Reddit reviews Designing Randomised Trials in Health, Education and the Social Sciences: An Introduction
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You can do a significance test. But 1 in 20 randomization checks will be statistically significant. So if your test is significant, you don't know if randomization failed, or if you were unlucky.
And if a randomization check is not significant, that doesn't tell you much. It just tells you that you failed to detect failures of randomization.
So don't try. Design robust randomization procedures that you believe in (using telephone randomization, opaque envelopes, whatever). If you don't have complete faith in your randomization, you don't believe it, no matter what.
Link (behind a paywall): http://www.bmj.com/content/319/7203/185.1.full I think the clinical trials book by Torgerson and Torgerson covers this: https://www.amazon.com/Designing-Randomised-Trials-Education-Sciences/dp/0230537359 (But so should any decent book on clinical trials).