Integrative factorization of bidimensionally linked matrices

Abstract

Advances in molecular “omics” technologies have motivated new methodologies for the integration of multiple sources of high‐content biomedical data. However, most statistical methods for integrating multiple data matrices only consider data shared vertically (one cohort on multiple platforms) or horizontally (different cohorts on a single platform). This is limiting for data that take the form of bidimensionally linked matrices (eg, multiple cohorts measured on multiple platforms), which are increasingly common in large‐scale biomedical studies. In this paper, we propose bidimensional integrative factorization (BIDIFAC) for integrative dimension reduction and signal approximation of bidimensionally linked data matrices. Our method factorizes data into (a) globally shared, (b) row‐shared, (c) column‐shared, and (d) single‐matrix structural components, facilitating the investigation of shared and unique patterns of variability. For estimation, we use a penalized objective function that extends the nuclear norm penalization for a single matrix. As an alternative to the complicated rank selection problem, we use results from the random matrix theory to choose tuning parameters. We apply our method to integrate two genomics platforms (messenger RNA and microRNA expression) across two sample cohorts (tumor samples and normal tissue samples) using the breast cancer data from the Cancer Genome Atlas. We provide R code for fitting BIDIFAC, imputing missing values, and generating simulated data.

Two-stage Linked Component Analysis for Joint Decomposition of Multiple Biologically Related Data Sets

“…We compare 2s-LCA with several existing methods. First, we fix the dimensions of all subspaces at their true values, i.e., 2, and compare 2s-LCA with JIVE (Lock and others , 2013), AJIVE (Feng and others , 2018), SLIDE (Gaynanova and Li, 2019), BIDIFAC (Park and Lock, 2020), and BIDIFAC+ (Lock and others , 2020) for subspace estimation. Then, we do the same as above except that the data are generated such that the variances of scores associated with individual subspaces are much larger than those for common and partially shared subspaces.…”

“…We briefly mention a few other related works. Park & Lock (2020) jointly analyzed multiple data sets for heterogeneous groups of objects with heterogeneous feature sets. Gao et al (2020) and Wang & Allen (2019) considered clustering problems for multi-view data.…”

“…It is important to note that, for our method, it is fixed signal subspaces rather than latent random scores that are shared across data sets. Another significant difference is that the existence of partially-shared signal subspaces for the proposed model is determined by design, which is scientifically meaningful for our motivating data and somehow similar to Park & Lock (2020) for bi-dimensional data. To the best of our knowledge, joint de-composition of linked data sets with a general design has not been considered before in the literature.…”

“…We mention a few other related works. Lock and others (2020) and Park and Lock (2020) jointly analyzed multiple data sets for heterogeneous groups of objects with heterogeneous feature sets. Kallus and others (2019) developed a data adaptive method which allows structure in data to be shared across an arbitrary subset of views and cohorts.…”

“…While not designed explicitly for linked data and the use case described, other methods can also be used in this setting. The BIDIFAC+ method (Lock and others , 2020), an extension of the BIDIFAC method for bi-dimensional data (Park and Lock, 2020), could also accommodate partially shared subspaces by design for linked data. These methods were proposed for decomposing bi-dimensional data.…”