  The two-sample problem for Cronbach's coefficient $\alpha_C$, as an estimate
of test or composite score reliability, has attracted little attention,
compared to the extensive treatment of the one-sample case. It is necessary to
compare the reliability of a test for different subgroups, for different tests
or the short and long forms of a test. In this paper, we study statistically
how to compare two coefficients $\alpha_{C,1}$ and $\alpha_{C,2}$. The null
hypothesis of interest is $H_0 : \alpha_{C,1} = \alpha_{C,2}$, which we test
against one-or two-sided alternatives. For this purpose, resampling-based
permutation and bootstrap tests are proposed. These statistical tests ensure a
better control of the type I error, in finite or very small sample sizes, when
the state-of-affairs \textit{asymptotically distribution-free} (ADF)
large-sample test may fail to properly attain the nominal significance level.
We introduce the permutation and bootstrap tests for the two-group multivariate
non-normal models under the general ADF setting, thereby improving on the small
sample properties of the well-known ADF asymptotic test. By proper choice of a
studentized test statistic, the resampling tests are modified such that they
are still asymptotically valid, if the data may not be exchangeable. The
usefulness of the proposed resampling-based testing strategies is demonstrated
in an extensive simulation study and illustrated by real data applications.
