  Motivated by the importance of measuring the association between the response
and predictors in high dimensional data, In this article, we propose a new mean
variance test of independence between a categorical random variable and a
continuous one based on mean variance index. The mean variance index is zero if
and only if two variables are independent. Under the independence, we derive an
explicit form of its asymptotic null distribution, which provides us with an
efficient and fast way to compute the empirical p-value in practice. The number
of classes of the categorical variable is allowed to diverge slowly to the
infinity. It is essentially a rank test and thus distribution-free. No
assumption on the distributions of two random variables is required and the
test statistic is invariant under one-to-one transformations. It is resistent
to heavy-tailed distributions and extreme values. We assess its performance by
Monte Carlo simulations and demonstrate that the proposed test achieves a
higher power in comparison with the existing tests. We apply the proposed MV
test to a high dimensional colon cancer gene expression data to detect the
significant genes associated with the tissue syndrome.
