A distribution-free test of independence based on mean variance index

A new test based on mean variance (MV) index is proposed for testing the independence between a categorical random variable Y and a continuous one X. The MV index can be considered as the weighted average of Cramér–von Mises distances between the conditional distribution functions of X given each class of Y and the unconditional distribution function of X. The MV index is zero if and only if X and Y are independent. The new MV test between X and Y enjoys several appealing merits. First, an explicit form of the asymptotic null distribution is derived under the independence between X and Y. It provides an efficient way to compute critical values and p-value. Second, no assumption on the distributions of two random variables is required and the new test statistic is invariant under one-to-one transformations of the continuous random variable. It is essentially a rank test and distribution-free, so it is resistant to heavy-tailed distributions and extreme values in practice. Monte Carlo simulations demonstrate its excellent finite-sample performance. In applications, the MV test is used in two high dimensional gene expression data to detect the significant genes associated with tumor types.