific structure of the differential network. Theoretical justification is provided to ensure the validity of the proposed tests and optimality results are derived under sparsity assumptions. A simulation study demonstrates that the proposed tests maintain the desired error rates under the null and have good power under the alternative. The methods are applied to a breast cancer gene expression study.

Multiple Testing of Submatrices of a Precision Matrix With Applications to Identification of Between Pathway Interactions

“…We shall first define a standardized estimate W i,j for each individual entry of the precision matrix, which is the one-sample version of the estimates proposed in Xia et al (2015), then propose a novel test statistic S ℐ×𝒥 based on the sum of all possible $W_{i,j}^2$, for ( i, j ) ∈ ℐ ×𝒥.…”

“…As in Xia et al (2015), we first develop an estimator of ω i,j and then base the test on its bias corrected standardization. We begin by constructing estimators of r i,j .…”

“…Such estimators can be obtained easily via the standard methods such as the Lasso and Dantzig Selector, see, e.g., Xia et al (2015) Section 2.3. Specifically, if we use the Lasso estimator (see (18) in Section 5), then equations (4) and (5) can be satisfied under the condition (C1) in Section 3 and the sparsity condition max 1≤ i ≤ p | β i | 0 = o { n 1/2 /(log p ) 3/2 }.…”

“…Before studying the null distribution of S ℐ×𝒥 , we first introduce the following condition on the eigenvalues of Ω , which is a common assumption in the high-dimensional setting (Cai et al, 2013; Xia et al, 2015; Liu, 2013). …”

“…Liu (2013) proposed a multiple testing procedure with the FDR control for the partial correlations under a Gaussian graphical model. Xia et al (2015) considered the problem of identifying gene-by-gene interactions associated with a binary trait under a two-sample framework and proposed a procedure for testing the differential network by simultaneously testing entry-wise hypotheses with FDR control. These methods, which can identify the locations of individual gene-by-gene interactions, are however unable to detect the presence interactions between pairs of gene groups while controlling the FDR at the group level.…”