  In this work, we show that Spearman's correlation coefficient test about
$H_0:\rho_s=0$ found in most statistical software packages is theoretically
incorrect and performs poorly when bivariate normality assumptions are not met
or the sample size is small. The historical works about these tests make an
unverifiable assumption that the approximate bivariate normality of original
data justifies using classic approximations. In general, there is common
misconception that the tests about $\rho_s=0$ are robust to deviations from
bivariate normality. In fact, we found under certain scenarios violation of the
bivariate normality assumption has severe effects on type I error control for
the most commonly utilized tests. To address this issue, we developed a robust
permutation test for testing the general hypothesis $H_0: \rho_s=0$. The
proposed test is based on an appropriately studentized statistic. We will show
that the test is theoretically asymptotically valid in the general setting when
two paired variables are uncorrelated but dependent. This desired property was
demonstrated across a range of distributional assumptions and sample sizes in
simulation studies, where the proposed test exhibits robust type I error
control across a variety of settings, even when the sample size is small. We
demonstrated the application of this test in real world examples of
transcriptomic data of the TCGA breast cancer patients and a data set of PSA
levels and age.
