Statistics and Its Interface

Volume 11 (2018)

Number 4

Doubly regularized estimation and selection in linear mixed-effects models for high-dimensional longitudinal data

Pages: 721 – 737

DOI: https://dx.doi.org/10.4310/SII.2018.v11.n4.a15

Authors

Yun Li (Department of Statistics, University of Michigan, Ann Arbor, Mi., U.S.A.)

Sijian Wang (Department of Biostatistics & Medical Informatics, and the Department of Statistics, University of Wisconsin, Madison, Wi., U.S.A.)

Peter X.-K. Song (Department of Biostatistics, University of Michigan, Ann Arbor, Mi., U.S.A.)

Naisyin Wang (Department of Statistics, University of Michigan, Ann Arbor, Mi., U.S.A.)

Ling Zhou (Department of Biostatistics, University of Michigan, Ann Arbor, Mi., U.S.A.)

Ji Zhu (Department of Statistics, University of Michigan, Ann Arbor, Mi., U.S.A.)

Abstract

The linear mixed-effects model (LMM) is widely used in the analysis of clustered or longitudinal data. This paper aims to address analytic challenges arising from estimation and selection in the application of the LMM to high-dimensional longitudinal data.We develop a doubly regularized approach in the LMM to simultaneously select fixed and random effects. On the theoretical front, we establish large sample properties for the proposed method under the high-dimensional setting, allowing both numbers of fixed effects and random effects to be much larger than the sample size. We present new regularity conditions for the diverging rates, under which the proposed method achieves both estimation and selection consistency. In addition, we propose a new algorithm that solves the related optimization problem effectively so that its computational cost is comparable with that of the Newton–Raphson algorithm for maximum likelihood estimator in the LMM. Through simulation studies we assess performances of the proposed regularized LMM in both aspects of variable selection and estimation. We also illustrate the proposed method by two data analysis examples.

Keywords

diverging rate, regularization, random-effects selection, variable selection

2010 Mathematics Subject Classification

Primary 62J05, 62J07. Secondary 62F12.

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The research of Y. Li and N. Wang is partially supported by an NCI grant CA74552.

The research of Y. Li and J. Zhu is partially supported by NIH R01-GM-096194 and NSF DMS-1407698.

The research of S. Wang is partially supported by NIH R01-HG-007377.

The research of P. Song and L. Zhou is supported in part by NIH R01-ES-024732 and NSF DMS-1513595.

Received 12 November 2015

Published 19 September 2018