In the study of gene expression data, network analysis has played a uniquely important role. To accommodate the high dimensionality and low sample size and generate interpretable results, regularized estimation is usually conducted in the construction of gene expression Gaussian Graphical Models (GGM). Here we use GeO-GGM to represent gene-expression-only GGM. Gene expressions are regulated by regulators. gene-expression-regulator GGMs (GeR-GGMs), which accommodate gene expressions as well as their regulators, have been constructed accordingly. In practical data analysis, with a "lack of information" caused by the large number of model parameters, limited sample size, and weak signals, the construction of both GeO-GGMs and GeR-GGMs is often unsatisfactory. In this article, we recognize that with the regulation between gene expressions and regulators, the sparsity structures of a GeO-GGM and its GeR-GGM counterpart can satisfy a hierarchy. Accordingly, we propose a joint estimation which reinforces the hierarchical structure and use the construction of a GeO-GGM to assist that of its GeR-GGM counterpart and vice versa. Consistency properties are rigorously established, and an effective computational algorithm is developed. In simulation, the assisted construction outperforms the separation construction of GeO-GGM and GeR-GGM. Two The Cancer Genome Atlas data sets are analyzed, leading to findings different from the direct competitors.
Keywords: assisted estimation; gene expressions; graphical models; hierarchy.
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