Shrinkage-based regularization tests for high-dimensional data with application to gene set analysis

Traditional multivariate tests such as Hotelling's test or Wilk's test are designed for classical problems, where the number of observations is much larger than the dimension of the variables. For high-dimensional data, however, this assumption cannot be met any longer. In this article, we consider testing problems in high-dimensional MANOVA where the number of variables exceeds the sample size. To overcome the challenges with high dimensionality, we propose a new approach called a shrinkage-based regularization test, which is suitable for a variety of data structures including the one-sample problem and one-way MANOVA. Our approach uses a ridge regularization to overcome the singularity of the sample covariance matrix and applies a soft-thresholding technique to reduce random noise and improve the testing power. An appealing property of this approach is its ability to select relevant variables that provide evidence against the hypothesis. We compare the performance of our approach with some competing approaches via real microarray data and simulation studies. The results illustrate that the proposed statistics maintains relatively high power in detecting a wide family of alternatives.