In this paper, a generalized version of a U-statistics-based test for MCAR developed by Aleksi\'c (2024) is presented. The novel test, similar to the original, tests for MCAR by calculating and combining the covariances between the response indicators and the data variables. However, unlike the old test, it is able to utilize partially observed variables, resulting in a significantly larger class of detectable alternatives. The novel test appears to be well calibrated, much better than the Little's MCAR test that was used as a benchmark. For the alternatives that were detectable for the old test, the novel test has comparable, although slightly lower, power as the old one, but is still able to outperform Little's test in all of the studied scenarios. For alternatives that were previously undetectable or barely detectable, the novel test performs the best of three. The novel test has the same assumption of finite fourth moments of the data, the same assumption necessary for Little's test. The results indicate that the novel test is more robust to this assumption, although both tests have similar limitations.

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