Simple tests on multiple correlation coefficient in high-dimensional normal data: Simple tests on multiple correlation coefficient

The multiple correlation coefficient (MCC) quantifies the maximum correlation between a variable and a linear combination of a set of variables. Within the context of multiple regression and correlation analysis, testing for the null hypothesis of zero MCC has garnered significant attention. However, in high-dimensional data settings, where the data dimension ( p ) far exceeds the number of observations ( n ), this testing problem poses a challenging task that renders classical testing methodologies practically infeasible. This study proposes three testing procedures to assess the null hypothesis of zero MCC under the assumption of multivariate normality. Simulations are conducted to evaluate the performance of the proposed tests and compare them with an existing competitor. The simulation results are highly encouraging. Finally, the proposed tests are applied to two publicly available datasets to demonstrate their practical utility.