Question

A CRD is properly randomized when all permutations of treatment labels to experimental units are equally likely. Suppose you have 6 experimental units
and two treatments, A and B, assigned to 3 units each. Starting with EU 1, you flip a coin to decide which treatment to assign. If heads, you assign A,
otherwise you assign B. You keep doing this until all three replicates of one of the treatments have been assigned. All remaining units are assigned the
other treatment.
Show that the above randomization scheme is NOT valid. Hint: find two possible assignments, e.g. ABABAB, that are not equally likely.

Answer 1

To show that the above randomization scheme is not valid it is enough to find any two assignments which are not equally likely.

We find the probability of two possible assignments, ABABAB and AAABBB, and find the respective probabilities of both the assignments.

ABABAB

According to the question, to get the ABABAB assignment, we need to toss the coin five times(since by the fifth toss all three replicates of treatment A have been assigned) and clearly we need to get the following result 'HTHTH' on the five tosses to get the ABABAB assignment.

Thus,

AAABBB

According to the question, to get the AAABBB assignment, we need to toss the coin three times(since by the third toss all three replicates of treatment A have been assigned) and clearly we need to get the following result 'HHH' on the three tosses to get the AAABBB assignment.

Thus,

Since, , the above randomization scheme is not valid.

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