Joint Mean and Covariance Estimation with Unreplicated Matrix-Variate Data

ABSTRACT

It has been proposed that complex populations, such as those that arise in genomics studies, may exhibit dependencies among observations as well as among variables. This gives rise to the challenging problem of analyzing unreplicated high-dimensional data with unknown mean and dependence structures. Matrix-variate approaches that impose various forms of (inverse) covariance sparsity allow flexible dependence structures to be estimated, but cannot directly be applied when the mean and covariance matrices are estimated jointly. We present a practical method utilizing generalized least squares and penalized (inverse) covariance estimation to address this challenge. We establish consistency and obtain rates of convergence for estimating the mean parameters and covariance matrices. The advantages of our approaches are: (i) dependence graphs and covariance structures can be estimated in the presence of unknown mean structure, (ii) the mean structure becomes more efficiently estimated when accounting for the dependence structure among observations; and (iii) inferences about the mean parameters become correctly calibrated. We use simulation studies and analysis of genomic data from a twin study of ulcerative colitis to illustrate the statistical convergence and the performance of our methods in practical settings. Several lines of evidence show that the test statistics for differential gene expression produced by our methods are correctly calibrated and improve power over conventional methods. Supplementary materials for this article are available online.

KEYWORDS:

• Generalized least squares
• Genomics
• Graphical modeling
• Sparsity
• Two-group comparison

Acknowledgments

The research is supported in part by NSF under Grant DMS-1316731 and the Elizabeth Caroline Crosby Research Award from the Advance Program at the University of Michigan. The authors thank the Editor, the Associate editor and three referees for their constructive comments that led to improvements in the article. Roger Fan, Kerby Shedden, and Shuheng Zhou are listed alphabetically. This work was done while all authors were affiliated with the Department of Statistics at the University of Michigan.

Additional information

Funding

Division of Mathematical Sciences [NSF13-16731]; University of Michigan [Elizabeth C. Crosby Research to S. Zhou].