## Randomization.com Help File: Subjects Receiving Treatments In Random Order

The Mechanics of Using the Program

Up to 6 treatments can be permuted. The randomization plan is not affected by the order in which the treatments are entered or the particular boxes left blank if not all are needed. The program begins by sorting treatment names internally. The sorting is case sensitive, however, so the same capitalization should be used when recreating an earlier plan.

Suppose treatments are called A,B,C... If there are two treatments, there are two ways the treatments can be ordered (permuted): AB and BA. For three treatments, there are six permutations: ABC, ACB, BAC, BCA, CAB, and CBA. In general, when there are 'k' treatments, there are k! (read "k factorial" and equal to k(k-1)(k-2)...1) ways in which the treatments can be ordered.

Permutations can be selected at random or balanced. Random permutations are selected independently from the set of k! possibilities. Each permutation is equally likely to be any of the k! possibilities, regardless of what the other random permutations might be. That is, random permutations are selected at random with replacement. Even when the number of subjects is a multiple of the number of possible permutations, it is likely that some permutations will occur more often than others and some permutations might not occur at all.

When "balanced permutations" is chosen, permutations are selected at random without replacement until the set of all permutations is exhausted. In general, the number of subjects, nsubj, and the number of possible permutations, npermcan be expressed as

nsubj = nsets * nperm + nextra,
for some values nsets and nextra. For example if there are 3 treatments, nperm is 6. If there are 15 subjects, then
15 = 2 nperm+ 3 .
In this case, each of the 6 possible permutations will appear twice in random order. Then nextra (here, 3) permuations are chosen at random without replacement from the full set and tacked onto the end of the list. The entire set of 15 permutations could have been randomized, but I decided against it in order to insure better balance if the study is terminated early.

If the number of subjects is an exact multiple of the number of possible permutations, the randomization can be carried out in blocks to insure against imbalance should the study be terminated prematurely. For example, with three treatments and 36 subjects, the randomization might be carried out in three blocks so that every permutation will have been used twice every time 12 subjects are recruited.

The seed for the random number generator (Wichmann and Hill, 1982, as modified by McLeod, 1985) is obtained from the clock of the local computer and is printed at the bottom of the randomization plan. If a seed is included in the request, it overrides the value obtained from the clock and can be used to reproduce or verify a particular plan.

Initial subject ID number: By default, the randomization plan will number subjects starting with 1. The starting value can be changed by placing the chosen value in the box labeled Initial subject ID number. A starting value different from 1 might be entered when plans call for a particular set of consecutive subject IDs to be used. The starting value would be the first ID number of the set. It is also helpful when a study is extended and an additional set of random assignments is needed. The additional assignments can now be numbered starting where the original set left off.

References

Fleiss, JL (1986). The Design and Analysis of Clinical Experiments. New York: John Wiley and Sons.

McLeod, A. Ian (1985), "Remark AS R58. A remark on algorithm AS 183. An efficient and portable pseudo-random number generator," Applied Statistics, 34, 198-200.

Wichmann BA and Hill ID (1982), "Algorithm AS 183. An efficient and portable pseudo-random number generator," Applied Statistics, 31, 188-190.