Low-Rank Regression Models for Multiple Binary Responses and their Applications to Cancer Cell-Line Encyclopedia Data

Abstract

In this article, we study high-dimensional multivariate logistic regression models in which a common set of covariates is used to predict multiple binary outcomes simultaneously. Our work is primarily motivated from many biomedical studies with correlated multiple responses such as the cancer cell-line encyclopedia project. We assume that the underlying regression coefficient matrix is simultaneously low-rank and row-wise sparse. We propose an intuitively appealing selection and estimation framework based on marginal model likelihood, and we develop an efficient computational algorithm for inference. We establish a novel high-dimensional theory for this nonlinear multivariate regression. Our theory is general, allowing for potential correlations between the binary responses. We propose a new type of nuclear norm penalty using the smooth clipped absolute deviation, filling the gap in the related non-convex penalization literature. We theoretically demonstrate that the proposed approach improves estimation accuracy by considering multiple responses jointly through the proposed estimator when the underlying coefficient matrix is low-rank and row-wise sparse. In particular, we establish the non-asymptotic error bounds, and both rank and row support consistency of the proposed method. Moreover, we develop a consistent rule to simultaneously select the rank and row dimension of the coefficient matrix. Furthermore, we extend the proposed methods and theory to a joint Ising model, which accounts for the dependence relationships. In our analysis of both simulated data and the cancer cell line encyclopedia data, the proposed methods outperform the existing methods in better predicting responses. Supplementary materials for this article are available online.

Keywords:

• Generalized information criterion
• Logistic regression
• Low-rank models
• Multiple binary responses
• Smoothly clipped absolute deviation

Supplementary Materials

Supplementary materials include discussions about our assumptions in the theory, our theoretical results for the estimation error, the numerical implementations, some simulations, additional application results, and technical proofs of theorems.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

The authors thank the Editor, Associate Editor, and referees for suggestions that significantly improved the article.

Additional information

Funding

Seyoung Park is supported by the National Research Foundation of Korea grant funded by the Korean government (no. NRF-2022R1A2C4002150). Eun Ryung Lee is supported by a National Research Foundation of Korea grant funded by the Korean government (no. NRF- 2022R1A2C1012798). Hongyu Zhao is supported by the NIH grants R01 GM134005 and P50 CA196530, and NSF grant DMS 1902903.