A loop-counting method for covariate-corrected low-rank biclustering of gene-expression and genome-wide association study data

A common goal in data-analysis is to sift through a large data-matrix and detect any significant submatrices (i.e., biclusters) that have a low numerical rank. We present a simple algorithm for tackling this biclustering problem. Our algorithm accumulates information about 2-by-2 submatrices (i.e., ‘loops’) within the data-matrix, and focuses on rows and columns of the data-matrix that participate in an abundance of low-rank loops. We demonstrate, through analysis and numerical-experiments, that this loop-counting method performs well in a variety of scenarios, outperforming simple spectral methods in many situations of interest. Another important feature of our method is that it can easily be modified to account for aspects of experimental design which commonly arise in practice. For example, our algorithm can be modified to correct for controls, categorical- and continuous-covariates, as well as sparsity within the data. We demonstrate these practical features with two examples; the first drawn from gene-expression analysis and the second drawn from a much larger genome-wide-association-study (GWAS).Author summary: An important problem in genomics is how to detect the genetic signatures associated with disease. When a disease is caused by a single well-defined biological mechanism, the genetic signature often involves a handful of genes present across the majority of the diseased patients. On the other hand, when a disease is complicated or poorly understood there may be many possible biological mechanisms at play. Any genetic signatures associated with such a disease may involve multiple genes, and each signature might only manifest across a subset of the diseased patients. Finding the signatures responsible for this heterogeneity requires searching through subsets of genes and subsets of patients—a problem referred to as ‘biclustering’. In this paper we present a new biclustering method which can scale up efficiently to handle large genomic data sets, such as GWAS-data. Our method is quite accurate, outperforming current ‘spectral’ biclustering methods for many problems of interest. Perhaps most importantly, our method can be corrected for many features of experimental design, such as controls, covariates and sparsity—all of which are especially important when analyzing real data sets.