High-dimensional two-sample precision matrices test: an adaptive approach through multiplier bootstrap

Precision matrix, which is the inverse of covariance matrix, plays an important role in statistics, as it captures the partial correlation between variables. Testing the equality of two precision matrices in high dimensional setting is a very challenging but meaningful problem, especially in the differential network modelling. To our best knowledge, existing test is only powerful for sparse alternative patterns where two precision matrices differ in a small number of elements. In this paper we propose a data-adaptive test which is powerful against either dense or sparse alternatives. Multiplier bootstrap approach is utilized to approximate the limiting distribution of the test statistic. Theoretical properties including asymptotic size and power of the test are investigated. Simulation study verifies that the data-adaptive test performs well under various alternative scenarios. The practical usefulness of the test is illustrated by applying it to a gene expression data set associated with lung cancer.

[1]  Jiashun Jin,et al.  Robustness and accuracy of methods for high dimensional data analysis based on Student's t‐statistic , 2010, 1001.3886.

[2]  T. Cai,et al.  A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation , 2011, 1102.2233.

[3]  Biao He,et al.  Wnt signaling in lung cancer. , 2005, Cancer letters.

[4]  Han Liu,et al.  A Unified Framework for Testing High Dimensional Parameters: A Data-Adaptive Approach. , 2018, 1808.02648.

[5]  C. Fenoglio-Preiser,et al.  beta-Catenin mutation is a frequent cause of Wnt pathway activation in gastric cancer. , 2002, Cancer research.

[6]  Quanquan Gu,et al.  Identifying gene regulatory network rewiring using latent differential graphical models , 2016, Nucleic acids research.

[7]  Jun Yu Li,et al.  Two Sample Tests for High Dimensional Covariance Matrices , 2012, 1206.0917.

[8]  M. Yuan,et al.  Model selection and estimation in the Gaussian graphical model , 2007 .

[9]  Jinchi Lv,et al.  Innovated scalable efficient estimation in ultra-large Gaussian graphical models , 2016, 1605.03313.

[10]  Roman Vershynin,et al.  High-Dimensional Probability , 2018 .

[11]  Louise R Howe,et al.  Wnt Signaling and Breast Cancer , 2004, Cancer biology & therapy.

[12]  Leena Peltonen,et al.  Finding disease candidate genes by liquid association , 2007, Genome Biology.

[13]  Tianxi Cai,et al.  Testing Differential Networks with Applications to Detecting Gene-by-Gene Interactions. , 2015, Biometrika.

[14]  A. Sala,et al.  Non-canonical WNT/PCP signalling in cancer: Fzd6 takes centre stage , 2017, Oncogenesis.

[15]  T. Cai,et al.  Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings , 2013 .

[16]  L. Isserlis.  ON A FORMULA FOR THE PRODUCT-MOMENT COEFFICIENT OF ANY ORDER OF A NORMAL FREQUENCY DISTRIBUTION IN ANY NUMBER OF VARIABLES , 1918 .

[17]  T. Ideker,et al.  Differential network biology , 2012, Molecular systems biology.

[18]  Ming Yuan,et al.  High Dimensional Inverse Covariance Matrix Estimation via Linear Programming , 2010, J. Mach. Learn. Res..

[19]  T. Cai,et al.  Direct estimation of differential networks. , 2014, Biometrika.

[20]  Harrison H. Zhou,et al.  Asymptotic normality and optimalities in estimation of large Gaussian graphical models , 2013, 1309.6024.

[21]  Kengo Kato,et al.  Central limit theorems and bootstrap in high dimensions , 2014, 1412.3661.

[22]  Harrison H. Zhou,et al.  Estimating Sparse Precision Matrix: Optimal Rates of Convergence and Adaptive Estimation , 2012, 1212.2882.

[23]  Patrick Danaher,et al.  The joint graphical lasso for inverse covariance estimation across multiple classes , 2011, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[24]  Weidong Liu.  Gaussian graphical model estimation with false discovery rate control , 2013, 1306.0976.

[25]  J. Pongrácz,et al.  WNT signaling – lung cancer is no exception , 2017, Respiratory Research.

[26]  N. Meinshausen,et al.  High-dimensional graphs and variable selection with the Lasso , 2006, math/0608017.

[27]  James R. Schott,et al.  A test for the equality of covariance matrices when the dimension is large relative to the sample sizes , 2007, Comput. Stat. Data Anal..

[28]  R. Tibshirani,et al.  Sparse inverse covariance estimation with the graphical lasso. , 2008, Biostatistics.

[29]  Hirokazu Yanagihara,et al.  Testing the equality of several covariance matrices with fewer observations than the dimension , 2010, J. Multivar. Anal..