Abstract
Motivated by the pressing needs for dissecting heterogeneous relationships in gene expression data, here we generalize the squared Pearson correlation to capture a mixture of linear dependences between two real-valued variables, with or without an index variable that specifies the line memberships. We construct the generalized Pearson correlation squares by focusing on three aspects: variable exchangeability, no parametric model assumptions, and inference of population-level parameters. To compute the generalized Pearson correlation square from a sample without a line-membership specification, we develop a K-lines clustering algorithm to find K clusters that exhibit distinct linear dependences, where K can be chosen in a data-adaptive way. To infer the population-level generalized Pearson correlation squares, we derive the asymptotic distributions of the sample-level statistics to enable efficient statistical inference. Simulation studies verify the theoretical results and show the power advantage of the generalized Pearson correlation squares in capturing mixtures of linear dependences. Gene expression data analyses demonstrate the effectiveness of the generalized Pearson correlation squares and the K-lines clustering algorithm in dissecting complex but interpretable relationships. The estimation and inference procedures are implemented in the R package gR2 (https://github.com/lijy03/gR2).